Combinatorics '90Recent Trends and Applications, Proceedings of the Conference on Corn binatorics,...

COMBINATORICS '90 Recent Trends and Applications

ANNALS OF DISCRETE MATHEMATICS

General Editor: Peter L. HAMMER Rutgers University, New Brunswick, NJ, USA

Advisory Editors:

C. BERGE, Universite de Paris, France R.L. GRAHAM, AT&T Bell Laboratories, NJ, USA M.A. HARRISON, University of California, Berkeley, CA, USA V. KLEE, University of Washington, Seattle, WA, USA J. H. VAN LINT California Institute of Technology, Pasadena, CA, USA G.C. ROTA, Massachusetts Institute of'Technology, Cambridge, MA, USA T: TROlTER, Arizona State University, Tempe, AZ, USA

52

CO M B I N ATORICS '90 Recent Trends and Applications Proceedings of the Conference on Corn binatorics, Gaeta, Italy, 20-27 May, I990

Edited by

A. BARLOlTI Universita di Firenze Firenze, Italy

A. BICHARA Universith de L'Aquila L'Aquila, Italy

PV. CECCHERlNl and G. TALLlNl UniversitS di Roma ' l a Sapienza' Roma, Italy

1992

NORTH-HOLLAND -AMSTERDAM LONDON NEW YORK TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 21 1, 1000 AE Amsterdam, The Netherlands

ISBN: 0 444 89452 7

Q 1992 Elsevier Science Publishers B.V. All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers 6. V., Copyright & Permissions Department, RO. Box 521, 1000 AM Amsterdam, The Netherlands.

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No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained ir: the material herein.

pp. 519-528: Copyright not transferred

This book is printed on acid-free paper.

Printed in The Netherlands

V

Foreword

Combinatorial methods have been used with great advantage in several diverse branches of pure and applied mathematics. This is not the place to analyze the many factors which have contributed to the increasing pace of developement of Combinatorics in the last few decades. Clear signs of the great interest for the progress in this wide area of Mathematics are the increming number of Journals and Conferences devoted to different aspects of combinatorial theories.

The present. volume contains the proceedings of the intermtional Conference “Coin- binatorics ’90” held i n Gaeta (the ancient Caieta, a nice small town northwest of Naples) from May 20th to 2.ith 1990.

The meeting was a “State of the Art” conference and the papers read were both of survey and research tfypes.

The topics considrred concern latest developments in combinatorial geometries, their links with foundations of geometry and algebra, graph theory and various applications.

We are indebted to the following institutions and sponsors for financial support: Uni- versity of Rome “La Sapienza”, “Presidenza della Facolt& di Scienze” of the University “La Sapienza” , “Consiglio Nazionale delle Ricerche” , “Gruppo Nazionale per le Strut- ture Algehriche e Geometriche e loro Applicazioni” of the C.N.R., “Azienda Autonoma di Soggiorno e Turisino di Ga.eta” , “Comune di Gaeta.” , and “Ca.sa. Editrice Liguori” (Na.ples).

We also wish to express our warmest thanks to the referees for their invaluable assistence.

Adriano Barlotti Alessandro Bichara Pier Vittorio Ceccherini Giuseppe Talliiii

This Page Intentionally Left Blank

vii

Contents

FOREWORD

M. AIGNER, E. TRIESCH and Z. TUZA Irregular assignments and vertex-distinguishing edge-colorings of graphs

A. ASTIE’-VIDAL, V. DUGAT and Z. TUZA Construction of non-isomorphic regular tournaments

W. BENZ On structures T( t , q , r , n)

D. BETTEN and M. BR.AUN A tactical decomposition for incidence structures

A. BICHARA and S. INNAMORATI Note on a. chamcterimtion of Segre variety in PG(r, q )

A. BONISOLI and G. KORCHMAROS A property of sharply 3-transitive finite permutation sets

A. BOWLER. Faithful orbits in symmetric designs

F. BUEKENHOUT Minimal flagtransitive geometries

V. CAVACCINI and A. LETTIERI Some results on hyperarchimedean MV-algebras

W. CHU Three combinatorial sequences derivable from the lattice path counting

J.R. CLAY Compound closed cha.ins in circular planar nearrings

M. CORDER0 p-primitive semifield planes

M.J. DE RESMINI The fractal-like Steiner triple system

V

1

11

25

37

45

49

61

69

71

81

93

107

119

Contents ... V l l l

A. DEL FRA, D. GHINELLI and A. PASINI Locally partial geometries with different types of residues

A. DEL FRA, D. GHINELLI and S.E. PAYNE (0, n)-sets in a generalized quadrangle

F. EUGENI Combinatorics and cryptography

127

130

159

G. FAINA Recent intrinsic characterizations of ovoids and elliptic quadrics in PG(3, I<)

F. GAETA A natural association of PGL(V)-orbits in the Segre variety (P(V))’” with

175

flags and Young tableaux 191

T.S. GRIGGS, M.J. DE RESMINI and A. ROSA Decomposing Steiner triple systems into four-line configurations

H. GROPP Non-symmetric configurations with deficiencies 1 and 2

E. HAHN Chamcterizing 1inea.r spaces by blocking sets

D.R. HUGHES Partial geometries of rank n

215

227

24 1

249

A.A. IVANOV The minimal paxabolic geometry of the Conway group Col is simply connected 259

2. JANKO Coset enumeration in groups and constructions of symmetric designs 275

V. JHA and N.L. JOHNSON On the ubiquity of Denniston-type translation ovals in generalized Andld planes

N.L. JOHNSON Transla tion planes and related com binatorid structures

H. KARZEL Finite reff exion groups and their corresponding structures

W.-F. KE On nonisomorphic BIBD with identical para.meters

279

297

317

337

Contents ix

M. MARCH1 Incidence loops and their geometry

D.B. MEISNER Families of Menon difference sets

N. MELONE On the characterization problem for finite linear spaces

K. METSCH Linear spaces in which every line of maximal degree meets only few lines

A. PASINI and S. YOSHIARA Flag-transitive Buekenhou t geometries

S.E. PAYNE Collineations of the generalized quadrangles associated with q-clans

S. PIANTA Projective embedding of fibered groups and the Suzuki groups

G.F. PILZ Codes, block designs, fiobenius groups and new-rings

A.R. PRINCE The flag-transitive f i n e planes of order 27

D. SENATO and A. VENEZIA Symmetric functions and bijective identities

H. SHEN On the existence of newly Kirkman systems

J. SIMONIS Codes and semilinear spaces

J.A. THAS Old and new results on spreads and ovoids of finite classical polar spaces

Z. TUZA Large partial parallel classes in Steiner systems

B.J. WILSON Minimal line distinguishing colourings in graphs

Q. XIANG Some results on -1 multiplier of difference lists

347

365

381

39 1

403

449

463

471

477

501

511

519

529

545

549

559


Combinatorics '90 A. Barlotti et al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved.

Irregular assignments and vertex-distinguishing edge-colorings of graphs

M. Aigner", E. Trieschb, and Z. Tuza'

"11. Math. Institut, Freie Universitat Berlin, Arnimallee 3, D-W1000 Berlin 33

bLehrstuhl fur Unternehmensforschung, RWTH Aachen, Templergraben 64, D-W5100 Aachen

'Comp. Autom. Institute, Hungarian Academy Science, Kende u. 13-17, H-1111 Budapest

Abstract A coloring g : E(G) + C of the edge-set of a graph G into a color-set C is called

verter-distingllishing if g ( S t ( u ) ) # g ( S t ( v ) ) for any two stars. Let c(G) be the minimal number of colors necessary for such a coloring. For k-regular graphs G we clearly have c(G) 2 where n is the order of G. We prove c(G) 5 C,nllk, and for k = 2, c(G) I $4 + C.

1. Introduction

A well-known fact in graph theory states that any (simple) graph with at least two vertices contains two vertices of the same degree. Hence the following problem, initiated in [5], arises: Consider a weighting w : E(G) + (1,. . . , m} of the edge-set E(G) of a graph G and denote by w(v) = C w(e) the weighted degree of the vertex ZI. We call the

weighting w admissible or an irregular assignment if all weighted degrees are distinct. What is the minimum number m for which an admissible weighting exists? This number s(G) is called the irregularity strength of G . Our observation above shows that s(G) 1 2 for any graph G, and it is an easy matter to show that s(G) < 00 iff G has no isolated edges and at most one isolated vertex. The number s(G) has been studied in a variety of papers (see e.g. [1,9] and [lo] for a survey).

A natural variant of our problem asks for the smallest size c(G) of the image of an irregular assignment. That is, c(G) is the smallest number r such that an irregular assignment w : E(G) -+ N exists with lim(w)l = r. Clearly c(G) 5 s(G) and c(G) < DC)

iff s(G) < 00. It was shown in [2] that the study of c(G) reduces to an interesting edge-coloring problem. Consider an edge-coloring g : E(G) + C into some color-set C. The star St(u) with center u is the set of all edges incident with the vertex u. Let us

uEe

2 M . Aigner et al.

identify St(u) with the multiset of colors assigned to the edges of St(u) . We call the coloring g vertex-distinguishing or irregular if St(u) # St (v) as multisets for any two vertices u # v. Then c(G) is precisely the minimal number of colors necessary for an irregular edge-coloring of G. Note that irregular edge-colorings can be viewed as the dual notion of line-distinguishing vertex-colorings, investigated e.g. in [8,14].

As for examples, it is shown in (21 that c(Kn) = 3 for any complete graph I{,, (n >_ 3), and c(Kn+) = 3 for any complete bipartite graph I<,,,,, (n 2 2).

In the following, we investigate in section 2 c(G) for k-regular graphs, k-fixed, and determine its asymptotic growth. Of particular interest is the case k = 2 which reduces to packings of complete graphs by connected Eulerian subgraphs, in the vein of the famous Oberwolfach problem of G. Ringel. This shall be the content of section 3.

2. The irregular coloring number c(G) for k-regular graphs

Let us consider a k-regular graph G ( k 2 2 fixed) on n vertices, and suppose we have an irregular edge-coloring of G with c colors. Any multiset St(u) has size Ic taken from c elements. By a well-known formula there are ('+:-') different multisets of size Ic with c entries, hence

(c+;-l 12%

i.e.

c(G) 2 C1 nilk

where C1 is a constant only dependent on k.

The aim of this section is to show that nilk is the correct growth, i.e.,

c(G) 5 Cz nilk (3)

holds for some constant C,.

To prove this we use random colorings. Consider all colorings g from E(G) into a set of c colors, each coloring with equal probability ~ / c I ~ ( ~ ) I . For a pair p = { u , v } of vertices u,v, we denote by A, the event that St(u) = St (v) .

Lemma 1. Let p = {u,v}.

a.

b .

Proof. Suppose u and v are not joined. Look at all k! bijections from St(u) to St (v) . For a bijection cp, the probability that g(e;) = g(cp(e,)) , for all e l , . . . , ek in St(u) , is f , hence P r ( A , ) 5 $.

If u and v are not joined in G, then P r ( A , ) 5 $, If u and v are joined in G, then P r ( A , ) 5 w.

Irregular assignments and vertex-distinguishing edge-colorings of graphs 3

The proof of b. is analogous, taking into account the edge uv. 0

By the definition of A, our aim is to find c large enough to ensure Pr(Apl A A,, A .. . A A,(:)) > 0, where the pi’s run through all (;) pairs of vertices. The key to the proof is a variant of the well-known Erdos-LovLz local lemma (see [4,7,13]). Let A l , . . . , A,,, be events in a probability space and, for all i, suppose Ai is independent of a set {A j : j E J } if and only if Ai is independent of each A j , j E J . The dependence graph r has as vertices (1, . . . , m} with i and j joined iff A; and Aj are dependent. r(i) shall denote the set of neighbors of i. Then the following holds [4, p. 211:

Lemma 2 . Let A l , . . . , A,,, be events and r the dependence graph. Suppose there exist numbers y l , . . . ,T,,, between 0 and 1 with P T ( A ; ) 5 “y; n (1 - ~ j ) for all i. Then

je:r(i)

PT(A1 A A 2 A .. . A A,) > 0.

In our situation two pairs p and q are dependent iff they share a vertex or are distance 1 apart in the graph G. Set 7, = e PT(A, ) , then

Clearly, all pairs q = {z, y} dependent on p are covered by at most 2k + 2 vertices in G. Hence the number of such q’s is at most 2k + 2 k ( k - 1) = 2k2 if z, y are joined in G, and at most ( 2 k + 2 ) . if z, y are not joined. Now from Lemma 1 we infer that (5) is satisfied if

( 6 ) 1 5 e(1- e$&) k-1 ! 2k2 (1 - e$)(zk+Z)n.

From the well-known inequality

(1 - z)+-l > e-1 (0 < z < 1)

we deduce for y > 0,

4 M. Aigner et al.

(9) - 1 ~ z 2 k 2 + &(2k + 2). I 1.

2&(2k + 2)n I 1.

For large n the second summand in (9) dominates the first, hence (9) is satisfied if

(10)

Substituting the expression for d, (10) is equivalent to

2e&(2k + 2). I 1

which is equivalent to

4 e ( k + l ) !n I ck - ek! .

Now (12) is certainly satisfied if

5e(k + l ) !n 5 ck,

and Lemma 2 and (13) now yield the desired result.

Theorem 1. Let G be a k-regular graph on n vertices. Then f o r jixed k ,

c(G) 5 Cz n'Ik,

where CZ is a constant only depending on k . In fact, Cz 5 ( 5 e ( k + l ) ! ) l / k .

3. 2-regular graphs and packings by Eulerian subgraphs

Of particular interest is the case k = 2. A 2-regular graph G is a disjoint union of cycles C,, , . . . , C,,, where C; is the cycle of length i. Suppose g is an irregular edge- coloring of G with r colors. Denote by M,. the complete graph K , with a single loop attached to each vertex, and identify the vertices of M,. with the colors of g. Look at the first cycle C,, , and choose a sense of traversal around C,, . For any two colors appearing in some St(u) of C,,, , we draw an edge (or a loop) between the corresponding vertices of M,. Notice that we never draw an edge twice since St(u) # St(v) . Accordingly, as we go around C,,, this construction yields a closed trail, i.e. a connected Eulerian subgraph with n1 edges in M, (see figure 1) .


1

3 Figure 1

By the same procedure, we obtain connected Eulerian subgraphs of sizes n2, . . . , nt corresponding to the other cycles, and all these Eulerian subgraphs are edge-disjoint by the irregularity of the coloring g . Clearly, the converse holds as well, and we have thus reduced the determination of c(G) to the following packing problem:

Let G = C,, -t . . . + C,,. Then c(G) is the smallest number r such that we can pack edge-disjoint connected Eulerian subgraphs of sizes 1 2 1 , . . . , nt in M,.

This packing problem is reminiscent of the famous Oberwolfach problem of Ringel (see [3,12]):

Given KT1 r odd, which sets of cycles C,, , . . . , C,, can be packed edge-disjoint such that every edge of I<, is in some cycle?

Notice that in contrast to the Oberwolfach problem where the connected Eulerian subgraphs must be cycles, in our situation we have complete freedom as to the structure of the Eulerian subgraphs - only the size ni matters. Hence we expect that our problem might be easier to handle, and this is indeed the case. The aim of this section is to give a direct construction of such a packing, proving again c(G) 5 C2n1f2 for 2-regular graphs.

Suppose we are given a Steiner triple system (STS) on v points with triples TI,.. . , Tb, b = (v(v - 1) ) /6 . Let H denote the graph with the triples TI , . . . , Tb as vertices, joining two triples whenever they have a point in common. The following lemma is well-known and follows by an application of a theorem of Chvital-Erdos [6]:

Lemma 3. The graph H is Hamiltonian for v 2 7 .

t

i=l Theorem 2. Suppose nlr . . . , ni are natural numbers such that C ni 5 ( z ) where v = 1

or 3(mod 6),and each I Z ; is divisible b y 3. Then the complete graph I<, contains edge- disjoint connected Eulerian subgraphs HI , . . . , Ht with nl, 1 2 2 , . . . , resp., nt edges.

Proof. Choose an STS on v points. By Lemma 3, there exists a labelling of the triples TI,. . . , Tb such that IT; n Ti+ll = 1 (1 5 i < 6 ) . Set mi = %, and denote by Ti(*) the

6 M. Aigner et al.

set of 2-element subsets of Ti. The edge-sets E(G;) of the subgraphs G; are defined as follows:

Then it is clear that the graphs Gj have the desired properties. CI

t

i=l Corollary 1.

c(G) 5 6 t C.

Let G = C,, + . . . + C,,, n = C n; with all n; = O(mod 3), then

Proof. Solving (;) = n we obtain v = 4 2 n + f + $. By the previous proposition, we conclude

c (G) 5 v + 4 = &+C. 0

To treat the general case we need the following

Lemma 4. Suppose v = 3(mod 6). Then there exists a graph H = ( V ( H ) , E ( H ) ) on (5v - 3)/2 vertices, and a bijection ‘p : E(K,) + E ( H ) such that incident edges in I(, are mapped onto independent edges of H .

Proof. We make use of the existence of Kirkman triple systems (KTS) on v points (see [ll]). That is, there exists an STS whose (v(v - 1))/6 triples can be partitioned into (v - 1)/2 classes Ul , . . . , Uv-l such that each class consists of v/3 disjoint triples

-5-

ui = {T;J ,T iJ , . . . ,Tl,v,3} ( i = 1,. . . , (v - 1)/2).

From each class U; we obtain 3 matchings Mi,1, M,,z, Mi,3 of I(, of size v /3 such that 4 3

MIJ u MiJ u 4 3 = u Tip). j = 1

Thus E(Z(,) is the disjoint union

For 2 E V(K”) denote by ML the set of all pairs {y,z} C V ( & ) such that {z,y,z} is a triple of the KTS. Again, M: is a matching of K,,, and E(K,,) is the disjoint union of the

Irregular assignments and vertex-distinguishing edge-colorings of graphs I

ML, z E V(K,,). The Mi,j and ML form the 3 q + v = vertices of our graph H . Two matchings are joined in H if and only if they have an edge in common. Notice that two matchings can have at most one edge in common. This is clear for two matchings of the first or second type. Furthermore, if ML n M;,j # 0, then any edge in the intersection must belong to a triple containing 2. But the edges in Mi,j belong to disjoint triples. Now we define cp : E( K,,) 4 E( H ) by cp( e ) = { M , M} iff { e } = M n a. cp is well-defined since each edge is contained in exactly two matchings of V ( H ) and the remark above. It is also clear that cp is bijective. Finally, if e , e' are incident in K v , and cp(e) = {M,fi}, cp(e') = { M ' , f i ' } , then it is impossible that one of M,A? contains e' by the matching property, and similarly that M' or MI contains e. Hence H and cp have the desired properties. 0

Theorem 3. Let G be a 2-regular graph on n vertices and let v be the smallest natural number congruent to 3(mod 6) such that (g) 2 n. Then

c(G) 5 i(9v - 3).

Proof. Suppose G = C,,; + . . . + C,,;, then it suffices to show that the complete graph 1'(9,,-3)/2 contains edge-disjoint connected Eulerian subgraphs G i , . . . , G: with IE(G:)I =

n! 1) i = l , . . . , t . Set ni = 3[$], 1 5 i 5 t , and define the graphs G; as in the proof of Theorem 2. So, the vertices of G; are a subset of (1,. . . , v}. With each vertex 2 E { 1,. . . , v} we associate a new vertex d, and choose a graph H as in Lemma 5 for K,, = Kt1 ,..., ,,I such that V ( H ) U {l,. . . , v } U {1', . . . ,v'} is a set of i ( 9 v - 3) points on which we consider the complete graph.

Denote by M, the set of 2-element sets

{T - z : T triple of our KTS on{ 1,. . . , v}which contains z}

Now we construct the graphs G: as follows:

(i) If n: = 0 (mod 3), we set G: = Gi.

(ii) If n: = 1 (mod 3), we choose some edge of Gi, say e;. The edge e; is contained in some M,. Now we replace ei by the two edges joining the endpoints of e; with 5'.

The new graph is, again connected, Eulerian, and has n: edges.

(iii) If n: = 2 (mod 3), we choose again some ei E E(G;) . Now we replace ei by a 3-edge path consisting of cp(ei) E E ( H ) and two edges joining the endpoints of ei with the endpoints of cp(ei). Again, the new graph is connected, Eulerian and contains n: edges.

It remains to show that the graphs G: are pairwise edge-disjoint. The only non-trivial cases are

8 M. Aigner et al.

a) n: n: 1 (mod 3)

b) n: n: E 2 (mod 3).

In case a, assume that an edge e is added to both Gi and Gj. Then the removed edges e, and e j must have an end-point in common. But then e; E M, and ej E Mu with x # y, and we have a contradiction.

In case b, assume again that e E E(G:) n E(G:). As 'p is a bijection, we have cp(e;) # cp(ej), and ei and ej must be incident. But then cp(ei) and cp(ej) are independent by Lemma 5, whence E(G:) n E(G$) = 0, contradiction. 0

Theorem 4. Let G = C,,, + . . . + C,, be an arbitrary 2-regular graph. Then c(G) 5 $++c. Proof. With (!j') = n (j v = + i, we obtain by the previous theorem

c(G) 5 !j(9(v + 6) - 3) = 5fi + C.

References

M. Aigner and E. Triesch: Irregular assignments of trees and forests. To appear in SIAM J. Disc. Math. 3(1990), 439-449.

M. Aigner and E. Triesch: Irregular assignments and two problems la Ringel. In: Topics in Combinatorics and Graph Theory, dedicated to G. Ringel (Bodendiek, Henn, eds.). Physica Verlag Heidelberg (1990), 29-36.

B. Alspach, P.J. Schellenberg, D.R. Stinson and D. Wagner: The Oberwolfach problem and factors of uniform odd length cycles. J. Combinatorial theory A 52 (1989), 20-43.

B. Bollobis: Random Graphs. Academic Press 1985.

G. Chartrand, M. Jacobson, J. Lehel, 0. Oellerman, S. Ruiz and F. Saba: Irregular networks. Proc. of the 250th Anniversary Conf. on Graph Theory, Fort Wayne, Indiana, 1986.

V. Chvital and P. Erdos: A note on Hamiltonian circuits. Discrete Math. 2 (1972), 111-113.

P. Erdos and L. Lovisz: Problems and results on 3-chromatic hypergraphs and some related results. In: Infinite and Finite Sets (Hajnal, Rado, S ~ S , eds.). Coll. Math. J. Bolyai 11 (1975), 609-627.


8 0. Frank, F. Harary and M. Plantholt: The line-distinguishing chromatic number of a graph. Ars Combinatoria 14 (1982), 241-252.

M. Jacobson and J. Lehel: A bound for the strength of an irregular network. To appear.

J. Lehel: Facts and quests on degree irregular assignments. To appear.

D.K. Ray-Chaudhury and R.M. Wilson: Solution of Kirkman’s schoolgirl problem. AMS Proc. Symp. on Pure Math. 19 (1971), 187-203.

9

10

11

12 G . Ringel: Problem No. 25. In: Theory of Graphs and its Applications. Proc. Symp. Smolenice 1963 (Fiedler, ed.). Publ. House Czechoslovak Acad. Sciences, 1964.

13 J. Spencer: Asymptotic lower bounds for Ramsey functions. Discrete Math. 20 (1977), 69-76.

14 B.J. Wilson: Line distinguishing and harmonious colorings. In: Graph Colorings (Nelson, Wilson, eds.). Pitman Research Notes in Math. Series 218, 1990.


Combinatorics '90 A. Barlotti et al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. 1 1

Construction of non-isomorphic regular tournaments

b A. AstiC-Vidala, V. Dugata and Z. Tuza

a IRIT, Universiti Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse, France.

b Computer and Automation Institute, Hungarian Academy of Sciences, Kende u. 13-17, H-11 1 1 Budapest, Hungary.

Abstract Reversing the arcs of any 3-circuit of a tournament, the score vector is unchanged ;

therefore the class of regular tournaments is closed under this operation. Here we prove that the number of non-isomorphic, non-symmetric tournaments obtained by reversal from a particular

n2-9 n2-1 - 2 regular tournament on n vertices is equal to - - 1 for n I 0 (mod 3) and -

otherwise. Moreover, we generate all the non-isomorphic regular tournaments of order 9 and present their interchange graph.

24 24

1. PRELIMINARIES

The definitions not given here can be found in [l], [2] and [3].

A tournament is a directed graph in which every pair of vertices is joined by exactly one arc. Throughout, T =(X,U) will denote a tournament where X is the set of vertices, and U is the set of arcs. A tournament is regular if all vertices have equal scores (i.e., with in-degree =

out-degree = ( IXI-1) /2 ). A tournament is called rotational if its vertices can be labelled 1,2,

..., n in such a way that, for some subset S of (1, ..., n-11, vertex i dominates vertex i+j (mod n) if and only if j E S . In this case, S is said to be the symbol of T. If x is any vertex, we let O(x) = ( y / (x,y) E U ) and I(x) = [ z / (z,x) E U ) .

An automorphism of a tournament is a permutation of the vertices which preserves the orientation of arcs. The set of automorphisms of a tournament T forms a group, a(T), called the automorphism group of T. This a(T) acts on the set of arcs U, and thus defines a partition of U

12 A. Astie'-Vidal et al.

into arc orbits. Thus, two arcs (x,y) and (x',y') of U belong to the same arc orbit if and only if there is a a in a(T) with a(x) = x' and a ( y ) = y'.

A tournament T is said to be vertex symmetric if for every pair x, y of vertices there is an automorphism that takes x to y. The term 3-circuit will mean a circuit of length 3.

2. DELTA INTERCHANGE

Bruali and Li [4] studied the class of tournaments having the same score vector. They call delta-interchange (A-interchange) a transformation which reverses the direction of a 3-circuit of a tournament. The resulting tournament has the same score vector. Let $(R) denote the class of all tournaments having the same score vector R. They proved that for any two tournaments T and T in $(R), there exists a finite sequence of A-interchanges that transforms T to T'. Moreover, they defined an undirected graph called the interchange graph of $(R) and denoted by 5 (R), whose vertices are the tournaments of $(R). There is an edge joining T and T in 5 (R) iff T can be obtained from T by a A-interchange. Bruali and Li studied interchange graphs for special score vectors in the case when the vertices of the tournaments in question were labelled.

We are interested in non-isomorphic regular tournaments. Our problem is to determine how many non-isomorphic regular tournaments of order n (n odd) can be generated by A-interchange.

3. RESULTS

The main results of this section are Theorems 6 and 7, describing all regular tournaments obtained from a particular one by A-interchange. Throughout, arithmetic is done modulo n. We choose the rotational tournament C(n) with symbol S=[ 1,2, ..., m) where m = nA, and n is the number of vertices. In the sequel, m will always deno ted . We call this 2 2

tournament the "cyclonic tournament" or the "cyclone". For this tournament C(n) = (X, U) we have X = [ 1, 2, 3, ..., n ) and (i, j ) E U iff (j - i) (mod n ) I m.

1 It is well-known [2] that a regular tournament of order n has c3(T) = z;in(n-l)(n+l) different 3-circuits. So one can reverse cj(T) 3-circuits ; however, many of them will give

Construction of non-isomorphic regular tournaments 13

isomorphic tournaments. We shall now study the structure of the resulting tournaments according to the position of the reversed 3-circuit. Let us fiist classify the 3-circuits of different types.

Proposition 1

(mod n). The number of 3-circuits of C(n) containing the arc (i, j), (i, j E X) is exactly j-i

DQe A vertex k forms a 3-circuit with (ij) if and only if j dominates k and k dominates i. There are

m+l vertices not dominating i, m+l vertices not dominated by j, and d:= j-i+l vertices which do not dominate i and are not dominated by j ; the latter are i,

i+l , ..., j. Hence, the number of vertices k forming a 3-circuit with ( i j ) is equal to n-2(m+l)+d=j-i, since n-2m=1. 0.

We classify the 3-circuits according to the arithmetic differences of their vertices. In the sequel, we shall refer to the cyclic sequence of these arithmetic differences by the word A-set, and use it instead of the vertices composing a 3-circuit. If i, j, k E X form a 3-circuit (in this order), then the A-set of this circuit is defined as fi-i, k-j, i-k] where the differences are taken modulo n. Hence, [a, b, c], [b, c, a] and [c, a, b] denote the same A-set, and for any A-set [a, b, c], we have a+b+c = n.

Proposition 2 Let E and E' be two 3-circuits having the same A-sets, and T be the tournament obtained from C(n) by a A-interchange of E, and T be the tournament obtained from C(n) by a A-interchange of E'. Then T and T are isomorphic.

UQQt Assume that the 3-circuits E = (ij,k) and E' = (i',j',k') with the same A-set [a, b, c] are labelled in such a way that we have a = j-i = j'-i', b = k-j = k'-j', c = i-k = it-k (all equations modulo n).Then the rotation by i'-i, assigning x+i'-i to each XEX, is an isomorphism between T and T'. 0.

Considering any one particular vertex, by Proposition 1 we see that x is contained in 1+2+ ...+ m = (n+l)(n-1)/8 distinct 3-circuits. Their A-sets can be arranged in the following

14 A . Astie'-Vidal et al .

mangle:

11, m, ml [2, m, m-11, 12, m-1, ml [3, m, m-21, [3, m-1, m-11, [3, m-2, m]

[k,m, m-k+ll, [k, m-1, m-k+2], ..........., [k, m-k+2, m-11, [k, m-k+l, m] . . . . . . . . . . I

In this structure we intend to eliminate those A-sets which yield isomorphic tournaments after A-interchange.

Proposition 3 The A-interchange of a 3-circuit corresponding to the A-set [l,m,m] leads to a tournament isomorphic to C(n).

preaf; Say the 3-circuit (l,m,m+l) is reversed, yielding a tournament T. Then the mapping CT that fixes all i E X\(m,m+l) and puts a(m) = m+l, o(m+l) = m, is an isomorphism between T and C(n).U.

Proposition 4 If T is the tournament obtained from C(n) by the A-interchange of a 3-circuit of A-set

[2, m, m-11, and T' the one obtained from a A-set [2, m-I, m], then T is isomorphic to T'.

Brsef; Let T and T' be the tournaments obtained from C(n) after reversing the 3-circuits (m+l,m+3,1) and (m+l,m+3,2), respectively. It is easily seen, e.g. from Fig. 1, that the mapping a with o(1) = 1, a(2) = 2, a(m+l) = m+2, o(m+2) = m+3, o(m+3) = m+l is an isomorphism between the subtournaments induced by the set Y = (1,2,m+l,m+2,m+3) in T and i n T . Let o(x) = x for all x ~ x \ Y . Then a maps T onto T', since each x ~ x \ Y either dominates all of m+l, m+2, m+3 and is dominated by both 1, 2 or otherwise x is dominated by m+l, m+2, m+3 and dominates 1 ,2 (according as 3 I x I m or m+4 S x < n).U


1 2 1 2

m+2 m+2

T T'

Figure 1.

Proposition 5 If T is a tournament obtained from C(n) by the A-interchange of a A-set different from [ 1, m, m], then T is not vertex-symmetric.

n f : Let [a,b,c] be any A-set different from [l,m,m], and T the tournament obtained from

C(n) by the A-interchange of a triangle (i,j,k). Say, min(a,b,c) = a =j-i (mod n). Then the vertex set O(i+l) induces a transitive tournament of order m.

On the other hand, 2 I a I Ln/31 I (n-3)/2 = m-1 for m 2 7, so that the circuit (i,i+l, ..., i+a) of length a+l I m is contained in the subtournament induced by the set O(i-1). Thus, O(i-1) and O(i+l) are not isomorphic. This fact implies that T cannot be vertex- symmetric.0

Theorem 6 Let T and T' be obtained from C(n) = (X, U) by A-interchanges of mangles (i,j,k) of A-set [a,b,c] and (i',j',k') of A-set [a',b',c'], respectively. If T and T' are isomorphic, then either [a,b,c] = [a',b',c'] or [ [a,b,c] , [a',b',c'] ) = ( [2,m-l,m] , [m,m-1,2] 1.

Recall that the A-sets [a,b,c], [b,c,a] and [c,a,b] are considered to be identical. For convenience, we assume that b is the "middle" element, i.e., a I b I c or a 2 b 2 c holds in


every A-set . By Propositions 3 and 5 , [ l,m,m] is the unique A-set generating C(n), therefore we

can assume that 3 2 a I b 5 c I m. In particular, n = 2m+l 17, and if c = m then b I m-2. Let f : T + T' be an isomorphism. Denote by T(x) and T'(x) the subtournaments

induced by O(x) (the vertices dominated by x) in T and T', respectively. Then T(x) is isomorphic to T(f(x)) for each x E X. First we shall apply this fact for the circuit structures of the T(x).

Let d E (a,b,c) and suppose that d = j - i (mod n). Then it is easily seen from the arrangement of arcs in C(n) that the i - k (mod n) tournaments T(x) for x E (k+l,k+2, ..., i-1,i) have the following properties:

(i) Precisely m - d of those T(x) (namely, T(i-11, T(i-21, ..., T(i-m+d) ) contain the arc (i j) which induces a circuit (longest in T(x) ) of length d + 1 and the reversal of (i,i) makes T(x) transitive.

(ii) The other i - k - 1 - m + d tournaments T(k+l), T(k+2), ..., T(i-m+d-1) are transitive.

(iii) T(i) is transitive if m = i - k (mod n), and T(i) is Hamiltonian otherwise.

Repeating the same observations starting from the other two vertices of the reversed triangle, we find 2(a + b + c) - 3m -3 = n - m - 2 transitive subtournaments, m - a tournaments with a longest circuit of length a+l, m-b tournaments with a longest circuit of length b + 1, and m - c tournaments with a longest circuit of length c (provided that c < m). Moreover, the number of Hamiltonian tournaments among the T(x) is determined by [a,b,c] (we consider them later).

From (i) we conclude that the smallest length a + 1 of a longest circuit is determined by the T(x) (for a 2 Ln/3J 5 m - 2 when n 2 7), and therefore min (a',c') = a holds. Moreover, b + c and the corresponding sum in [a',b',c'] is known, too. Thus at least one of b and b' is less than m - 1. (The reason is that [3,m-l,m-1] = [m-1.m-1,3] is the unique A-set with min (a,b,c) = 3 and b = m-1). Say, b I m - 2. In this case, however, the m - b tournaments with maximum circuit length b + 1 I m - 1 are not Hamiltonian. Hence, their circuit structure determines the value of b as well, so that b' = b must hold. Since a + b + c = n = a' + b' + c', we conclude that the multisers [ a,b,c) and (a',b',c'] are the same.

On one hand, this fact implies that a' = c and c' = a, i.e., the A-sets of the reversed triangles in T and in T' are [a,b,c] and [c,b,a]. On the other hand, we can see that a < b < c holds. Indeed, if say, a = b, then the A-sets [a,a,c] and [c,a,a] are the same by definition, and we have nothing to prove. Also, recall that b I m - 2, since [a,b,c] # [2,m-l,m] by


assumption. From this point, our aim is to derive the contradiction that T and T' are non- isomorphic.

The proof will be completed in three different ways, according to the value of c. In each case, the two basic ideas are as follows.

(iv) Let H be the set of those x E X for which T(x) is Hamiltonian, and let H' be the set of those x' for which T'(x') is Hamiltonian. Then f is an isomorphism between the subtournament induced by H in T and the one induced by H in T'.

(v) Let x and y be two arbitrary vertices. The number of vertices dominating x and dominated by y in T is equal to the number of vertices dominating f(x) and dominated by f(y) in T'.

Properties (iv) and (v) follow from the facts that the properties "being Hamiltonian" and "being dominated" are preserved under isomorphism, and f is supposed to be an isomorphic mapping from T to T .

Case I : c Im-2. In this case, in T - as well as in T' - there are precisely three Hamiltonian

tournaments among the T(x). Namely, T(i), TQ), and T(k) - and TV), T(j'), and T(k') - are Hamiltonian. Hence, f is an isomorphism between the reversed triangles (ij,k) and (i',j',k'). Say, f(i) = i', fQ) = j', f(k) = k'. Note that f cannot reverse the cyclic permutation of the mangle since the orientations of the arcs are fixed. Since a c b 4 c but a' > b > c', there are two vertices of the triangle (i,j,k) whose distance is different from the distance of their images under f, say j - i # f(j) - f(i) (mod n). Now, applying (v) with x = j and y = i, we have a contradiction.

Case2: c = m - I . Say, j - i = j' - i' = m - 1. Put q = i - 1 and q' = i' - 1. By properties (i)-(iii) there are

precisely four Hamiltonian subtournaments T(x) (in T they are T(i), TQ), T(k), T(q) ; in T they are T'(i'), T'Q'), T'(k'), T(q') ). In the induced subtournaments on those four vertices, k and q (k' and 9') have out-degree 2, while i and j (i' and j') have in-degree 2. Hence, f maps (k,q) onto {k',q') and { i j ) onto (i',j']. Since k, k', j, j' dominates q, q', i, i', respectively, we obtain that f(i) = i', f(j) = j', f(k) = k'. However, k - j # k' - j' (for a z b), contradicting (v).

Case3:c=m. Say, j - i = j' - i' = m. Recall that a < b I m - 2. Then there are precisely two

18 A. Astiel-Vidal et al.

Hamiltonian tournaments, T(i) and T(k) in T , and T(i ') and T'(k') in T'. Moreover, i dominates k in T and i' dominates k' in T'. Thus, f(i) = i' and f(k) = k'. Since k - i # k' - i', we obtain a contradiction by (v).O.

The above proof yields that each [a,b,c] with min (a,b,c) 2 3 is determined by the circuit structures of the tournaments T(x). The other three A-sets [l,m,m], [2,m-l,m], [m,m-1,2] are specified as follows:

[l,m,m] is the unique A-set for which all subtournaments T(x) are transitive ; [2,m-l,m] and [m,m-1,2] are the only A-sets for which some of the subtournaments

T(x) contain 3-circuits but do not contain 4-circuits.

Now we are in a position to calculate the number of non-isomorphic tournaments generated by A-interchange.

Theorem 7

The number , N, of non-isomorphic non-symmetric tournaments that can be obtained from C(n) by A-interchange is

24

24

N=-- n2-9 1 i fnzO(mod3)

N=- n2-1 - 2 otherwise.

First we show that the number, A(n) of distinct A-sets is

A(n) = if n I 0 (mod 3) 24 n2- 1 24 A(n) = - otherwise.

Indeed, we have already seen that each vertex x is incident to precisely (n+l)(n-1)/8 3-circuits. Observe that each A-set [a,b,c] corresponds to three distinct 3-circuits containing x, unless n is a multiple of 3 and [a,b,c] = [n/3,n/3,n/3] ; in the latter case [a,b,c] represents just one 3-circuit at x. Thus, the formula for A(n) follows. Certainly, A(n) is an upper bound on N. Moreover, we have to subtract 1 for [l,m,m] and 1 for the pair [2,m-l,m], [2,m,m-1], by Propositions 3 and 4. Thus, N = A(n) - 2.0.


4. THE INTERCHANGE GRAPH OF REGULAR TOURNAMENTS OF ORDER 9

We have constructed the A-graph 2 (R) of regular tournaments of order 9. In this case R = (4,4,4,4,4,4,4,4,4) since the tournaments are regular. We consider 5 (R) because it is known that there are exactly 15 non-isomorphic regular tournaments of order 9 (see [5,6]),

so 5 (R) has 15 vertices. On the other hand, the number of non-isomorphic regular tournaments of order greater than 9 is not known. For the order 11 it seems to be more than one thousand. We calculated the graph 5 (R) starting from the cyclone with 9 vertices. For each of its sons (2 tournaments), we inverted all the 3-circuits and we tested the isomorphisms between the resulting tournaments. We applied the same procedure to the remaining tournaments until we obtained the entire interchange graph 2 (R). The resulting undirected graph is shown in Fig. 2.

Figure 2. 5 (R) : 15 vertices, 46 arcs. Centre : T9-9, diameter: 4, minimum degree: 1,

maximum degree: 11, chromatic number: 6. An arc between two vertices means that the corresponding tournaments can be obtained from each other by A-interchange.

20 A. Astie’-Vidal et al.

The 15 non-isomorphic tournaments corresponding to the vertices of 5 (R) are exhibited in Fig. 3 .

C9 (cyclone) w- 1

4

9

5

I

n. h.(near-homogeneous) T9-2

4

9

5

I

w-3 T9-4


4

9

5

I

T9-5

4

9

5

7

T9-7

4

9

I

T9-6

nhRot (rotational near- homogeneous)

4

9

5

I

T9-10 T9-9


4 4

9 9

5 5

7

T9-12 T9- 1 1

C3oC3

Figure 3. The 15 non-isomorphic regular tournaments with 9 vertices.

Three of these tournaments a e venex-symmetric ([7]) : c9 : C3oC3 :

nhRot : the rotational near-homogeneous.

the cyclone with 9 vertices, the wreath product of two 3-circuits,

These vertex-symmetric tournaments are rotational. (It was shown in [7] that every vertex- symmetric tournament of order less than 21 is rotational). Let us recall from [8] (see also [9]) that a tournament Tn = (X, U) of order n is near-homogeneous if n = 4X+l,h21, and every arc belongs to 31, or X+1 3-cycles.

There are two non-isomorphic near-homogeneous tournaments with 9 vertices : nh and nhRot.


The tournament called nh is not vertex-symmetric and has 3 vertex-orbits: ( 1,4,7) , (2, 5 , 8) and (3 ,6 ,9].

5. CONCLUSION

By using the elementary operation of A-interchange, we can construct O(n2) non- symmetric regular tournaments from the cyclonic tournament of order n, that are mutually non- isomorphic. The method is easy to implement, and we have installed a computer program that generates the different tournaments. The difficulty is to avoid redundant calculi, and detect isomorphic tournaments. This method needs a good isomorphism test algorithm. Up to this condition it can lead to an algorithm generating all regular tournaments of order n.

6. REFERENCES

C. Berge, Graphes, 3rd Ed., (Bordas, Paris, 1983). J. W. Moon, Topics on tournaments, Holt, Rinehart and Wilson, New York, 1968. L.W. Beineke and K. B. Reid, Tournaments, in "Selected Topics in Graph Theory", (L.W. Beineke and R.J. Wilson Eds) chap 7, Academic Press, London (1978), 169-204. R. Bruali and Li Qiao, The interchange graph of tournaments with the same score vector, in "Progress in Graph Theory", (J.A. Bondy and U.S.R. MURTY Eds), Academic Press Canada, 1984. A. Kotzig, Cycles in complete graph oriented in equilibrium, Mat. Fyz. casopis Sav. 16

A. Kotzig, Des cycles dans les tournois, in "ThCorie des graphes", journtes internationales ditude, Rome 1966, Dunod Sr Gordon and Breach, New-York (1966),

(1966) 2, 175-182.

203-208. B. Alspach, On point-symmetric tournaments, Canad. Bul. of Math., 131 (3) (1970),

C. Tabib, Caractirisation des tournois presqu'homogknes, Annals of Discrete Math. 8,

A. Astii-Vidal and V. Dugat, Near-homogeneous tournaments and permutation groups, to appear in Discrete Math.

317-323.

(1980), 77-82.


Combinatorics '90 AT Barlotti et al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. 25

On Structures T ( t , q , T , n)

Walter Benz

1. In a previous paper [2] we introduced the notion of a structure T ( t , q , r , n ) , the definition of which will also be given here, however, iii an equivalent but slightly different form (s. sciction 2). 3-nets for instance, also affiiie planes and Lagucirrc planes, Minkowski planes, optimal geometries among other structures arc esariiplcs of geometries T ( t , q , r , n ) as far as they are finite (s. section 3). In [2] wc p r o v d a structure theorem representing a T ( t , q , T , n ) equivalently by a T ( t , A, r , 1) and by n - 1 sharply t-transitive subsets of the symmetric group S,. Nevertheless, ill-

teresting examples are difficult to obtain. In section 4 we give sonic new structurcs 2'(3,7(;)(7 + l)7i-1,y + l , n ) ( n 2 2,y a prime power ) which are based on cl-iaiii geometries.

2. Let y , v , n be posit.ive integers and let

be a matrix of 7 ' columns aiid T L rows. following threr properties hold true:

We say that PI is a ( q , I , , I , ) iiiatris iff tlic

( i ) The entries Clzk of A4 ar y-sets,

(ii) The set P := Ctl U Ci2 U . . . Ci, does not depend on i E { 1 , 2 , . . . ,71} and it is a q . w e t ,

(iii) The cardinality uj of the intersection of j(1 5 j 5 n) entries C;k does not dcp(t11d on the special selection of the Cjk provided that no two of them belong to t,lie same row of M .

Remarks: Suppose that M i s a (q ,~ ,n -n ia t r ix . Then T."-' iiinst be a divisor of ( s .

[2]). There exists, on the other hand, a ( q , r , 12)-matrix in casc tha.t, lq. - T I i ( ~ elements of P are called points. The sets C,k are called the coinponents of M . - \Vc, observe that (i) and (ii) imply C,, n C,p = 0 for 01 # /I. - In [2] we proved tliat tlie number aj of (iii) is given by 5 for every j E { 1,. . . , n}.

26 W. Benz

Defiiiitioii: Aii iiitcbr;i.ctioii -1- o f ii (g , I , , ti)-inatrix M is i i . /,-sc:t, sucli t,li;i,t, S 1-1 ( ' t k # v1 for every coniponcnt Cik of' 01.

Remarks: We obscrvc that card ( X fl C,k) = 1 for evcry iiitcra.ct,ioii X aiid w c ~ y coinponeiit Cil; of M . - In [2] we proved t,liat

cardl(M) = (z)r. Q (T!)'+',

where I ( M ) denotes the set of all iiiteractions of Ad.

Example: Lct, X lie the 'rt( 2 2)-diiiiensional afine gcomctry ovcr tlie G1;ilois fic.ltl

G F ( y ) = { u , , . . . , a?} Iiused on tlic affiiic coordinate syst,r?iii . I , , , . 1 ' 2 , . . . , . I , , > . Dcfiiirl iiow C;k to be t,lie liyperplarie (as a poilit set) of ecluatioii .r, = (LA.. We t h i i gc,t, i i

(Y '~ - ' , y, ri)-inat,rix.

Dcfinitioii: Let A/r be c2 ( q , T , n)-matrix and let t 5 r bc a positive integer. Sulil)(ise t,li:Lt B is it subset of I ( A J ) . We say that (111, B ) is a geometry T ( t , (I, 'I,, 1 1 ) iff t,lie followiiig property holds true:

( iv) Let A be a f-subset of P si~cl i tliat iio two eleiiiciits of A iii(' iii t l i r 4iiiiie coi i ipoi icxt of A d . Then tlicre csists exactly oiie X E B with A c S.

Esaniples. Class 1) Lct Ad bc a ( q , r.,7i)-matrix. Put t .= 7 itiitl l? = I (A1) Tlicil obviously (Ad, B ) is a T ( r , q , T , n).

Class 2 ) Talie tlie special (y7'-', 7 , n)--matrix which we preseiited befoie and wliich was based on tlie affine space C. Let here B be the set of all lines { ( p l ) + X ( u,)IX E G F ( - / ) } such that n:=, t i , # 0 Then ( M , B ) is a T(2 , yn-l,y, 72.).

R w m k Let (A{, B ) lie a T ( f , q , T , n ) Then

:is we proved iii ['I.

3. W. Heise urid H. Iiarzcl [ 5 ] iiitroduced the so-called L L I ~ L I ~ T I Y - a.iitl Miillio\vslti- n-structures, R. Periiiutti [6 ] defined tlie Mobius-nL-struct,ur~s. It turiis uiit t.lint 0111

T ( ?+in, (I, ( I+ , I IZ , 1) are ex:xct,ly the La.guerre-rrL-structures iiiid t l in t t,lic, T ( I I I + 2 , y, (1. '7) are pi-c>cisely the hiIinliowslii-~/~-str.uctures. The classes of examples wc' woLIlcl lilic t,o

prcscnt in this scctioii itIe pu t ly tliose /ii-structures. Wc iievcrtlirlcbs 1)I here iii order to have t,lieni availiible iii t,he !anguage of our geoiiiet-rics T ( t , (I, I , , / I ) . Tlio optimal ( r , t)-geonict,ries of order q a.re exactly the T( t , g, r , 1).

Let P # @ be a set aiid let L1,Lz,L3 be pairwise disjoint sets of subsets of 1'. Tlir: chrieiits of P a.rc called points and tlic eleirients of L I U L2 U L3 lines. Tlic stiiictiiw

On structures T (t,q.r,n) 27

(P,L1, Lz,L:l) is called a 3-net ( s . Aczd [I]) provided that every line contailis a t lcast two points, t,lia.t two lilies of distinct L, meet in exactly one point, tliat to evcry Po E P and to every L; there exists exactly one g E Li such that Po E y .

Obviously, two lilies of a 3-net have the same cardinality y 2 2 and obviously, card Li = q for evcry i E { I ,2 ,3} . Putting

Li =: {Cil,. . . , Ciq}

for i = 1 , 2 in case that y is finite we get a (y,q,2)-matrix Ad. It is Ls ii set of interactions of Ad aiid we hence get a T( l ,y ,q ,2) . Starting on the other hand with a structure T(1, q , (I, 2 ) , y 2 2, say ( M , B), we can derive a 3-net b y defining

Li =: { C,1,. . . , C,,]

for z = 1 , 2 and by defining L3 := B. We hence have the

Examples: C ~ ~ Y Y 3 ) Exactly the T(1, q , q , 2) with y 2 2 are tlie finite 3-nets Let us turn over to the ca.se of a finite affinc plane A of order q. Take t8wo distiiict parallel bundles rI1, 112 of A and put

{ G I , . . . , Ctq} := ni

for z = 1, 2. We 1ienc:e get a ( q , q , %)-matrix M . Every line now of /I wliich is iiot. i n II1 U r I 2 must be aii interactioii of Ad. Deiiote by B the set of tlicse 1iiit:s. Tlius (121, 13) is a T(2 , y , q , 2). Suppose on the other hand that ( M , B) is a structure T( 2, q , q , 2 ) with q 2 2. We define t,lie components of M and also the elements of B to l x the 1iiic.s. Observe a2 = 1. So having two distinct points in the same component, this conilionent turns out to be tlie oiily line containing both points. If they do not lielong to the same componeiit there is exactly one interaction in B containing both ]mints. Therc exist at, least lwo components. So there exist three points not on a. comiiion line. - Let, g Iic a line and Po $! g be a point. In case g = Cik take h = C;, :3 Po. Tlieii h is the oiily line through Po which ist parallel to y. We now count the lilies t,lirougli Po. For tliis purpose take a coinponent C 9 Po. A line through Po meets C or it, is pa.rallcl t,o C . There are hence q + 1 lines through Po. Consider now the case that y E 13. I get, q 1iiic.s by joining Po with a point in B. The (y + l) th line through Po is lielice parallel to y .

This leads to the Examples: Clu.ss 4 ) Exactly the T(2, y , y, 2) with q 2 2 are tlie finite a.ffinc plmics.

Let now ( P , Z ) be a finite Laguerre plane (s. Halder, Heisc [4]), P tlic set, of poiiits (spears), 2 the set of cycles (bloclts). We t,hen go over to tlic class of

Examples: Cluss 5 ) Exactly the T(3,q,y -t 1, 1) with q 2 2 arc tlic fiiiitc Luguc~rc. plaiies. The generators of ( P , 2) are all of tlie same cardinality q 2 2. Wc clefiiie tlic C'Lk t o I)c the generators. Therct are q+ 1 generators. We thus get a (y, q+ 1, l)-matrix Ad. Every

28 W. Benz

cycle of (P , 2 ) is an interaction of M . Define B to be the set of all cycles. Hence (Ad, B ) is a T(3, q, q + 1,l) siiice through three pairwise non parallel points there is exactly oiic, cycle. - On the other hand let ( M , B ) be a T(3, q, q + 1, l ) with y 2 2. Define 13 to be the set of cycles and the C1k to be the generators (the parallel classes of points). 111 order to verify tlii\.t, ( P , B ) is a Laguerre plane the only crucial point will be the proof of the tangent axiom: If z is a cycle and if Po, Qo are points with Po E z 9 QO and PO W QU

then there exists exactly one cycle z 3 Po, Qo such that z n T = {PO}. - Suppose t h t Po E C11 and Q O E Cl2. Because of y + 1 2 3 there also exists C13 and Lec:i.usc~ of ca.rd CIS = y and f = 3 there exist exactly q elements 2 of B through Po and (2". T l i ~ problem iiow is to show that precisely one of those elements 5 sa.tisfies z n :r = {Po}. Let Q, be the point on z which is parallel to Qo. Then there are exx t ly q - 1 points in z\{Po,Ql}, say Rl, . . . ,R,-l. The cycles z ; 3 Po,Qo,Ri(i = 1, , y - 1) are pairwisc. distinct since zi = zj for ,i # j would imply z , 3 Po, Ri, Rj and .nee z = z , 3 Qo. A cycle z 3 Po, Qo with ca.rd ( z n x) 2 2 must, be a suitable zj. Since tliere are q cyclw .I'

through Po, Qo there hence exists exactly one with z n 2 = {Po} .

Furtlierinore we would like to consider the

Examples: Clavve 6) Exactly the T(3 , q , y, 2) with q 2 3 a.re the finite Minkowski p 1 a . n ~ ' ~ .

Suppose that C is a. Minkowski plane (for a definition s. Halder, Heise loc. cit.). Now defirie the Clk to be the equivalence classes concerning tlie relation ( I t axid thc c.11;

to be the classes with respect to [ I _ . The circles are the interactions iii B. The11 we get a T(3 , y, y, 2) with y 2 3. Having on the other hand a structure T(3 , q , y, '7) with y 2 3, again the ta.ngeiit axiom is the crucial point of the proof Let z be R circle, (i.ci. a11

element of B ) a.iitl let, Po, Qo be pointjs with Po E z 9 Q o and Po WQo ( ix . Po, Qo arc iiot,

in the same component). We have to prove that there exists e x i d y one circle T 3 P,,, (20

with z n r = {Po}. III fact! Take the points &of,&; E z with Q: [ I t (20 1 1 - Q;. Ol.,sc:rvr that. the points Po, Q o f , 62, are pa.irwise distinct. There are hencc exactly q - 3 pairwisc. disthct poiiits Rl , . . . , Rq-3 in ,\{Po, Q;, Q;}. Because of q 2 3 thcrc, are at, 1( coniponents of tho forin C l k . A s s ~ i i i e that Po E Cll n C21r Q,, E C',? n C X L . 111 t l m c are y points. A circle y 3 el, Qo has exactly one point Yo wit,li CI3 ill coniiiion. 1'" must be in C13\(C21 U C22). We ha.ve therefore precisely q - 2 circles through P,, Q,, I)y observing that throiigli 3 pairwise 11011 parallel points there is exactly one circle. q - 3 of those circles contain a I&. There is hence one circle z 3 Po, Qo left wit.h z f l .r = { P o } .

The following class contains class 2 of section 1:

Exa.inples: Clas.~ 7) Take a finite near field ( F , +, .), i.e. (F, +) is a.11 alxtliaii groiil> wit,li neutral element 0, (F\{O}, .) is a group and

C L ' ( b 3 . C ) = n . ' b + C L ' C ,

0 . C L = 0

hold true for all u, h,c E F.

On structures T (t,q,r,n) 29

Put

Define B to be the set of all n

(pi , . . . ,pn) + ( ~ 1 , . . . , un) .F with n vi # 0 k = l

Then (Ad, B ) is a T( 2, y n - l , y, n) .

4. In [2] we presentcd examples of type (T(3,(y + l)"-',y + 1,n), wherc y is a priiiir. power. The aim of this section is t,o construct geometries (T(3 ,7 ( ; ) .(-!+ l ) n - l , ~ + I,?!) with IZ > 1 and y a prime power.

Suppose that, A is an associative algebra with ideutity elenlent, 1 # 0 over t,he f ie l t l E . We identify n E F with a 1 E A. We then assume F # A a i d tliat F is i n tlic cctiitc'r of A . It stands IT for the group of units of A. The sets

u ' (Z1,ZZ) := { ( U Z I , U Z 2 ) I U E U }

are called points in case Z ~ , Q E -4 such that the right ideal < 3:1,x2 > R generntx~d by x l , x 2 is equal to the whole ring A. The point sets

{U . ( k l a l l + k Z ~ 2 1 , k l a i z + kzoizz)lk1, kz E F, not both 0 )

are called chains in case al l , 0121, a12, aZ2 E A such that the matrix

is regular from both \ides. Tbc set of points is denoted by 2. This chain geometry AF will now be specialized in the following way: Lrt F he tlic Galois field G F ( y ) and put

where n > 1 is suppo5ed to be an iiitcgei. Again we are usiiig tlie notation F =: {al , . . . ,a,}. By I2(z = 1,. ideal

, n ) wt' denote tllc'

I, := { a E Al(a),, = 0)

30 W . Benz

of A. Here ( a ) t k staiids for the entries of the matrix a ,

Examples: Cluvv 8) For 2 E { 1 , 2 , . . . , n } put

Every chain of A!' is then an interaction. If B is the set of all chains thcii ( M , B) wit11

is a geometry ~ ( 3 , 7 ( ; ) ( 7 + l)n-' ,y + I,??).

Rernark: In case = 2 for instance we get a T(3,y(y + l ) , y + 1,2) with cai t l 2 = 18 arid card B = 4s.

The proof that class 8) consists of geonietries T(3 , y(:) . (y + l ) lL- i , y + 1 , 1 / ) will lw given in several steps, ( A ) , ( B ) , . . . , ( H ) .

Proof: Let U ( x l , 2 . 2 ) be a point. Then we get ((z~),,, ( z ~ ) , ~ # (0,O) for all L = l,?, . . , I I

because of < x1,z2 > R = A. It is now possible to find an element u in l 7 such tliat 71 -1.2

is a matrix with

in case (z~),; # 0. Tliis reduces the still free entries of u 011 t,hose ( u ) , ; ( v 5 2 ) for which (22 ) ; ; = 0. Since (x1)ii # 0 in case (x2) i i = 0 we are using the still free ent,rics ( U ) ~ ; ( U 5 i ) of u in order to get

u < i (uq), , = { 1 for

u = 2

Then I L is uniquely tlet,t:riiiined and we get a standard foriii (y,, y2) = ( u.r l , / / . I , ? ) for t , l r point U ( z l , 2 2 ) as follows:

(1) (Y2)tzE{O11)foriE { l , . . . , l Z ) ,


(2) (Y~)~, = 0 for v < i in case ( ~ 2 ) ~ ~ = 1,

(3) ( Y l ) u t = { y for u < i

in case ( ~ 2 ) ~ ~ = 0. v = 2

If now (ZI,ZZ) := (wyl ,wy2) ,w E U , also satisfies (I), ( 2 ) , (3), then to must be the identity element of A.

These considerations will now be used in order to determine the number of points of 2. So the question is how many ordered pairs (y1,y2)(yI,y2 E A ) exist which sat,isfy ( l ) , ( 2 ) , (3). Take a y2 such that exactly t of the elements (g2)iZ are equal to 0. Theii there are - t free entries in y1 and y2. We hence get y(( 2 )-' points in this sit,uatioii.

i.e. ( y ) . Y('';')-~ poi1it.s in all cases that, exactly t of the eleiiients (yz )z i arc 0. If WP iiow add up those numbers from t = 0 ~ i p to t = n we get the wanted iiunibcr.

7 ' t l

(B) C,k n C,l = 0 for a11 i E (1,. . . , n } and k , l E (1,. . . ,y + 1) with X: # 1. Proof: Let U(z l , Q) he a point. Then there exist a , /3 E A with x c l a + z2P = 1 because of < ~ l , 5 2 >R= A. Assume now U(xl , x2) E C,k n Cll, k # 1.

This implies E I , for suitable points U ( p 1 , p 2 ) # U ( y ~ , y , )

with p1,p2,q1 ,q2 E F . Now

leads to the contradiction U(p1 ,p2) = U ( q 1 , q ~ ) .

(c) Let 2 1 , . . . , a 3 Ije j distinct elements of (1,. . . , n } . The11

card (C,l~l n . . . n C,,kk,) = y(;) . (7 + l)"-J

for all k l , . . . , k, E { 1,. . . , y + 1).

Proof: Points u(zl , x 2 ) in nj,=, clYk, must satisfy equations

32 W. Benz

with ( p u , q u ) = (0 , 1) or p v = 1 and q,, E F . For (x1,xZ) we take the st,nlitla.rd forili i ~ s it was described earlier. In order now to count the st,ill free eiitries of .L', a,llcl .c2 IVY I iavc to pay attention to all places ( k k ) of 5'2 which are not in {(il21), . . . , ( i , 7 i J ) } . Tlicre are r~ - j such places wliicli we will call the special places. In case that there a r ~ cxactly t (0 5 t 5 I I -- j ) special places contaiiiiiig the element 0 we get

frce entries in .c1 a n d ~ 2 . We hence get

point,s ( T I , ~ 2 ) in this situation, i.c

(" j ) . y [ ( " $ 1 ) - + 1

poiiits iii all cascs that c.xact,ly t spccial places contain 0. Now

firiislics the proof of (C) .

7 t 1

IJ C,k = A k= I

for d l i E {I , . . . , I L } . Obviously, Ti is a. q . r-set. Property (iii) of a ( ' ~ , , / ' , ' / , ) ~ i i i i i . t , i , i ? c is consequence of (C) .

(E) Every cliaiii of Al., is ail interaction of M .

Proof Chaiiis are (y + l)-scts. Suppose now that c is ii chaiii of - 4 ~ . Tlicii WI' IKI~v(: t o

show c n Clk # ld for every conipoueiit, C,k of M . Let, c be giveii b y


c = {u ' ( P I ~ I I + ~ 2 ~ 2 1 , ~ 1 0 1 2 + ~ 2 ~ ~ 2 2 ) ( p l , ~ 2 E F, not botli 0},

where 0 1 1 1 , 0 2 1 , ~ 1 2 , a21 E A such that

( :i: zi: ) is regu1a.r froin both sides. We are now looking to the ideal li

and to the point U(dl,d2) with d l , d 2 E F . The equation

is solvable in pi ,pi E F such that 110th are not 0. This implies

(F) Let [ [ ( T I , q), I'(y1, y2) be distinct points. Then thcre is 110 coiiipoiieiit of M containing both points if and only if

f o r a l l i E (1, . . . , ' I 1 ) .

Proof This is a. consequence of tlie following statement: Suppose that i is ~ I I cl(mi(wt of { 1,. . . , PL}. Theii tliere exists k E { 1,. . . , y + 1) with U ( z 1 , .u2 ), I!( y, y r ) E C z ~ . i f ai id

oiily if

In order to prove t l i i h stateineiit we will assume that

uz1 + bz2 E I,,

ay1 + by, E I ,

holds true for a pair u, b E F with (a , b ) # (0,O). This iniplies (*). If 011 tlie otliei liaiicl (*) holds true, we find a, b E F , not both 0, such that

a(Zl),* + b(Z2)i; = 0

4 Y l ) l z + 4 Y 2 ) i I = 0

is satifietl. But tliis iiiiplies the existence of a k E { 1, . , . , y + 1) sucli tliat. Iiotli l)oiiit,s are in Czk.

34 W. Benz

( G ) Let U(z1 , x2), U(1~1, y 2 ) be distinct points. Then there is no componcnt of Ad containing: both points if and only if the matrix

is regular from Imtli sides.

Proof: Having ( 0 ) for all i E { 1,. . .} we can solve

uiiiquely in a , E E A:

tleterniines uniquely a,, and E ~ , . From place (ii) we go over to place (i-1)) a r i d so on up to place (l i) . This deterinines and &,-1,, uniquely and so on up to a , , t ind c^lc

hecause of ( 0 ) .

We also can solve

y1/1 + y 2 6 = 1

Z I P + 226 = 0

uniquely in / j ,6 E A by just, applying the result ahead. We tlius get

(;; ;:)(: : )=( ; ;) This especially iinplics

for all i = 1,. . . , n. With t,lie sanie arguments we thus find [, rl , ( , p E A siicli that

Multiplying this cquatioii from the left with


So we have regularity from both sides. Starting on the other side with

we get a t once (0).

Proof What remains to prove is the following: Let U ( s l , z2), "(y1, y z ) , U( z l , ~ 2 ) I J C three distinct points such that no two of them belong to the same coniponent of Ail. Then there exists exactly one chain containing these three points. Applying ( G ) we go along the proof of A.2.2 in [3], page 229: Instead of " U ( ( z 1 , s 2 ) is not pa.rallc1 to

U(y1, yz)" there, read now "the iiiatrix ( ;: ;: ) is regular froni botli sicIes". Also

make use of the fact t1ia.t

is an endomorphism o f A.

References

[l] .4czd, J.: Quasigroups, Nets and Noiiiogranis, Advances in Math. 1 (19G5), 383-350

[2] Benz, W.: On a. test of Dominance, a strategic decomposition aiid Struct,ures T ( t , q , T , ,n), Annals of Discrete Math. 30 (1986), 15-30

[3] Benz, W.: Uher cxine Cremonasche Raumgeoinetrie, Math. Nnchr. 80 (1977), 225- 243

36 W . Benz

[4] Ha.lder, H.R.., Heise, W.: Einfuhrung in die Kombinatorili, Hanser-Verla.g, Miii iclim - Wien 1976

[5] Heise, W., I<a.rzel, H.: Laguerre- und Minkowski-n-Strulituren, Rend. 1st. tli Ma- them. Univ. Trieste 4 (1972), 139-147

[CJ] Permutti, R.: Uiia geiieralizzazione dei piani di Mobius, Le Matematiche 22 (13G7), 360-374

Walter Benz Department of Mathematics University of Hamburg Bundesstr. 55 D-2000 Hainburg 13

Combinatorics '90 A. Barlotti et at. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. 31

A TACTICAL DECOMPOSITION POR INCIDHNCE STRUCTURES

MeterBettcn and Mathlaa Bram

Math. Sem. Univ. Ludewig-Meyn-Str. 4, D 2300 Kiel

Stet t iner Str . 6, D 2080 Pinneberg

For every finite incidence s t ruc ture we define parameters by s o m e ordering process. Refining this process will give a tactical decomposition of t h e incidence matrix which we call TDO (tactical decomposition by order ing ). This decomposition generally is coarser than t h e tactical decomposition defined by the au tomorphism group ( T D A ) ; bu t it has t h e advantage t h a t it can be computed rather quickly. We point o u t t h a t t h e TDO in combination with the TDO's of the point derivations is a s t rong invariant. For instance it suffices to characterize a l l 5250 linear spaces on 10 points.

1 ) Studying incidence s t ruc tures one of ten assumes t h a t there is a large group of automorphisms. For instance one may suppose tha t the automorphism group ac ts transitively on points ( homogeneity). One also likes to have regularity conditions, f o r instance t h a t every block has t h e same number of points and every point is incident with the same number of blocks. But since most incidence s t ruc tures have small automorphism groups or a r e ra ther irregular we are interested a l so in these cases. We observe t h a t irregularity has t h e advantage of providing some information concerning t h e s t ructure . In the following we describe a sequence of parameters which ar ises f rom irregularity in a natural way.

2 ) We explain t h e procedure by an explicit example: Let there be given t h e following incidence matrix

1 2 3 4 5 6 7 8 9

X x x x x x x X x x

X x x X X X

x x X x x x

x x X

38 D. Betten. M. Braun

E

having 9 points and 8 blocks. This is a linear space on 9 points, where the 9 blocks of length 2 have been omitted. We collect al l blocks with maximal length ( in t h e example, one block of length 4 ) to one block domain. By permutation of blocks we put this domain a t the top. We separate by a horizontal line the next block domain consisting of 7 blocks of length 3.

1 2 3 4 . 5 6 7 8 9

x x x x X x x X x x

X x x X X X

x x X x x x

x x X

In general there are more than two block domains which we order by t h e length of their blocks. We call t h e length of a block the block type of t h e f i r s t kind , and we a l so call the block domains j u s t defined the block domains of the f i r s t k ind .

Now we look a t points and the way in which their incidences a re distributed to the various block domains j u s t defined. For instance point no. 3 has one incidence in t h e upper block domain and one incidence in t h e lower block domain, whereas point no. 4 has one incidence in t h e upper domain and t w o incidences in the lower domain. So, point no. 3 and point no. 4 have different _type. We now collect all points having t h e same type to one point domain and we order t h e different point domains lexicographically, separating t h e various domains by vertical lines.

1 2 4 3

x x x X X

X X

x x

X x x

8 9

- K

X K

X -

We speak of Doint t ypes and poin t domains of the f i r s t kind.

Now we look a t t h e blocks again and t h e way in which their incidences are distributed to t h e point domains of f i rs t kind. We call these distributions

A tactical decomposition for incidence structures 39

the block types of second kind. In every block domain of first kind we collect all blocks having the same type of second kind and order these domains of second kind lexicographically, separated horizontally by a line.

1 2 4 3 S 6 7 8 9

Now it is clear how to go on : We define point types and point domains of second kind, then block types and block domains of third kind and so on.

This procedure will stabilize because t h e number of points and blocks is finite. In t h e example t h e next stages are:

1 2 4 3 . 4 6 7 8 0 1 2 4 3 5 6 7 8 9

2 -+ 2 1 1

4 4

6 6

7 7

3 3

5 s 8 8

7' 1 2 4 3 5 7 6 8 9 1 2 4 3 5 7 6 8 9

The tactical decomposition by ordering ( TDO ):

The stable situation we reach by the procedure described above defines a de-

40 D. Betten, M. Braun

composition of t h e incidence matrix into rectangles, where every rectangle is a configuration. By a configuration we mean a n incidence s t ruc ture consisting of vi points, bi blocks with every block incident with ki j points and every point incident with rij blocks. If we define by f . .= v i - r.. = b: k. t h e number of incidences in t h e rectangle ( i , j ) , we g e t t h e following scheme, giving the parameters of the TDO:

11 'I I 11

. . . . . . . * . .

Here I is t h e number of final point types and J is t h e number of final block types.

There a re t w o extremes: a.) In the TDO all final point domains and all final block domains have

only one element ( point or block ) as in t h e example above. In th i s case t h e TDO and the incidence s t ruc ture coincide, when the f . .= t denote t h e incidences.

b.) The incidence matrix is a configuration from the beginning. Then of course t h e TDO is of no help.

'1

Usually the procedure of refinement will s t o p somewhere in between.

If t w o incidence matrices are isomorphic, then t h e related TDO's a re congruent . Therefore, if for t w o matrices t h e related TDO's a re not congruent , then the incidence matrices cannot be isomorphic.

3 ) We will use t h e notion of TDO in order to s tudy linear spaces on small point number. A linear space ( P , B ) consis ts of a set P of points and a family B of subsets of P called blocks, such tha t f o r every pair p * q of different points there is exactly one block b c B with p , q c b . In order to exclude degenerate cases we assume I P I 2 2 and I b I 2 2 for every b c B.

The number N ( n ) of pairwise non-isomorphic linear spaces on n points, 2 5 n 5 1 0 , is

n 2 3 4 5 6 7 8 9 10 N ( n ) 1 2 3 5 10 24 69 384 5250

The numbers for n 9 can b e found in C 11; t h e number N( 10) has been calculated by David Glynn ( Christchurch, New Zealand ), who a lso determined


t h e size of the automorphism group in every case. Calculating the TDO's with t h e computer shows t h a t f o r n i 8 all isomorphism types can be distinguished by t h e TDO's. The f i rs t situation where t h e TDO d o e s not suffice occurs f o r n = 9 . Example:

(i)

( i i )

1 2 3 4 5 6 7 8 9

x x x x x x

x x x X X X

X X X X x X

7 a 9

2 3

6

The 2-blocks a re omitted. These t w o examples are not isomorphic but they have t h e same TDO:

0

Another example consis ts of t h e following linear spaces on 9 points:

( iii )

x x X X x x x x x

x x x x x X X x x

x x x x x x

x x x X X xx

X X x x

x x x x

x x X X

x x

( i v ) ( V )

< x x < x x < x x x x x x x X X x x

x x X x x x x x x

< x < x x x

X X x x

x x x x

x x x x

x x x X x x x x x

x x X x x x X x x

x x X x x X x x x

x x x x

x x x x x x

x x x x x x

x x

42 D. Betten. M. Braun

These three linear spaces are pairwise non-isomorphic as can be seen from the cycles i n the 2-block domains. B u t in all three cases we get the same TDO:

This example shows that the tactical decomposition defined by the orbits under the automorphism group ( TDA) may be finer than t h e TDO.

4)Since the TDO is not sufficient for the isomorphism problem, we look for further invariants. One easily sees that the two linear spaces ( i ) and ( i i ) can be distinguished by taking point derivations. By t h i s we mean omitting one point and all blocks incident with this point. In example ( i ) all points are equivalent and derivation gives :

B u t in example ( i i ) w e can find a point having the derivation:

This example proposes to use besides the TDO also the derived structures for all points.-Here we do not look a t the isomorphism type of the derived structures but we calculate their TDO's again. In addition we take into account t h e TDO of t h e total structure already found. The TDO together with t h e TDO's of t h e point derivations is now a stronger invariant.

5 ) Observation: TDO together with TDO's of point derivations distinguish all linear spaces on n points, ns10. Namely with help of t h e computer, using TDO and TDO's for point derivations we get 384 linear spaces on 9 points and 5250 linear spaces on 10 points. These numbers coincide with the numbers listed above.

We note that the computer constructs and distinguishes 232923 linear spaces o n 11 points. Question: Is t h e number of isomorphism types bigger?


Remark: Of course this s t ronger invariant ( TDO and TDO's f o r point derivations ) will no t suffice in general. The smallest example we know, where th i s invariant does not suffice, a r e t h e t w o Steiner Systems S ( 2,3,13) o n 13 points ( C 2 1 page 27, table 1 ).

References

C 11

C 2 1

C 3 1

C41

j. Doyen, Sur le nombre d'espaces lineaires non isomorphes d e n points, Bull. SOC. Math. Belg. 19 ( 1967) , pp. 421-437 J. W. di Paola and H. Gropp, Hyperbolic Graphs from Hyperbolic Planes, Congressus Numerantium 68 ( 1989 ) , pp. 23-44 M. Braun, Erzeugung Linearer Raume auf kleinen Punktanzahlen am Computer , Diplomarbeit Kiel 1990 W. Page and H. L. Dorwart, Numerical pat terns and geometrical configurations, Math. Mag. 57, (1984), pp. 82-92


Combinatorics '90 A. Barlotti et al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. 45

NOTE ON A CHARACTERIZATION OF SEGRE VARIETY IN P G P A

.llessandro R T C € W and Stefano INNAMMATI

1. - Intrcduct ion.

Let. S be a f i n i t e p ro jec t ive spr-.t. of dimns1i.m 2h+L, h r l , over t h e Galois f i e l d GF(q) of order q, where 111 i s a p w e r of * i prine. Let f, = zf.li

O<d52h+l. A k-set I<, i .e. a si.kiset of S havinq k p i r i l '=, is sa id t-ci t c

I I J f class [ 0 , l , ? , a l 1 i f each l i n e which has at. least t.hree p i n t s of K LS

contained i n K . Moreover. a k-set K is said t o be of t y p (m,n) , where 0gn<ns82h, if each h y p r p l a n e i n t e r s e c t s K i n m or n p i n t s , hu t t h e r e is

at. least one m-serint h y p r p l a n e and at least. m e n-secant hflxrplane. We

call an S cont-ainpcll i n K an S of I< sunply. We r e f e r t o [31. I61 and [ 71 .

W e recall t h a t a Seyre v a r i e t y S xS is t.he lixus of t h e p i n t s m t.he l i n e s which in t e r sec t . t h r e e S ' s skew t.wo hy t w o i n a S . The reader des i r inq a nwre txxiplete t - r e a t m n t is re fe r r ed t o [ l l , I41 and [51.

3 h t l

he t h e "dJe1- CJf t1E LOi1lt .S IIjf cl d-dilTM31SlOMl S L A h 5 ~ k t z Y 5 Of S , d 2 h t l

2 h t l

2 h

d d

1 h

h 7 h t l

I n d previous p q w r of t-he aut.hr:ii-s [ 2 1 , k-set.s of class [ 0, 1, 1, and t y ~ ~ e ( 8 , 28 -t, ) i n a projeckive space of dinension 2h+l. where h i1 , w e r e 1nVeStl~jat.rd. In p a r t i c u l a r w e proved t-he folluwinq:

1 h - 1 h I3 Z h

46 A . Bichuru, S. lnnamoruti

2 ) the numlwr of n-secant h-ypx-planes t h m w h '3 p i n t of K is n:

3 ) the n m k r of n-secant hr'perplanes through a l i n e of h is m;

4 ) through an-v p i n t of h there piss exac t l y 8 +1 lines of h:

51 l e t /L be a line of It f i o t cor!tained in dnv plane of h , t h ~ n the lines of

k, iriters~-t L ~ Q z md othei. than 1 , cover n.

h - 1

In order t.o charact.erize a Segre va r i e ty S xS as concerns its twliavior

wit.11 t.he l i n e s arid w i t h t he t i nx rp lanes . w e pr i~ve, tisinq Result 1 .1, t h e 1 h

f o l low1ng :

Theorem 1 . 2 .- In a project ive spce of dimension 2 h + l , h r l , and order q,

greater than 3 , a k-set I< of class [n,1,2,@ 1 ax1 type ( P @ ,20 -ti )

s~tisfyirig the fi~lloctrinq c.ondition: 1 1 1 h - 1 h 11 2 h

I * ) there is a t l e a f mi+ line of K t l i m t q h which tliere psses 110 plane

of K ;

is a Seqre w x - i c t y S sS . 1 h

2.- The proof of t h e theorem.

Let. u s asstm that K is a k-set of S and t y p

(p1ph- , ,7@ -8 ) sat . isfyjny t.he condi t ion ( * ) and let R denote a l i n e of K through which t.hera passes no p lane of K. By Result 1.1. through each p i n t . of 1, there pss C) l i n e s of E; d i f f e r e n t from /L. Since ncl plane of

K psses through a , t h e 8 @ l i n e s of K, d i f f e r e n t from q., thrnuqh p i n t . s of 4, met. i n no p i n t rxitsLde q. k t R he a p i n t of /t. &wh h q x r p l a n e t.hrouqh R bu t not through T has q(3 p i n t s i n connun wi th t.he l i n e s of K, di f fe ren t . from R, through t h e p i n t s of /L di f fe ren t . € r m R.

Let us cons ider t h e ( 2~ -0 )-secant. hyperplanes. T;lking int.o ar~coimt. Result 1.1, t h e n m h r I>€ such S 's through q. is 0 8 and t h e number of those through R is 17p -0 . Then the ( 2 6 -8 )-secant h y p r p l m e s thrciugh R but. not. through are 0 -e . The in t e r sec t ion of t.he (28 -0 ) - secant

hyperplanes thnxigti R hut. not. thrcxiqh 'L is a 1-d imnsional s~~&piiace S . Since, i n view o f Result . I. 1 , t.he 1 inm of K wkiLch int-ersert. 7 i'c)vt'l- K ,

each ( 2 8 -@ )-secant hyperplane has t h e ot.her 20 - @ " - ~ q @ ~ - , -0 p3int.s on t.he 1 ines of K throuqh R d i f f e r e n t from q,, so on t-he 8 1 ines r3f K

t-hrrslgh R d i f f e r e n t from q.. Thus S must. w n t a i n the p lines O € i(

throiiqh R d i f fe ren t . from 2 . Then it r e s u l t s i2h. It. remains to prove that.

of C ~ l L I S S [ O , l , L , . ' el I , 2 h t 1

h 0 2 h

h - 1

1 h - l

h - 1

h CI

3 h 1 h - 1

h O h i l

h h - 1 h CI

1

h 0 h

h - 1

i h - 1

Note on a characterization of Segre variety in PG(r,q) 41

i<h. Let us a s s m , on t h e c o n t r a r y , iihtl and ccmsidei- the star T o f L2h- I

h t h e h y p x p l a n e s of dxis S . c o n t a i n s a t kist eh-oh- =q

1 L - I

(-70 -H ) - secant h y p x p l a n e s . I f irh+L t h e n it results h '1

and so

a cont rad ic t . ion . Then throiiqh each p i n t . of 'L t h e r e passes rsacrt.ly one S

of K . None o f such S ' s c a n c o n t a i n t w c d i f f e r e n t . p i n t . s of 4. and so +hey h

are skew t w o k ) v two. Let S' , S' nnd S 3 denote t.hr-ee i > f t h e s e S Is. The

t c j t a l i t y of t h e l i n e s which i n t e r s e c t S', S2 arid S ' is a Srqre varLety

S SS which w e denote hy V. S i n c e K is d set o f of class [ 0 , 1 , 2 , @ 1 , 1 h 1 1

each l i n e which n lce t - s S ' , S' and S:' is cont.aLntu3 i n K . T h e r e f o r e VcK.

h

h

References

12 1 4 . R I C m - S . IW3mT1, On Seqre variety, m t t e i l u n g e n Mathem. Sen. Giessen , 201, 1991. 19-24.


Combinatorics '90 A. Barlotti et al. (Editors) 0 1992 Elscvier Scicnce Publishers B.V. All rights reserved. 49

A property of sharply 3-transitive finite permutation sets (*)

A. Bonisoli and G. Korchmiros

Dipartimento di Matematica, Universitb della Basilicata, via N. Sauro 85, 85100 Poten- za, Italy

Abstract Let G be a sharply %transitive permutation set on PG(l,p"), p odd, p" > 9.

Suppose G contains all the involutions in PGL(2,p") and is such that if a permutation g lies in G then so does every power of g. We have that G necessarily coincides with PGL(2, p"' ).

1. INTRODUCTION A N D PRELIMINARY RESULTS

The study of sharply 3-transitive finite permutation sets arises in connection with those geometric structures known as Minkowski planes: every finite Minkowski plane can be described by a sharply 3-transitive permutation set on a finite set fl and, conversely, every such set of permutations yields a finite Minkowski plane, cf. [2], [3, III54], [9].

Among sharply 3-transitive finite permutation sets a special role is played by the projective general linear groups PGL(2, p" ) (in their natural permutation representations on the projective lines PG(1,p")). These groups are characterized by the fact that "Miquel's configuration" holds universally in the corresponding Minkowski planes, which are in turn isomorphic to the geometries of plane sections of hyperbolic quadrics in 3-dimensional finite projective spaces, cf. [2], (3, III!j4], 191.

The following well-known result has been proved independently by a number of authors (cf. [l], [7], [8], 191, [ll], [13], [14]): if G is a sharply 3-transitive permutation set on the finite set fl of odd cardinality such that idn E G, then G is a group and it is thus possible to identify the elements of fl with the points of the projective line PG(1,2") in such a way that G = PGL(2,2") holds. We have thus a complete classification when the underlying set f l has odd size: the projective general linear groups are the unique examples in this case.

When the underlying set f? has even size we do have examples for sharply 3-transitive permutation sets which are "essentially different" from the projective general linear groups, i.e. the corresponding Minkowski planes are "non-miquelian" (cf. (131: in all such examples hl is the projective line PG(l ,pm) for some odd prime p). Therefore the problem of recognizing the projective general linear groups among sharply 3-transitive

(*) work done within the activity of GNSAGA of CNR and supported by the Italian Ministry for Research and Technology.

50 A. Bonisoli, G . Korchmdros

permutation sets arises in this case. The recent papers [6] and 1151 have started an approach which is typical of the theory of finite groups, namely the attempt to find characterizations in terms of properties of the involutions, and we shall continue along this line.

will denote the projective line PG(1,p") = GF(p") U {m} for an odd prime p and a positive integer rn. By Syrn(n) we denote the full symmetric group on n, i.e. the group of all permutations on 0. By definition a permutation on n is a special subset of the Cartesian product n x 0; on the other hand, the usual functional notation for mappings also has some advantages; therefore if g is a permutation on f-2 and x, y are elements of n the relations (x,y) E g resp. g(x) = y will have for us the same meaning and we shall use either one of them according to our convenience.

A non-empty subset of Syrn(fl) is called a permutation set on a, Throughout the paper G will denote a sharply 3-transitiue permutation set on n, that means, if (xi , 21 , z3), (y, , , y3) are triples of distinct elements of n, there exists a unique permutation g in G mapping xi to yi for i = 1,2,3.

We denote by M(G) the Minkowski plane associated to G, i.e. the incidence structure constructed by the following well known procedure. The points of M(G) are the elements of the Cartesian product n x n. The blocks (or circles) of M(G) are the elements of G. We distinguish further subsets of n x n, namely, if a is any element of n we define (a)+ := {(a,y)I y E n}, (a)- := { (%,a) / x E n}; we set t+ := { ( . ) + I a E n}, 1- := {(a)- la E n}, L := L+ u L- ; the elements of t+ resp. Lf- resp. t will be called positive generators resp. negative generators resp. generators. Point-block incidence and point-generator incidence is simply given by E in the natural way.

If P = (x,y) is a point of M(G) we define the derived afine plane M(G)p as follows. The points of M(G)p are the points of M(G) with the exclusion of the points lying on the two generators through P . The lines of M(G)p are the generators of M(G) not through P together with the circles of M(G) through P (i.e. the permutations g in G such that g(x) = y). Point-line incidence is that induced by incidence in M(G).

By a square in a finite field GF(p") we shall simply mean an element u E GF(p") such that there exists an element v E GF(p") with va = u, If p is an odd prime then the non-zero squares form a subgroup of index 2 in the multiplicative group GF(p" )* and its unique coset is the set of non-squares.

PROPOSITION 1 . For an odd prime p and a positive integer rn let e(z) E GF(p")[z] be a quadratic monic polynomial which is not the square of a linear polynomial. The polynomial mapping GF(p") -t GF(pm); x I+ e(s) takes up at least one non-square value.

PROOF: Set e(z) = z2 + bz + c with b,c E GF(p"). We write e(z) = [z + (b/2)la + [(4c-ba)/4]. For any element t E GF(p") the mapping GF(p") -t GF(p"); x H x + t is bijective. It is thus sufficient to prove the assertion for quadratic polynomials of the form za + c with c # 0.

If c is a non-square then e(0) = c and the assertion holds. If c is a square, say c = ua, then write xa + c = xp + ua = U ~ [ ( X / U ) ~ + 11. The mapping GF(pm) -+ GF(p"); x x/u is bijective and since the product of ua by a non-square is again a non-square, it suffices to prove the assertion for the quadratic polynomial za + 1.

Throughout the paper

A property of sharply 3-transitive finite permutation sets 5 1

Suppose the assertion were not true for e(z) = za + 1. Then to each z E GF(p") there corresponds an element y E GF(p") with ya = xa + 1, i.e. (y - z)(y + z) = 1. In other words, to each element x E GF(p") there corresponds an element y E GF(p") with y - x # 0, y + 5 = (y - x ) - ' . Setting u := y - x we have y + z = u-l, whence z + u = y = -z+u-l andconsequentlyalso2x=u-'-u,~= (1-ua)/(2u) . Wecanthus rephrase the property as follows, with no reference to y: to each x E GF(p" ) there exists an element u(z ) E GF(p")' with 5 = [I - u(z) ' ] / [2u(z ) ] . The mapping G F ( p m ) + GF(p")*;z H u(z) is injective: u ( z l ) = u(za ) implies z1 = [l - u(z1) ' ] / [2u(z1) ] = [I - ~ ( z , ) ~ ] / [ 2 u ( z , ) ] = 5 2 ; an injective mapping from a set with p" elements to a set with pm - 1 elements is clearly a contradiction. 0

PROPOSITION 2 . Let p be an odd prime. For a non-zero element u of GF(p") define p, := I{(z,y) E Q x N; z + y = .}I. We have p, = (p" - 1 ) / 4 or p, = (P" - 3 ) / 4 according as p" = 1 or - 1 mod 4 respectively. 0

PROPOSITION 3. Let p be an odd prime and m an integer with p" 2 5 . If w is a non-square in G F ( p ) , then there exists an element t E GF(p")* such that ta + w is a non-square in GF(p") \ { w } .

PROOF: Write w as the sum of a non-zero square up and a non-square u (see the previous Proposition): w = u2 + u. There exists s E GF(p")* with u = saw. Then we have w/sa = (u/s)' + w. We cannot have s1 = 1 otherwise u should be zero; hence w / s a is a non-square and is distinct from w. Setting t := u/s we have the assertion.

0

PROPOSITION 4. Let p be an odd prime and m an integer with pm 2 5 . Let a,P, 7 be elements of GF(p"). Define

D ( z ) := z4 + 2 ( p - 2 7 ) ~ ' + (-4a + p2 + 2 7 ) ~ ~ + 2(2a - P ~ ) z + y2 E GF(p")[z]

and assume that for each element t E GF(p") the element D(t) is a square. Let C be the quartic curve over GF(p") defined by the equation

ya = D(z ) . (1)

Precisely one of the following two cases occurs: I) D ( z ) is the square of a quadratic monk polynomial E ( z ) E GF(p")[z] and C splits

11) D ( z ) hm only simple roots and C is an absolutely irreducible curve possessing at

PROOF: Let GF(p"") be the splitting field of D ( z ) over GF(pm).

with D ( z ) = E(z)' . Comparing coefficients we obtain the equations

into the two parabolas y = E ( z ) , y = -E(z ) ;

most p" + 2@ - 1 points in AG(2,p").

Suppose there is a quadratic monic polynomial E ( z ) = z p + B z + C E GF(p"")[z]

B = p - 2 7 , B a + 2 C = - 4 a + p a + 2 7 , B C = 2 a - p 7 , C2 = y p . ( 2 )

The relations B = /? - 7 and C = f 7 show that the coefficients of E ( z ) necessarily lie in GF(p") and this is precisely the situation of case I).

52 A. Bonisoii, G . Korchmdros

Suppose ( E GF(p"") \ GF(p") is a multiple root of D(z ) . There exists a non- identical automorphism E Gal(GF(p"") : GF(p")) with q := (" # ( (otherwise ( should lie in GF(p") , the fixed-field of Gal(GF(pmn) : G F ( p m ) ) ) . As q is again a root of D ( z ) of the same mutliplicity as (,we have D ( z ) = ( z - ( ) a ( z - q ) z = [ ( z - ( ) ( z -q ) la and so D ( z ) is the square of the polynomial z' - (( + q)z + ( q E GF(p")[z] , leading us back to case I).

Suppose D ( z ) has a multiple root ( E GF(p"). Then there exists a quadratic monic polynomial E ( z ) E GF(p")[z] with D ( z ) = ( z - ( ) 'E ( z ) .

In case E ( z ) has two distinct roots in GF(pmn) then by Proposition 1 there exists an element p E GF(p") such that E(p) is a non-square in GF(p"): we want to show that we can always choose p # 6.

If this were not the case, then ( should be the unique element of GF(p") such that E ( ( ) is a non-square, say E ( ( ) = w . Then ( should be the unique root in GF(p") of the quadratic monic polynomial F ( z ) := E ( z ) - w and in GF(p")[z] we would obtain F ( z ) = ( z - i.e. E ( z ) := ( z - ()a + w . Since pm 2 5, the existence of an element p E GF(p") \ {(} such that E(p) is a non-square would then follow from Proposition 3, a contradiction.

The polynomial E ( z ) has thus a double root q ; from E ( z ) = ( z - Q ) ~ and E ( z ) E GF(p")[z] we obtain q E GF(p"), D ( z ) = [ ( z - ( ) ( z -q)I2 and we are back to case I).

We conclude that if D(z ) is not the square of a quadratic monic polynomial in GF(p")[z] then D(z ) possesses only simple roots. The quartic curve C is then absolutely irreducible (cf. 112, 6.541). Let N be the number of points of C in AG(2,p") i.e. the number of all pairs ( z , ~ ) E GF(p") x GF(p") satisfying equation (1); then [16, $81 yields IN + 1 - p" I 5 2 F whence in particular N 5 p" + 2@ - 1.

0

2. A GEOMETRIC INTERPRETATION OF A PROPERTY OF CERTAIN PERMUTATIONS

Consider the group PGL(2,p") in its natural permutation representation on the elements of II and denote by J the subset of PGL(2,p") consisting of all involutions.

Let f E Syrn(n) be a permutation with f(00) = 1, f(0) = 00. Let a, by c be elements of GF(p") with -(a' + bc) # 0. The transformation j : (uz + b ) / ( c z - a) is an involution in PGL(2,p"). We have

+ fl; z

f (w)=j (w) +=$ u = c , (3)

f(0) = j ( 0 ) w u = 0. (4)

If z E GF(p")' is a solution of the equation

f ( x ) = i(4 (5)

then the equality f ( z ) ( c x - u) = uz + b holds in GF(p").

A property of sharply 3-transitive finite permutation sets 53

Let (X, Y, 2) denote homogeneous coordinates of ponts in PG(2, p" ); define the following lines:

4 : em: x-z=o, 4 : z=o.

Xf(t)t - Y - Z(f(t) + t ) = 0, t E GF(P")'Y

For t E n the point (X,Y,Z) with -Z1 - XY # 0 lies on the line 4 if and only if the permutation f acts on t as the involution in J given by n + n; z I-+ (Zx+Y)/(Xz-2).

PROPOSITION 5 , If the permutation f E Sym(n) is such that for each involution j E J the inequality I{. E 0; f (z) = j(z)}I 5 2 holds then theset { l , ,&}U{4; t E GF(p")*} consists of precisely p" + 1 lines no three of which are concurrent.

PROOF: The lines t!, , 4, are certainly distinct from each other and from each line 4 , t E GF(p")*. Suppose s, t are distinct elements of GF(pm)* such that l , = 4 . Then we have f(s)s = f ( t ) t , f(s) + s = f ( t ) + t. These equalities in turn imply s = f ( t ) , t = f(s). If h denotes the transformation in PGL(2,p") defined by h(s) = t , h(t) = s, h(0) = 00, then h is an involution and the permutations f and h coincide on s, t , 0, a contradiction.

Suppose now s, t, u are distinct elements of GF(p")' such that the lines 4, , 6 , l , meet at the point (X, Y, 2). We have Xf (s)s - Y - Z( f (s) +s) = 0, Xf(t)t - Y - Z ( f ( t ) +t) =

If 2 = 0 then X # 0 and we obtain f(s)s = f ( t ) t = f(u)u = w for some w E GF(p") which is necessarily non-zero. The involution h : + 0; z I+ w/x lies in PGL(2,p") and coincides with f at s, t , u, a contradiction.

If X = 0 then 2 # 0 and we obtain f(s) + s = f ( t ) + t = f(u) + u = w for some w E GF(p"). The involution h : n --t n; x H -x + w lies in PGL(2, p") and coincides with f at s, t , (I, a contradiction.

We have the following

0, Xf(.). - Y - Z(f(u) + u) = 0.

Both X and Z are thus non-zero and we have

f(t)[(X/Z)t - 11 - t = f(4[(X/Z)s - 11 - 8, { f(U)[(X/Z)U - 11 - u = f(s)[(X/Z)s - 11 - 8 .

yielding f(s)[(X/Z)s - 11 - s = f(t)[(X/Z)t - 11 - t = f(u)[(X/Z)u - 11 - u = w for some element w E GF(p"). We cannot have w = -Z/X, otherwise take among s, t, u two values which are distinct from Z/X, say for instance 5 , t , and obtain

z x Z z x x z X x z f(s) = (s - -)/(-s - 1) = - = (t - -)/(-t - 1) = f ( t ) ,

a contradiction since f is injective. The mapping h : n --t n; y I-+ (y + w)/((X/Z)y - 1) is thus an involution in PGL(2,p") coinciding with f at s, t , u, a contradiction. This shows that the lines t!, , 4 , l , are not concurrent.

Suppose s, t are distinct elements of GF(p")* such that the lines l , , 4 , l , meet at the point (X,Y,Z). We have then X = 2, X # 0, whence f(s)s - (f(s) + s) = f ( t ) t - ( f ( t )+t) , i .e . f(s)(s-l)-s = f ( t ) ( t - l ) - t =wforsomeelement w E GF(p").

54 A. Bonisoli, G. Korchmdros

We cannot have w = -1 otherwise, if we choose one element different from 1 in {sit}, say for instance 5 , we would obtain f(s) = ( 8 - l)/(s - 1) = 1 = f(00), a contradiction. The mapping n 4 n; z I+ (z+w)/(z-l) is thus an involution in PGL(2,p") coinciding with f at s, t , 00, a contradiction.

Suppose 8 , t are distinct elements of GF(pm)* such that the lines L,, 4, lo meet at the point ( X , Y , Z ) . We have then 2 = 0, X # 0 whence f(8)s = f ( t ) t = w for some element w E GF(pm); we cannot have w = 0 otherwise both f(s) and f ( t ) must be zero, a contradiction. The mapping n -t n; z I+ w / z is thus an involution in PGL(2,p") coinciding with f at s, t, 0, a contradiction.

Finally, it is clear that if s is an element of GF(pm)' then the point of intersection cl

Under the assumption of Proposition 5 the pm + 1 lines l , , l o , 4, t E GF(pm)' form thus an oval 9 in the dual desarguesian plane: by Segre's Theorem [17] 9 is an irreducible conic, the equation of which we shall now determine.

We denote homogeneous coordinates in PG(2,p") by [z,y,z]; we have & = [O,O, 11, 1, = [ l , O , -11 and 4 = [ f ( t ) t , -1, - ( f ( t ) + t)] for t E GF(pm)'. The equation of 9 has the form

of the lines l , and &, namely ( O , l , O ) , does not lie on 1,.

allza + ally' + aSSZa + a,'z$/+ a1szz + h3yz = 0. (7)

The conditions & E 9, l , E 9 imply aaa = 0, a,, + ass - aIs = 0 respectively, whence a,, = 4 1 s . We cannot have a,, = 4 1 3 = 0 otherwise the conic should be reducible. We may therefore set a,, = a13 = 1 and defining a := % a , P := ala, 7 := aas equation (7) becomes

zZ+ffy' + p z y + z z + 7 y z = o (8)

for elements a, P , 7 E GF(p") not all zero. The conditions I$ E 9 for t E GF(p")' yield

f(t)'(t' - t ) - f(t)(t ' + Pt - 7) + (a + r t ) = 0. (9)

The element f ( t ) is thus a solution in GF(p") of the quadratic equation (in the unknown W )

Wa(P - t ) - W(t' + P t - 7 ) + (ff +7t ) = 0, (10)

the discriminant of which is given by

D ( t ) := t' + 2(P - 27)t' + (-4a + @' + 27)ta + 2(2a - P7)t + 7'. (11)

For each element t E GF(pm)* the element D(t) ie thus a square in GF(p"). This property also holds for t = 0, since D(0) = 7l.

PROPOSITION 6. Suppose the permutation f E Syrn(n) is such that for each involution j E J the inequality I{. E n; f(z) = j(z)}I 5 2 holds. Assume pm > 9. There

A property of sharply 3-transitive finite permutation sets 5 5

exists then a transformation h E PGL(2,p") such that for each h-orbit C on s1 we either have f(z) = h ( z ) for all x E C or f(x) = h-'(x) for all x E C. Remark. The orbits of f on s1 are thus the same as those of h: in the previous statement we could therefore replace (' . . . for each h-orbit . . . " by ' . . . for each f-orbit . . . ". PROOF: The permutation f cannot be an involution, otherwise if (a, b) is one of the 2-cycles of f and c E \ {a, b} then, since J[,,b) is sharply 1-transitive on s1 \ {a, b}, there should be an involution j E J(,,b) with j(c) = f(c) and so f and j should coincide at a, b, c, a contradiction.

As a consequence f must contain a cycle of length no less than 3, say (a, b, c,. . .). With no loss of generality we may assume a = 0, b = 00, c = 1, otherwise, since PGL(2,p") is 3-transitive on n, we replace f by gfg-' where g E PGL(2,p") is such that g(a) = 0, g ( b ) = 00, g(c) = 1.

Let C be the quartic curve over GF(p") defined in Proposition 4 and assume Case 11) occurs. Let M denote the number of roots of D ( z ) in GF(p"). If ( E GF(p") is a root of D(z ) then ((,O) is a point of C. If ( E GF(pm) is not a root of D ( z ) then there exists an element q E GF(p")' such that D(() = qa holds and ( ( , v ) , ((, -q) are thus two distinct points of the curve C. We have thus N 2 2 ( p m - M ) + A4 2 2p" - 4 , which together with N 5 p" + 2 p m I a - 1 yields p" - 2 p m f a - 3 5 0. This last relation is false for p" > 9.

We may therefore aasume that Case I) of Proposition 4 occurs. Considering the equations (2) we distinguish the following two cases. Case C = Y.

We obtain Q = r(P - 7); solving equation (9) for f(t) with t E GF(p")' we have one of the following two relations:

r(t) = ( t - P - 7)/( t - 11, ( 1 2 )

f ( t ) = 7/t. ( 1 3 )

If (12) occurs with /3 - 7 = -1 then f( t ) = 1 which is impossible since t # 00. If 7 = 0 then (13) occurs for at most one value of t E GF(p")* . If P - 7 # -1 and 7 # 0 then the transformations

ji :n+n, Z H ( Z + ~ - ~ ) / ( Z - ~ ) ; j a : n + n , Z H ~ / Z ;

are involutions in PGL(2,p") and we have jl(00) = 1, jl(0) = 7 - P , ja(O0) = 0, &(O) = 00. In any case we see that since pm > 3 then there exists an involution j in PGL(2,p") with I{. E s1; j (x) = f(x)}l 2 3, which is in conflict with our assumption.

We obtain Q = y2, $27 - P - 1) = 0, whence either 7 = 0, which is already included in the previous case C = 7, or P = 27 - 1. We assume therefore P = 27 - 1 with 7 # 0 and solve equation (9) for f ( t ) with t E GF(p")'. We have one of the following two relations:

ase =-Y .


f ( t ) = 7 / ( t - 1). (15)

Since 7 # 0, the mapping h : fl + n; x w ( x + 7 ) / s lies in PGL(2,p") and we have h-' : n + n; x H 7 / ( x - I). We also have f (w) = 1 = h(oo), f(0) = 00 = h(0).

We want to show that if t, f are distinct elements of fl with orbh (t) = orbh (f) then precisely one of the following two relations holds:

f ( t ) = h(t) and f ( f ) = h( f ) , (16)

f ( t ) = h- ' ( t ) and f ( f ) = h-'(f). (17)

With no loss of generality we may assume f = h(t) (otherwise we consider inductively the pairs ( t ,h( t ) ) , (h(t),h'( t)), ..., (k(t),f).

The fact that the relations (l6), (17) cannot hold together derives from the fact that h is not an involution.

If the property is not true then (exchanging eventually the roles of t, f) we may set f ( t ) = h(t) and f ( Z ) = h-'(f). We have t = h-'(h(t)) = h-'(f) = f ( f ) = f(h(t)) = f ( f ( t ) ) = fa(t) . Since f ( t ) = h(t) = f # t this shows that f has a cycle of length 2 , namely ( t , f ) . The set is sharply 1-transitive on n \ {t,F). Hence for each x E fl \ { t l f } there exists j E J(t,c)

0

PROPOSITION 7. Let G be a sharply 3-transitive permutation set on fl with J G. If g is a transformation in PGL(2,p") the order of which is either pm - 1 or pm + 1 then g must lie in G .

PROOF: Let g be a transformation in PGL(2, p"') of order p" -1 or p" +1 respectively. Let C be its unique orbit of length pm - 1 or p" + 1 respectively. There exist three distinct elements a, b, c E C such that g contains the cycle (a, 6, c , . . .). Let f E G be the unique permutation determined by f(a) = g ( a ) , f ( b ) = g(b ) , f ( c ) = g ( c ) . In particular a , b , c lie in the same f-orbit A. By Proposition 6 there exists h E PGL(2,p") such that for each f-orbit A on n either f ( x ) = h(z) holds for all x E A or f(z) = h-' ( x ) holds for all x E A. We may with no loss of generality assume f ( x ) = h ( x ) to hold for all x E A (otherwise we exchange the roles of h and h- ' ) . Since a , b , c are in A we have in particular f(a) = h(a), f ( b ) = h(b), f ( c ) = h ( c ) , whence also g(a) = h(a) , g(b ) = h ( b ) , g ( c ) = h ( c ) ; from g , h E PGL(2,p") and the fact that PGL(2,p") is sharply 3-transitive on n we obtain then g = h. We conclude g ( x ) = f(s) for all elements x E C; if the order of g is p" - 1 and u,v are the fixed points of g on fl we also have g(u) = f(u), g ( u ) = f (u) . Therefore we conclude g = f and g E G.

0

PROPOSITION 8. Let ?r be a finite affine plane of order q. The lines of one parallel class are uniquely determined by the remaining lines.

PROOF: The assertion can be obtained as a conaequnce of the fact that a set of q - 2 mutually orthogonal latin squares of order q in standard form determine a unique further

0

of all involutions in PGL(2,p") exchanging t and

such that j ( x ) = f ( x ) and f coincides with j at t , f, x , a contradiction.

square completing the set, see [5 , 9.31.


PROPOSITION 9. Let G be a sharply 3-transitive permutation set on f l which contains J and satisfies the condition

g E G =+ g' E G for each positive integer t.

Then G = PGL(2,p").

PROOF: We know from the previous results that G contains all transformations of order p" - 1 or p" + 1 in PGL(2,p"). A transformation in PGL(,p") either lies in a Sylow psubgroup or in a cyclic subgroup of order p" - 1 or in a cyclic subgroup of order p" +1, cf. [lo, I1 8.51. If the transformation f E PGL(2,p") lies in a subgroup of order p" - 1 or p" +1 respectively then we have f = g' for a transformation g E PGL(2, p") of order pm - 1 or pm + 1 respectively and a suitable positive integer r: we have g E G by Propo- sition 7 and therefore f E G by condition (18). All transformations in PGL(2,p") are thus contained in G except possibly for the p-elements. The p-elements of PGL(2, p" ) are the transformations with exactly one fixed point on 0. For each element a E n there exist precisely p" - 1 transformations with a as a unique fixed point; in the derived affine plane M(PGL(2, p"))(,,,) these transformations together with the identity form a class of parallel lines. The lines of the derived affine plane M(G)(,,,) are thus the same as those of the plane M(PGL(2,pm))(,,,) except possibly for those belonging to the parallel class of i d n : Proposition 8 forces M(PGL(2,pm))(,,,) = M(G)(,,,), i.e. PGL(2,p"), = G,, showing that every p-element of PGL(2,p") fixing a actually lies in G; since this is true for every a E 0 we have PGL(2,p")

0

PROPOSITION 10. Assume p" > 9. Let G be a sharply 3-transitive permutation set on fl with J C G. A permutation f E Sym(fl) belongs to G if and only if so does its inverse f - ' . PROOF: We may assume that the order of f is greater than 2. Let A be an f-orbit with lAl 2 3. Let h E PGL(2,p") be the permutation determined by f according to Proposition 7. We may assume with no loss of generality f ( z ) = h(z) for all 5 E A. Let a, 6, c be three distinct elements in A with f ( a ) = 6, f(6) = c and let f be the unique element of G determined by f(a) = f-'(u), f(6) = [-'(6), f(c) =- f-'(c). From f(c) = 6, f(6) = a we see that a,6,c belong to the same f-orbit A. Let h E PGL(2,p") be the permutation determined by f according to Proposition 7. We may assume with no loss of generality f(z) = h(z) for all z E A. We have in particular h-' (a) = f- ' (a) = !(a) = h(a), h-'(6) = f- ' (6) = f(6) = h(6), h-'(c) = f-'(c) = f ( c ) = h(c) . From h- ' , h E PGL(2,p") we conclude h = h-' . The permutations f , f have the property that their action on each h-orbit n coincides with the action of either h or h-' .

We have f # f since f ( a ) = 6 # c = f ( a ) holda. As a consequence we see that if E is an h-orbit with IEI > 1 (whence also 1x1 > 2) then f and f cannot act either as h or as h- on E, otherwise f and f should coincide on at least three elements of fl, contradicting the sharp 3-transitivity of G. We have thus f = f - ' and the assertion follows. 0

PROPOSITION 11. Assume pm = 1 mod 4 and let f E Syrn(0) be a fixed-point-free

(18)

G, whence equality.

58 A. Bonisoli, G. Korchmdros

involution. If the inequality I{x E fl; f ( x ) = j(x)}I 5 2 holds for each involution j E PSL(2,pm), then f lies in PGL(2,p").

PROOF: Represent by the points of an irreducible conic in PG(2,p"). The hyperbolic involutions in PGL(2,p") (i.e. the involutions in PSL(2,p") since p" = 1 mod 4) are represented by the interior points of 0. The elliptic involutions in PGL(2,p") (i.e. the involutions in PGL(2,p") \PSL(2,pm)) are represented by the (p" +1)/2 exterior points on an external line to 0.

For each 2-cycle (a,a') of f let A be the polar point of the line aa' (i.e. the point of intersection of the tangents to n through a and a' respectively). We obtain thus a set f of (p" + 1)/2 exterior points.

Let A and B be two distinct points in f and let t be the line joining them. We claim that P. is external to n.

Suppose .! is tangent to f l at x. The polar lines of A and B are distinct secants of n through x, whose further points of intersection with n we denote by y and z respectively. By definition we should have y = f(z) = z , a contradiction.

and let L be its pole. Then L is an exterior point determining an involution j E PSL(2,p"). The polar lines of A and B pass through L and if {a, a'}, { b, b'} are the corresponding pairs of points where these polar lines meet the conic we have {a, a'} n (6, V} = 0. By definition f coincides with j on a, a', b, 6' contrary to our assumption.

Since any two points of f lie on an external line the result of [4] applies, yielding that & consists of the (p" +1)/2 exterior points of an external line and f is the corresponding

0

PROPOSITION 1 2 . Let G be a sharply 3-transitive permutation set on with J G. Assume p" = 2s + 1 > 9 for an odd prime s. Then G = PGL(2,p").

PROOF: It suffices to show G, = PGL(2,p"),. This in fact implies M(G)(,,,) is desarguesian and the assertion would then follow from 118, 20.6.111.

Let U be the set of all elements of order p" - 1 in GF(pm)*. With the Euler 'p-

function we have IUI = p(p" - 1) = 'p(2s) = s - 1 = (p" - 1)/2 - 1 = (p" - 3)/2. If 6 E GF(p") and a E U then the mapping n --t n; z H ax + 6 lies in G, ; furthermore G, contains all involutions n --t n, x H - x + 6 for b E GF(p").

We want to show first of all that every hyperbolic permutation in G, actually lies in PGL(2,pm), . Assume the contrary and let g E G, \ PGL(%,p"), be a hyperbolic permutation. By Proposition 6 there exists h E PGL(2,pm), such that for each h- orbit A on n either g(z) = h(z) holds for all x E A or g(s) = h-'(s) holds for all x E A.

Write h in the form h : n -, n; x H cz+d where c E GF(pm)\{O, 1) has multiplicative order strictly less than p" - 1; we have h- : n --t n; x H c- x - c - l d. Definte the sets c, := {x E n; 2 # oo,d/(l - c),g(z) = h(z)} , cp := {z E II; x # oo,d/(l - c),g(z) = h-l(x)}, r1 := {(z,g(s)); x E El}, rl := {(x,g(z)); z E &}.

The set r := rl U ra U {(d/(l - c),d/(l - c))} is a line of the derived affine plane M(G), , ,, ). One of the sets rl , ra has cardinality no less than (p" - 1)/2, say rl . Fix a point P = (Z,g(Z)) E ra. Let P(P) be the set of all lines through P in the desarguesian

Suppose .! is a secant of

permutation in PGL(2, pm ).


plane AG(2,p") = M(PGL(2,pm))(,,,) other than the vertical line z = and the horizontal line y = g(3). We have ID(P)I = pm - 1. For u E U U (-1) the set l , := {(z,uz - uf f g(f)); z E GF(p")) is a line in P(P) which is also a line of the derived affine plane M(G)(m,m).

For P E ra consider the desarguesian line PIP E D(P). Since PIP meets r in two distinct points, this line cannot be a line of the derived affine plane M(G)(, , , I : we shall say that this is a forbidden desarguesian line. Another forbidden desarguesian line is the line joining P to the point (d/( l -c) ,d/( l -c)) , i.e. the line ((z ,cz+d); x E GF(p")). Hence there are at least (p" - 1)/2 + 1 = (p" + 1)/2 forbidden desarguesian lines in D(P). The number of desarguesian lines in D(P) which are also lines of the derived affine plane .M(G)(m,m) is thus at most (p" - 1) - (p" + 1)/2 = (p" - 3)/2. On the other hand l{lu; u E U U {-1))I = (p" - 3)/2 + 1, a contradiction.

Every permutation in G, lies thus in PGL(P,p"), except possibly for the permutations with no fixed points on n\ (00). These permutations interpreted in the derived affine plane M(G)(, ,, ) yield the lines which are parallel to idn. The affine planes M(G)(=,,) and M(PGL(S,p"))(,,,) constructed on the point-set GF(p") x GF(p") have thus the same set of lines except possibly for those of a single parallel class. Propo-

0 sition 8 shows that these planes coincide, whence G, = PGL(2,p")- .

3. REFERENCES

1

2

3 4

5

6

7 8

9

10 11 12

13

R. Artzy, A Pascal theorem applied to Minkowski geometry, J. of Geometry, 3 (1973)

W. Benz, Permutations and plane sections of a ruled quadric, Symposia Mathe- matica, Istituto Nazionale di Alta Matematica, V (1970) 325-339. W. Benz, 'Vorlesungen uber Geometrie der Algebren', Springer, Berlin et al. 1973. A. Blokhuis, A. Seress, H.A. Wilbrink, Characterization of complete exterior sets of conics, Combinatorica, to appear. J. Ddnes, A.D. Keedwell, 'Latin squares and their applications', Akadhmiai Kiad6, Budapest, 1974. G. Faina and G, Korchmkos, On finite sharply 3-transitive sets, Atti Sem. Mat. Fis. Univ. Modena, 37 (1989) 95-103. W. Heise, Minkowski-Ebenen gerader Ordnung, J. of Geometry, 5 (1974) 83. W. Heise, A combinatorial characterization of PGL(2, 2q), Abh. Math. Sem. Ham- burg, 43 (1975) 142-143. W. Heise, H. Karzel, Symmetrische Minkowski-Ebenen, J. of Geometry, 3 (1973)

B. Huppert, 'Endliche Gruppen I,, Springer, Berlin et al., 1967. H. Karzel, Symrnetrische Permutationsmengen, Aeq. Math. 17 (1978) 83-90. R. Lid1 and H. Niederreiter, 'Finite Fields', Cambridge Univ. Press, Cambridge et al., 1984. N. Percay, A characterization of classical Minkowski planes over a perfect field of characteristic two, J. of Geometry, 5 (1974) 191-204.

93-102, 103-105.

5-20.


14

15 16

17 18

P. Quattrocchi, Sugli insiemi di sostituzioni strettamente 3-transitiui finiti, Atti Sem. Mat. Fis. Univ. Modena, 24 (1975) 279-288. G . Rinaldi, A characterization ofPGL(2,q), q odd, Geom. Ded. 33 (1990) 331-335. W.M. Schmidt, ‘Equations over Finite Fields: An Elementary Approach’, Springer, Berlin et al., 1976. B. Segre, Ovals in a finite projective plane, Can. J. Math. 7 (1955) 414-416. B. Segre, ‘Istituzioni di Geometria Superiore, vol. III’, Lezioni raccolte da P.V. Ceccherini, Istituto Matematico “G. Castelnuovo” , Roma, 1965.


Faithful orbits in symmetric designs

Andrew Bowler

Department of Mathematics, RHBNC (University of London), Egham, Sur- rey TW20 OEX, U. K.

Abstract In this paper we consider conditions for an automorphisrn group of a

symmetric 2-design to act faithfully on at least one (point or block) orbit. We obtain restrictions on the order of the stabilizer of a maximal orbit of such an an automorphism group, and apply these to show that a cyclic automorphism group of a non-trivial triplane always acts faithfully on at least one orbit.

1. INTRODUCTION

A permutation group II acts faithfully on an orbit if the only element of lI which fixes every point of that orbit is the identity. Dembowski ([I] p181) has shown that an automorphism group of a finite projective plane always acts faithfully on at least one (point or line) orbit. Kimberley, Piper and Wild [2] have shown that, except for the cases k = 3,6 and possibly 8102, an automorphism group of a biplane acts faithfully on at least one (point or block) orbit. In this paper we consider generalisations of the problem to symmetric designs with X 1 3.

Section 2 consists of definitions, terminology and preliminary results from design theory and group theory.

In Section 3 we consider the stabilizer of a maximal orbit of an automorphism group of a symmetric design which does not act faithfully on any orbit. We also construct a family of symmetric 2-designs with 2 2 x 2 2 acting faithfully on no (point or block) orbit.

In Section 4 we apply the results to show that a cyclic automorphism group of a triplane acts faithfully on at least one orbit.

62 A. Bowler

2. DEFINITIONS AND PRELIMINARY RESULTS

A symmetric ,$?-design is an incidence structure with v points and v blocks such that (1) every block is incident with exactly k points (2) every pair of points is incident with exactly X common blocks. The following dual conditions hold: (1)' Every point is on exactly k blocks. (2)' Every pair of blocks is incident with exactly X common points. A simple counting argument gives the standard condition X(v - 1) = k( k - 1). The dual of a symmetric 2-design is obtained by interchanging points and blocks and is clearly also a symmetric 2-design. A symmetric 2-design is non-trivial if k > X + 1.

An automorphism of a symmetric 2-design is a incidence-preserving mapping of the points onto the points and the blocks onto the blocks. A group of automorphisms induces a permutation group on the points and blocks of the symmetric 2-design.

Let D be a symmetric 2-design and let II be an automorphism group of D. Let B be a point of D . The orbit B" of B under 11 is the set Bn = {B" I a E II). The point stabilizer IIB of B is the set of elements of II which fix B, that is IIB = {a E 11 I B" = B}. The orbit stabilizer IIEn of B" is the set of elements of II which fix B" pointwise, that is I I p = {a E 11 I Pa = P VP E B"}. IIB is a subgroup of II and I I g n is a normal subgroup of II. We define zn, II, and II,n for a block 2: analogously. A macimalorbit is one which has at least as many elements as any other point or block orbit.

We write (B",z") to denote the number of points of B" on each block of z", and dually (8, B") is the number of blocks of 2" on each point of B".

Lemma 2.1 27 and let B" be a point orbit and zn a block orbit. Then

The following results will be required.

Let 11 be an automorphism group of a symmetric 2-design

(I?",.") lz"l= ( 8 , B ' T ) lBnl

Proof

A centre of an automorphism a of a symmetric 2-design is a point B of P such that all the blocks on B are fixed by a.

Count incident pairs (P,y) with P E B" and y E z".

Faithful orbits in symmetric designs 63

Lemma 2.2 with k > X + 1 then a has at most one centre.

If a # 1 is an automorphism of a symmetric 2-design I)

Proof See [l] p82.

The following are results from group theory which we quote without proof.

(2.3) Let 11 be an abelian group acting on a set X. For B E X, IIp 2 a B n for aU P E B".

(2.4) ]Ant = lB"l then IIAn = I I B ~ .

(2.5) on a set X, and let A E X. Then lIIl = IA"I [HA[ .

(2.6) dividing IIII.

(2.7) dividing IIIl.

Note 27. If A" is a faithful orbit, then, by (2.3) and (2.5), lA"l = IIIl.

Let II be a cyclic group acting on a set X. For A , B E X, if

(The Orbit - Stabilizer Theorem) Let II be a group acting

For any group II there is an element of order p for all primes p

For any abelian group II there is a subgroup of order n for all n

Let II be an abelian automorphism group of a symmetric 2-design

3. RESULTS FOR GENERAL SYMMETRIC DESIGNS

Througout the rest of this paper ZJ will denote a non-trivial symmetric 2-design, II an automorphism group of D , and A" a maximal orbit. We assume that A" is a point orbit since the dual arguments imply the results for a maximal block orbit.

Lemma 3.1 Let p be a prime factor of ~ I I A I I I. Then p 5 A.

Proof By (2.6), there is an automorphism a E IIAn of order p. By Lemma 2.2, there is a block 2: which is moved by a and which is incident with

aP- 1 some point of A". Since a fixes A" pointwise, the blocks z,z", . . . , z are all on the same set of points of A". Suppose p > A. Since no two points are on more than A blocks, (An,zn) = 1. Every point of A" is on at least p blocks of z", since z,z ,..., z are all on one point of A". Hence ("*,A") 2 p. Thus by Lemma 2.1, lznl 2 plA"1, contradicting the maximality of A".

aP- 1

64 A. Bowler

We say that II is faithful if II acts faithfully on at least one orbit, and that II is unfaithful otherwise. We say that II is wayward if 11 is unfaithful, but every proper subgroup of II is faithful. Since every unfaithful automorphism group of D contains a subgroup which ie wayward it is sufficient to consider the case when II is wayward.

Lemma 3.2 Let Il be a wayward abelian automorphism group of D . Then IIIAnI = p where p is prime with p 5 A. Furthermore if II is cyclic then p < A.

Proof Suppose IIIAnI = p1p2 with p l , p 2 > 1. By (2.3) and (2.5), [IIl = plp2 [Ant. By (2.7) there is a subgroup r of order pz IAl. Since 11 is wayward, I' has a faithful orbit of size pzlAl , by the Note in Section 2. Thus II has an orbit of size at least pzIAl, contradicting the maximality of An. Hence IIIAnI = p where p is prime, and by Lemma 3.1, p 5 A.

Now suppose 11 is cyclic, generated by a, and that IIIAIII = A. By Lemma 2.2 there is a block 2 which is moved by a and which is on at least one point of An. Since 2 and za are on the same points of A" and two blocks are on at most X points, (A",,") 5 A, and since z,zp,. . . ,2 are all on the same points of A", (,",An) 2 A. Using Lemma 2.1, lz"1 2 [A"/, so that by the maximality of A", = IAnl. Thus by (2.4), z is fixed by I IAn, which is a contradiction. Hence if II is cyclic, IIIAn I < A.

*A - 1

As stated in Section 1, an automorphism group of a symmetric 2-design 2) with A = 1 or 2 must be faithful, with the exception of a 'few' values of k. It is natural to consider whether this is true for A > 2. It is true that for all A 2 2 a trivial symmetric 2-design always has an unfaithful automorphism group; for example it has one isomorphic to 2, x Z,, where p and q are primes with p + q 5 v. We now give a construction to show that there are infinitely many values of A for which there exists a non-trivial symmetric 2-design with an unfaithful automorphism group, namely one isomorphic to 2 2 x 2 2 . Note that the trivial symmetric 2-design and this construction give rise respectively to the exceptions k = 3,6 to the results for the biplane case ( A = 2) stated in Section 1. (The exception k = 8102 would correspond to a biplane which is not known to exist.)


Let A0 be the 4 x 4 identity matrix

1 0 0 0 (! H ;) For i a non-negative integer we define Ai+l recursively to be the matrix formed by replacing each 0 in A0 by Ai and each 1 in A0 by At , where At is the matrix formed by interchanging the 1’s and 0’s of Ai. Then, for each positive integer i, A; is the incidence matrix of a symmetric 2-design Di with v = 4t2, k = 2t2 - t and X = t2 - t, where t = 2’.

Let X be the set of integers (1’2,. . . ,4t2}. For i a non-negative integer and t = 2’ let ai and pi be the following permutations.

t ’ - 1

j=O

t ’ - 1

pi = -n- ( 2 + 4 j 3 + 4 j ) j=O

ai and p; act on the rows of A;, inducing the same permutation on the columns of A;. It is easy to check that a; and pi are automorphisms of the corresponding design Di. Observe that any point or block fixed by ai is moved by pi and conversely any point or block fixed by pi is moved by ai . Consequently ( a i , P i ) 2 2 2 x 2 2 is an unfaithful automorphism group of 27;.

4. TFUPLANES

A triplane is a symmetric 2-design with X = 3. In this section, D will denote a triplane with block size k. Since the trivial triplane has an automorphism group with no faithful orbit, we will suppose that k # 4. The unique triplane with k = 6 is the complement of a biplane with k = 5. By [2], a biplane with k = 5 always has a faithful orbit, so we will assume k # 6. We will show that a cyclic automorphism group always acts faithfully on a triplane with k > 6. We require the following result about the number of fixed points on a fixed block of an automorphism of order 3 of a triplane.

Lemma 4.1 a fixes 8 > 1 points on a fixed block then 9 5 f(k - 1).

Let Q # 1 be an automorphismof a triplane with a3 = 1. If

66 A. Bowler

Proof The number of fixed points of an automorphism of a symmetric 2-design is equal to the number of fixed blocks. (See [l] p81). Let f be the number of fixed points of a and suppose z is a fixed block with 8 > 1 fixed points. Suppose y is a moved block on two fixed points P,Q of 2. Then t,y,ya and yQ2 are all on both P and Q, contradicting X = 3. Hence a moved block contains at most one fixed point of z and consequently any pair of fixed points of z must be on three fixed blocks, one of which is 2.

Count incident triples (P, Q, y), P, Q E z, both P and Q fixed, y a fixed block different from 2. There are at most f - 1 choices for y, 3 choices for P and 2 choices for Q. Hence the number of triples 5 6(f - 1). There are 8 choices for P, 8 - 1 choices for Q and 2 choices for y. Hence s(s - 1) 5 3(f - 1). Rearranging gives

1 3

f 2 - 8 ( 8 - 1) + 1

Now count triples (P,Q,y), P,Q E z, P fixed, Q moved, y a moved block. There are at most v - f choices for y, one choice for P and 2 choices for Q. Hence the number of triples 5 2(v - f). There are s choices for P, k - s choices for Q and 2 choices for y. Hence 29(k - 8) 5 2(v - f). Rearranging and putting v = $k(k - 1) + 1 gives

1 3

f 5 -k(& - 1) - s ( k - 8 ) + 1

1 1

18(8 - 1) 5 i k ( k - 1) - s ( k - 8 ) 3 3

and rearranging gives

(k - a)(& - 28 - 1) 2 0

Now 8 < k, since every moved block is on at most one fixed point of z and every block of 'D is on exactly three points of z. Hence k - 29 - 1 5 0, and rearranging, 8 5 i(k - 1).

We now suppose that II = (a) is a wayward cyclic automorphism group of a triplane 2) with Ic > 6. Let [A"[ = t , so that n A n = (a*). By Lemma 2.2 there exists a block 2 of 'D moved by a*, with 1: on at least one point of An. By Lemma 3.2, we know that IIIAnI = 2 so that (a*)2 = 1. We now


proceed to show that such a II cannot exist, and consequently that all cyclic automorphism groups of 2) act faithfully.

Lemma 4.2 IznJ = t t and (A",$) = 3.

Proof Any point of An on z is also on 2"' since a' fixes A". Thus (A",z") 5 3 and (x",A") 2 2. Suppose (A",,") 5 2. Then Lemma 2.1 gives lz"1 2 IA"I. Thus by the maximality of A", (zn( = lan( and so, by (2.4), z" is fixed by a', which is a contradiction. Hence (An,z") = 3. Now Lemma 2.1 gives lz"l 2 i t , and by (2.5), Iz"I divides 2t. Thus Iz'I = ft.

Lemma 4.3 If B is a point of 2) moved by a* then B is fixed by a:'.

Proof Let IB"I = s. Since B is moved by a', s does not divide t . Suppose B is moved by a%'. By (2.5), s divides 2t, hence s < i t . Consider the three blocks through both B and a point of A". Suppose all such blocks were fixed by a'. Then each of them is on both B and B"'. Thus each of these blocks must contain A". However [A"/ 2 2 so there are three blocks on at least four points, contradicting X = 3. Thus there is a block z moved by a' with 2: on both B and a point of A". By Lemma 3.2, z is fixed by a%'. Hence B, BU3' and B"" are on z. Thus (z",B") 5 3, since all other blocks of 2:" on B are fixed by a$', and (B",z") 2 3. However z and z"' are on the same points of B". Thus (B",z") 5 3 and so (B",z") = 3. By Lemma 2.1, s = [Bnl 2 $ t , contradicting s < i t . Thus B is fixed by a l t .

Lemma 4.4 11 E ZS.

Proof Let p = crit and I' = (p) 2 2 6 . By Lemma4.3, all points of 2) are either fixed by p2 or p3 or both. Thus I' acts faithfully on no point orbit.

Let y be a block of 2) and suppose r acts faithfully on yr. Suppose y is on T I points fixed by p2 and on T Z points fixed by p3. Then TI + 7 2 5 k. Now y and yPa are both on the T I points of y fixed by p2, and hence r1 5 3. Similarly 1-2 5 3 and so T I +rz 5 6. Thus k 5 6 contradicting the assumption that k > 6. Thus I' is unfaithful so, since II is wayward, II = I' 2 2 6 .

Lemma 4.5 is faithful.

Suppose I' 2 Zp is an automorphism group of 2, . Then I?

Proof Suppose J? is unfaithful. Then there is a block x which is fixed by a2, Let B be a point of z not on zQ. Then B is fixed by a2 ; otherwise

68 A. Bowler

IBr I = 6 and Br would be a faithful orbit. Thus all the points of t not on P a r e fixed by a'. There are k - 3 such points, and since a' has order 3 and k > 6, by Lemma 4.1,

1 k - 3 5 -(k - 1).

2

Hence k 5 5 which is a contradiction. Thus I? acts faithfully on at least one orbit . Theorem 4.6 triplane D. Then 11 is faithful.

Let II be a cyclic automorphism group of a non-trivial

Proof

Theorem 4.6 is also true if D is a symmetric design with X = 4, the method of proof being similar.

For 11 a wayward abelian automorphism group of a triplane, similar methods to the ones in this section can be used to give the following result, which is stated without proof.

Theorem 4.7 Suppose 11 is a wayward abelian automorphism group of a triplane D , An is a maximal (point) orbit of II and t is a block on at least one point of A", with t moved by HAn. Then one of the following possibilities hold: (1) IIAn 2 2 , (2 " 1 - - 51 ' A"[ and (A" ,z")=3 (2) IIAn 2 2 , Itn[ = IA"I and (A",t") = 2 (3) IIAn G 2 3 , lznl = IA"I and (A",&) = 3

The result follows immediately from Lemmas 4.4 and 4.5.

Acknowledgement Studentship.

The work in this paper was supported by an SERC

References 1 Dembowski, P. Finite Geometries, Springer-Verlag, Berlin, Heidelberg,

2 Kimberley, M.E., Piper, F. C. and Wild, P. R. The Faithful Orbit Theorem New York, 1968.

for Biplanes, Ara Cornbinatoriu, 14, (1982), 87-98.

Combinatorics '90 A. Barlotti ei al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. 69

Minimal Flagtransitive Geometries

Francis Buekenhout

UniversitC Libre, Dept. Math., Boulevard du Triomphe, C.P. 216, B 1050 Bruxelles, Belgium

1. DEFINITIONS AND NOTATION

Let I be a finite set. Consider a finite geometry over I i.e. a triple ( X , * , t ) where X is a finite set, * is a symmetric, reflexive, binary relation on X and t is a mapping of X onto I such that a * b and t ( a ) = t ( b ) implies u = b.

A flag F of I' is a complete subgraph of the graph ( X , *). We assume that r is flrm i.e. every flag F with t ( F ) # I is contained in at least two flags of type I . The residue I ' F of a flag F is the firm geometry over I \ t ( F ) consisting of all a E X \ F such that a U F is a flag together with the relation (resp. mapping) induced by * (resp. t ) .

Let G be a group of automorphisms of r i.e. automorphisms of the graph ( X , * ) leaving t invariant. For each J C I we assume that G acts transitively on all flags F of I' such that t ( F ) = J . It is equivalent to require that G acts transitively on the flags F with t ( F ) = I , i.e. the maximal flags. In view of this property we call G flagtransitive.

The digon diagram of I' is the graph I = ( I , - ) defined by i N j if and only if there is a flag F with t ( F ) = I \ {i , j} such that I'p is not a complete bipartite graph i.e. r F has elements a , b with t ( a ) = i, t ( b ) = j and ( a , b ) $! *. We assume that the digon diagram is connected.

We call I' thin if every flag F with It(F)I = 111 - 1, has a residue containing exactly two elements.

We call I' residually connected if for any residue l?F with It(I'F)I 2 2, the graph induced on I ' F by *, is connected.

We call the pair ( r , G ) minimal if )GI 5 ( n + l)! where n = 11). Finally, we call the pair (I', G ) inductively minimal if for any connected subgraph

J of I and any flag f' of I' with t ( F ) = I \ J, ( r F , G F ) is a minimal pair.

2. RESULTS

We get a complete classification of the inductively minimal pairs (r, G ) thanks to the following theorems.

Theorem 1. Let I be a finite set of n 2 1 elements, I' a finite, firm geometry over I , with a connected digon diagram and let G be a flagtransitive group of automorphisms of r. Assume that (I?, G) is an inductively minimal pair. Then:

(1) I' is thin and residually connected; (2) G E Syrn(n $. 1) and for each i E I such that I \ i is connected, I' has n + 1

elements in t-'(i) on which G acts faithfully;

70 F. Buekenhout

(3) I has no minimal circuits of length P > 3; (4) every edge of I is on a unique maximal clique; (5) each vertex p of I is either on one or two maximal cliques. Theorem 2. For any graph I as in (3), (4), (5) of Theorem 1, there is (up to

isomorphism), one and only one inductively minimal pair (r, G ) a.dmitting I for diagram. The proofs will appear elsewhere.

Combinatorics ’90 A. Barlotti et al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. 71

SOME RESULTS ON HYPERARCHIMEDEAN MV-ALGEBRAS

V. Cavaccinia A. Lettierib

a Istituto di Matematica, Facolta di Architettura, Universita di Napoli, 80134 Via Monteoliveto n. 3 Napoli.

b Dipaa. di Matematica e Applicazioni “R. Caccioppoli” Napoli.

Abstract. In this paper we introduce the definition of quasi-boolean ideal in an MV-algebra that

allows us to characterize the hyperarchimedean quotients. Then we show a way to build hyperarchimedean subalgebras. Finally we classify hyperarchimedean MV-algebras in three classes: locally finite MV-algebras, boolean algebras and essentially hyperarchimedean MV- algebras.

Introduction.

MV-algebras were introduced by C.C. Chang [51 in I958 in order to provide an algebraic proof for the completeness theorem of the Lukasiewicz infinite valued propositional logic. Afterwards C.C. Chang [6] proved that there is a one-to-one correspondence between linearly ordered MV-algebras and linearly ordered abelian groups. Recently D. Mundici [8] extended such a correspondence to a functor r from lattice-ordered abelian groups with strong unit to MV-algebras. Moreover he proved [91 that MV-algebras are categorically equivalent to bounded commutative BCK-algebras.

Roughly, MV-algebras are a certain generalization of Boolean algebras because the elements in an MV-algebra are not, in general, idempotent: an MV-algebra in which each element is idempotent is a boolean algebra. Every MV-algebra contains a boolean algebra: the set of idempotent elements. In this paper we describe some properties of particular MV- algebras: the hyperarchimedean MV-algebras. They are such that every element of them, though no idempotent, has an idempotent multiple. Some way hyperarchimedean MV-algebras are closer to boolean algebras.

In Section 1 we give some basic definitions and properties of an MV-algebra and what we use later.

12 V . Cavaccini, A. Lettieri

In Section 2 we introduce the definition of quasi-boolean ideal in an MV-algebra that allows us to characterize hyperarchimedean quotient MV-algebras and hyperarchimedean MV- algebras as well as prime ideals do in linearly ordered MV-algebras and primary ideals do in local MV-algebras [71, [3].

In Section 3 we show a way to build hyperarchimedean sub-algebras.

Finally in Section 4 we classify hyperarchimedean MV-algebras in three classes: locally fmite MV-algebras, boolean algebras and essentially hyperarchimedean MV-algebras.

1. Preliminaries.

An MV-algebra is an algebraic structure (A, +, - , -, 0, 1) where A is a non-empty set, + and * are two binary operations, - is an unary operation, 0 and 1 are constant elements of A such that:

1) (A, +, 0) and (A, ., I) are commutative semigroups with identity.

2 ) x + X = l , x . x = o , O = l for all x E A.

3 ) x + y = x - y , x - y = x + y , x = x forall x , y ~ A . - _ _ _ _ _ _ _ _ _

4) Defining v and A by x v y = x + x - y, x A y = x ( x + y) we have that (A, v, 0), (A, A , I) are to be commutative semigroups with identity.

5 ) x(y v z) = x . y v x . z, x + (y A z) = (x + y) A (x + z) for all x, y, z E A.

From these axioms it follows that the structure (A, v, A, 0, 1) is a distributive lattice with least element 0 and greatest element 1.

In the sequel we will agree that Ox = 0, (n + I) x = nx + x and xo= 1.

Defiition 1.1

mx = 1. If no such integer m exists then ord(x) = =. The order of an element x E A - (0}, in simbols ord(x), is the least integer m such that

We agree to say that ord0 = =.

Defiition 1.2

i ) x , y E I - x + y ~ ' I i i ) x E I , y s x = , y E I .

An ideal of A is a non-empty subset I C A such that

Some results on hyperarchimedean MV-algebras 13

Defmition 1.3

y E P. An ideal P of A is called prime if P * A and for any x, y E A, x A y E P implies x E P or

Let P be the set of all prime ideals of A. Let S be any non-empty subset of P. On the MV- algebra A, L.P. Belluce [ 11, defines the following equivalence relation in A:

x-y(mod.S)- - (xEP iffyEPforanyPES).

From that two elements x, y E A are equivalent (rn0d.S) if and only if no prime ideal P E S can separate them.

Theorem 1 . 1 [ I ]

greatest element [ 1 Is. The structure ([Als, +, ., s,O, 1) is a distributive lattice with least element [Ols and the

The interest of this congruence is in the following:

Theorem 1.2[1]

homeomorphic. 0

The prime ideals spaces of the MV-algebra A and the distributive lattice [Alp are

Indeed L.P. Belluce points out that the structure of the set of all prime ideals of an MV-

In the sequel the distributive lattice as [A$ will be called y -quotient and if S =P we algebra is the same as that of some distributive lattice.

sometimes will obmit P.

Definition 1.4 131

p E P for some n. An ideal P of A is called primary if P * A and for any x, y E A, x - y E P implies fi E P or

Proposition 1 . 1 [ 3 I Every prime ideal of A is a primary ideal of A.

Defijtion 1.5 [ 3 ] An MV-algebra is called local if for every x E A, ordx < DC or ord X c 00.

A I

Theorem 1.3 [31 Let A be an MV-algebra . - is local iff I is a primary ideal of A.

Finally

Defmjtion 1.6 [ 1 I

maximal ideals of A. An MV-algebra A is called semisimple if RadA = (0} , where RadA is the intersection of all


2. Hyperarchimedean MV-algebras and quasi-boolean ideals.

We recall the following:

Defmition 2.1

such that nx = (n + I) x or equivalently x A z = 0. An MV-algebra is called hyperarchimedean if for each x E A then is a natural number n

It is known that in an hyperarchimedean MV-algebra A every prime ideal is maximal.

As for the linearly ordered and local MV-algebras, so we associate to hyperarchimedean Moreover: A is hyperarchimedean iff [A] is a boolean algebra.

MV-algebras a corrisponding type of ideal, namely:

Defmition 2.2

t h a t x h z E J. An ideal J of A is called quasi-boolean if for every x E A there is an natural number n such

Observe that from [5] every maximal ideal is a quasi-boolean ideal.

A J

Lemma 2.1

everyideal J ofA. If A is an hyperarchimedean MV-algebra, then - is an hyperarchimedean MV-algebra for

A Theorem 2.1

ideal. Let A an MV-algebra, J C A an ideal. Then is hyperarchimedean iff J is a quasi-boolean

- A x n x

Proof. Suppose hyperarchimedean. If x E A, then there is an integer n such that A = 0

Viceversa. Suppose J quasi-boolean there is an integer n such that

- ( n + I jx -nx

J ' J x n x x h K E J,thus - A - = O whichimplies J J

Theorem 2.2 If A is hyperarchimedean and local MV-algebra then A is locally finite.

Proof: Let x E A and ordx - = . Then x E M where M is the unique maximal ideal of A. Thus E Rand x S &, hence x A z = x for every integer n. On other hand, since A is

= 0, thus x = 0. Since 0 is the unique element of hyperarchimedean, for some m, x A

infinite order, A is locally finite. 0

Some results on hyperarchimedean MV-algebras

Corollary 2.1 If J is a quasi-boolean and primary ideal of MV-algebra A, then J is maximal.

Proof: It follows from Theorem 2.1, Theorem 1.3, Theorem 2.2 and Theorem 4.7 of [a. 0

Corollary 2.2 Let A be an MV-algebra and J a quasi-boolean ideal in A. Then J is primary iff J is

maximal.

Using the functor r to the corresponding lattice abelian group [8], from [3, 14.1.21 we have:

Theorem 2.3

homeomorphic images. An MV-algebra A is hyperarchimedean iff A is semisimple together with all its

In the sequel, for every ideal J of an MV-algebra A, M(J) will denote the set of maximal

ideals of A including J and P(J) will denote the set of all prime ideals of A including J. It is

known that fl

Theorem 2.4

p(J) H = J for every ideal J.

Let A be an MV-algebra. A is hyperarchimedean iff for every ideal J of A , n~ E M(J)H= J.

Proof: If A is hyperarchimedean, then thesis is obviom because P(J) = M(J) for every J of A.

Viceversa. A By hypothesis, for J = { O ] it follows RadA = (0). Let

A M J

be an homomorphic

image of A. Every maximal ideal of is of the type - with M E M(J). Then

Rad(S)=nM M(J) =I{O) by hypothesis. Thus by Theorem 2.3 A is

hyperarchimedean. 0

(“I At last we establish a link between quasi-boolean ideal and y -quotient lattice.

Proposittion 2.1 Let A be an MV-algebra and J an ideal of A. Then we have [A]p(j) .

Proof:

thesis. 0

P(J) is a saturated family of prime ideals of A. Then by proposition 4 of [2] it follows


Theomm 2.5

boolean algebra. Let A be an MV-algebra and J an ideal of A. Then J is a quasi-boolean ideal iff [A]P(J) is a

A J

Proof

Propsifjon 2.1 thesis follows. If J is quasi-boolean, then - is hyperarchimedean, so is a boolean algebra. By

A Viceversa.

hyperarchimedean and J quasi-boolean. Suppose [A]p(j) boolean, so by Proposifion 2.1 is boolean; hence - is IY J

3. Hyperarchimedean subalgebras.

In this paragraph we show a way to build hyperarchimedean subalgebras of a given MV- A Remember that every MV-algebra is a subdirect product of the family

algebra H that itself is not hyperatchimedean.

the linearly ordered MV-Algebra that is the quotient of A by P [6].

In the sequel for y E A we will denote by V(y) the set of all prime ideals of A that do not contain y, i. e.: V(y) = { P E P / y B P) .

Definition 3.1 . A family (Ap)p E~ will be called a-family of A if the following conditions are verified

A I) For every PEP, A,, is a subalgebra of

2) For each

all PEV (x A i n ) , for any n 2 0.

. - x x n p E p A p there is a natural number a such that -s a p for P

Fixed an a-family Fa = (Ap)pEp , setH(Fa) = (X E A / - E n p E p & i p ) . M P € P

Theorem 3.1.

of A. If Fa is an a-family of an MV-algebra A, then %(Fa) is an hyperarchimedean subalgebra

A Proof %(Fa) is a subalgebra of A, since A p is a subalgebra of

Let x E %(Fa). If P B V (X A in 1, then X A in E P. If P E V(X A

for every P E P.

1, then there is an - x x integer a such that - 5 a - , which implies (a + 1) x E P. P P

I1 Some results on hyperarchimedean MV-algebras

Let m = max(n, ( a + I)], we have x A E P for each P E P, hence X A 6 = 0 and

To search what hyperarchimedean subalgebras of A are born by any a-family Fa, namely H(Fa) is hyperarchimedean. 0

what are of the typeH(Fa), we premise the follow considerations: LetHasubalgebraofAandPaprimeidealofA.LetHp=

A subalgebraof -. P

Proposition 3.1 Let H be an hyperarchimedean subalgebra of an MV-algebra A, P a prime ideal of A and M

themaximalidealsuchthatPSM.ThengnH=finM X foreachp€ X H F .

Proof. X X hM x x hP - h M Let - = 5 and - = - where hP, hM E H. Since - S - , it is - - - P P M M P M M M

Let h E H n then &+ h~ E H n M = H n P, because H is hyperarchimedean, thus hP X X P P M h E - n H. From that and from - n H ‘G - n M thesis follows. 0

Proposition 3.2 Let A be an MV-algebra such that RadA is quasi boolean, H an hyperarchimedean

subalgebra of A. Then the family Ha=(Hp)pEp is an a-family of A and H S H(Ha).

Proof.

Let(z) p PEP -

Let P E V(x), M be &e maxima! ideal such that P E M and h E H such that

E np ,papS ince RadA is a quasi boolean ideal, there is an integer a

such that x h a x E RadA. x h = and

- A X

x h = fi .From 0 = ~ - - it follows h A ah E H n M = H n P, being H

hyperarchimedean. Thus also x A a x E P, which implies a x E P and - s - = -. That

x h a x - h h ah - - x a x 1 P P P

M M - 1

proves H, is an a-family.

It is obvious that H E H(Ha). 0

Theorem 3.2

with respect to be hyperarchimedean. Then H = H(Ha). Let A be an MV-algebra such that RadA is quasi boolean and H a subalgebra of A maximal

Proof:

follows H = H(Ha). By Proposition 3.2. H S; %(Ha) and, since %(Ha) # A (A is not hyperarchimedean), it


4. Classification of hyerarchimedean MV-algebras.

We shall show now that the class of hyperarchimedean MV-algebras (different from (0,l }) can be divided into three disjoint sub-classes. Raughly these are the locally finite MV- algebras, the boolean algebras and the MV-algebras which are niether, that we will define. To give latter formal status, we premise: Given an MV-algebra A, B(A) will denote the set of all idempotent elements. It is known

that B(A) is not only a subalgebra of A, but it is also the largest subalgebra of A, which is at the same time a boolean algebra with respect to the same operations + - and -. At last set H P A - B(A).

Definition 4.1. An hyperarchimedean MV-algebra wlll be called essentially hyperarchimedean (briefly EH

MV-algebra) if H+ @ and there are x, y E H such that x + y E B(A)- ( I 1. Let A+ be the class of all hyperarchimedean MV-algebras different from { 0, 1 ) . Denote by

IB E A h the class of boolean algebras, by BW S bah the class of essentially hyperarchimedean MV-algebras. Finally let LP S A h be the class of locally finite MV- algebras. Then we have:

Theorem 4.1.

i ) A E B ii) A E EW iii) A E ILP.

For every A E A h exactly one of the following holds:

Proof. First we show that B, EW, LP are pairwise disjoint. It is obvious that IB n LP =a.

Moreower if A E B, then H = 4, hence A B BIF, that is IB r l EN = 4. Suppose now A E ILP, then B(A) = (0 , l ) and H = A-{O,l]. If there are x, y E H such

that x+y E B(A) - (1) , it follows x +y = 0 which implies x = y = 0, absourd. So ILP n BW = 4. Let us show that A h = BW U ILIF U B. Suppose A B B U ILIF, then

H + 4 and B(A) - { I ) # 4. Let x E H such that ordx = 00, then there is an integer number n E w such that

x, 2x, ...,( n-1) x E Hand nx E B(A) - { I ) . Thus set y = (n-l)x we have x + y E B(A)-(I} and A E EW. Now let us show that there is really in H an element with infinite order. Suppose ordy <

Let x E H and z E B(A) - ( I ) , ord(x A z) = 00, hence x A z E B(A) - ( I ) . Let ordx = n, then we have x A = z which - - implies z zi x (*). Anagously set ordx=m we have X A Z = m'(irsA $- lllx A mz z, thus z S x h t is x S z (**).

By (*) and (**) x = z , absurd. This absurd show that A E EW. To show that EW is nonempty we exhibit the MV-algebra A = [O,l]n [I] where n is an

for every y E H.

n( A z)= nxn

integer positive.

We have B(A) = ( (x p...%) E [O, 11" / for each i = I. . . .n, xi = 0, or Xi = I ) , thus H f 4. If we consider x = (q ... XJ y = (y, ... y,> in H such that xio - Yio for any i, = I...n, then

x + y E B(A) - (1).

Some results on hyperarchimedean MV-algebras 79

REFERENCES

1 L.P. Belluce, Semisimple algebras ofinfinite valued logic and boldfuzzyset theory, Can. J. Math. vol. XXXVIII, 6, 1986, 1356-1379.

2 L.P. Belluce, A. Di Nola, A. Lettieri, On some lattice quotients ofMV-algebras, Ric. Mat. 39 (1990) 41-59.

3 L.P. Belluce, A. Di Nola, A. Lettieri, L d M V - algebras, submitted.

4 A. Bigard, K. Kmeil, S. Wolfenstein, Groupes et anneauxrbticuli, Lecture Notes in Math., Springer-Verlag, vol. 608, 1977.

5 C.C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. SOC. 88 (1958), 456-490.

6 C.C. Chang, A new proof of the completness o f the Lakasiewicz axioms, Trans. Amer. Math. SOC. 93 (1959).

7 C.S. HOO, MV--algebras id~sandsemisimplicity, Math. Japan 34,4, (1989), 563-583.

8 D. Mundici, Interpretation of AF-C*-algebras in Lukasiewicz sentential calculus, J. Functional Analysis, vol. 65, 1, 1986, 15-63.

9 D. Mundici, MV-algebras are categorically equivalent to Bounded Commutative BCK- Algebras, Math. Japonica, n. 6, (1986)) 889-894.



Three Combinatorial Sequences Derivable from the Lattice Path Counting

Wenchang Chu

Institute of Systems Science, Academia Sinica, Beijing 100080, China

Abstract This is a brief survey on three combinatorial sequences due to Catalan, Motzkin

and Schroder respectively. Various extensions of these sequences associated with lattice path counting and their convolutions are considered. Several major combinat orial interpretations of Catalan sequence and its generalized forms are presented, which concern non-associative algebra, random walks, triangulations, ballot problems and tree counting. At the end of the paper, we make a few remarks on the evaluation of Catalan sequence.

1. INTRODUCTION

The numbers C,, where [40]

C(Z) = 1 + ZCZ(2) = c C,Zn n>O

are known as the Catalan numbers. They were first studied by Euler, Catalan and others. This sequence has an extensive history and some 500 papers have appeared involving it. It has had about 50 different interpretations and occurs in a wide variety of combinatorial prcjblems. The bibliographies of Alter [l], Brown [2], and Gould [15] give an excellent account of the literature. It is frequently used for enumeration in the following aspects.

A. Non-associative algebra (Bracketing problem) (See Alter [l] and Cigler [7]). The number of non-associative products of n + 1 different factors with a binary

operation is equal to C,. B. Figure partitions. B1. The ways in which 2n points on the circumference of a circle can be jointed in

B2. The number of divisions of a fixed ( n + 2)-gon into triangles by diagonals

C . Random walks. Walks of 2n steps which begin and end at the origin and consist of unit steps to the

left or to the right but never going to the left of the origin is enumerated by C, (Chung & Feller [6], 1949; Good [14], 1958).

pairs without crossing is enumerated by C, (Motzkin [25], 1948).

without crossing equals C, (Polya [28], 1956; Euler).

Partially supported by NSF (Chinese) Youth Grant 019901033.

82 W. Chu

D. Ballot problems. In an election, two candidates, A and B , receive n votes respectively. In how many

ways can it happen that at no stage does A trail B? This number is also C,, (See Mohanty [24], 1979).

Extension of this result, such as the ballot theorem and its generalizations have wide applications in the statistics (Narayana [26], 1970; Takacs [41], 1962).

E. Plane trees. El . The number of isomorphic classes of planted plane trees with 71 + 1 vertices

equals C, (Knuth [20], 1968). E2. The number of ordered trees with n + 1 vertices is equal to C, (Dershowitz lk

Zaks [lo], 1980; Prodinger [29], 1983). E3. The collection of isomorphic classes of trivalent planted plane trees with 2n + 2

vertices is enumerated by C,, (See Alter [l]). In addition, Catalan numbers have important applications in enumerating decision

patterns (Wine lk Freund [44], 1957), in the theory of queues (Takacs [41], 1962), in enumerating flexagons (Oakley & Wisner [27], 1957), in domination of vectors (Carlitz St Scoville [3], 1968), in the enumeration of matrices (Jackson St Entringer [18], 1970; Leighton & Newman [22], 1980; and Shapiro [38-391, 1975/1984), and in similar relations on finite ordered set (Rogers [32], 1977). The details can be refered to the papers listed in the references.

Equivalent to the interpretations C or D, C, has a useful interpretation that arises in the lattice path problem: i.e., C, enumerates the lattice paths on the non-negative quadrant of the integral square lattice in two dimensional Euclidean space, from the origin to the point ( n l n ) , which remain on or below the diagonal x = y1 with unit up or right steps. In this direction, a lot of work has been done, and various extensions to higher and generalized Catalan numbers can be made. We shall discuss thein in the next sections.

Closely related to C,, there are two other interesting sequences. One is known as the Motzkin numbers m, defined by [40]

(1 2 )

Another is called the Schroder numbers rn defined by [40]

The former sequence has a number of combinatorial interpretations analogous to C, which have been collected by Donaghey & Shapiro (111 (1977). The latter sequence only has a few of these which have been known from the author (cf. Rogers & Shapiro [34], 1978). Hence for combinatorial interpretations, we confine ourselves to the Catalan sequence in this paper.

Rogers [33] has considered the following lattice path counting problem which can generate the higher Motzkin and Schroder sequences.

[R]-family: For some fixed integral s, v with s ,v 2 1 and all integral c, y, we take the lattice points to be ( x / s , y/s) in the non-negative quadrant of two dimensional

Three combinatorial sequences 83

Euclidean space, and in addition to unit horizontal and verticd steps, allow diagonal step from the lattice point ((x - v + l ) / s , (y - l)/s) to ( x / s , y/s). Then the number of lattice paths from the origin to the point (m/s , n / s ) which remain on or below the line x = (v - l ) y is that we want to compute.

If we take m = ti and s = 1, and exclude the diagonal steps in the above, then C, results. If we take nz = n, then nz, and r , are generated, corresponding to the cases of s = v = 2 and s = 1, v = 2 respectively.

Motivated by Rogers' work [33], the author will extend the Motzkin and Schroder sequences gradually, in the same way as the generalizations of the Catalan numbers.

2. THE MODIFICATIONS OF THREE SEQUENCES

In the end of last section, if we take m = n + s ( t - 1) instead of m = n, then tthe corresponding lattice paths are enumerated by the following slightly modified sequences respectively:

) C'"( k). n + k + t - 1 ( n - k

d t ) ( n ) = C k > O

(2.2)

(2.3)

Considering the last time at which lattice paths enumerated by C("+')(n) visit the line x = y + s - 1 , we find that

n

C(s+')(n) = c C'"'( Ic)C(')(n - I c ) k=O

On iteration, we obtain

Therefore we have the generating function

(2.6)

where C(Z) is defined by (1.1).

84

Correspondingly, we have

W. Chu

and

= c C(t)(k)Xk/(l - z)Zk+t = (1 - s)-tC( ')(z/(l - z)Z) k

= (1 - x ) - V ( x / ( l - z)2)

By using the fact that (the case of t = 1 in the above)

M ( s ) = (1 - s)-'C(xZ/(l - X ) Z )

and

R(s) = (1 - X)-1C(z/(l- s ) Z )

we also find that

and

From the above, we have the following convolution-type identities:

(2.10)

(2.11)

(2.12)

(2.13)

(2.14)

Their direct proof are not easy.


3. FURTHER GENERALIZATIONS OF THREE SEQUENCES

Further generalization of C( ' ) (n ) is the higher Catalan numbers CLt)(n) defined by (cf. Mohanty [24], p17)

t t + n u CL"(n) = -

t + n v ( n ) which is the number of outward directed walks (or lattice paths) on the non-negative quadrant of the integral square lattice from the origin to point ( t - 1 + n(v - l), n ) which remain on or below the line 5 = (27 - 1)y. In the same way as providing identity (2.4), we can obtain Rothe-Gould identity as follows:

Now let

(3.2)

(3.3) n

Then it is shown that (Rogers [33], 1978)

CV(5) = 1 + z c ; ( x ) (3.4)

and

For the [R]-family, taking s = t~ and m = n,(v - 1) + v ( t - 1) will provide the higher Motzkin sequence given by

Taking s = 1 and ni = n(o - 1) + t - 1 instead, will generate the higher Schroder sequence defined by

Similar to section 2, we find that

86 W. Chu

I p ( x ) = (1 - . I p C y ( r / ( 1 - r ) lJ ) = c r p ( 7 z ) x n I 1

When t = 1, we have

~ l 4 ~ , ( x ) = (1 - .r)- 'Ct,(x1'/(l - r)"') = Af!,')(.r)

R,,(T) = (1 - :t-)-'C,,(r/(l - 2)'') = R!,l)(r) .

From (3.5) and (3.8-3.11) we have

M/,')(T) = AfA(.r)

R:,')(r) = R:,(J+

Tlicse two relations call give the following combinatorial icleiititicw

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

Naturally, C$"( 1 7 ) has severa,l interpretations analogous to C,, in the non-associative algebra with wary opera7.tion (Sands [37], 1978), the divisions of convex polygon into ( P I + 1)-gons (Motzltin [XI, 1948), random walks with unequal step lenghts (Cong t!k %to [9], 1982), ballot problem with ( 1 , - 1)-time majorit.y (See Dvoretzky 8 Motzkin [12], 1947) and plane trees with ( 1 1 + 1)-valents (1ila.rner [19), 1970). The det,a.ils are oniittetl here and the interested readers can refer to the original papers.

4. THE MULTINOMIAL ANALOGUE OF THREE SEQUENCES

This scction will investigate the multinoniial foims of the three scciucnces discussed

Motivated by the woik of Molianty [24] (1979) and Sands [37] (1978), we fiist tlcfinc in the last scct ions

the geiieralizcd Catalan nuin1m-s c:)(E) as follows.

wlicre (E,Ti) denotes the scalar product of vectors E = ( ? 2 1 , ? ? z , . . * , ? ? k ) and V = (111, u z , . . . , u k ) , tlie coordinate bun1 for vector E , ant1 (I:[) the niultinoniial coef- ficiciit .

Three combinatoriat sequences 87

Corresponding to C,, this sequence has the following combinatorial interpretations

A. Non-associative algebra. Al. The number of ways of brackting a product for ( ( 5 , F ) - llzl + 1) terms from a

set with v,-ary operations used exactly n, times (1 5 i 5 k ) is equal to Ct’(5) (Sands [37], 1978). A2. The list of t words having composition type ( t ; X ) (where 7 is an infinite

sequence of natural numbers with v,th term equal to n, and others being zero, under the assumption that the coordinates of are all distinct) is equal to CF)(E) (Raney [30], 1960).

known from the author.

B. Generalized triangulations. B1. The number of divisions of (5,;) points on the circumference of a circle into n,

set of v,-point groups (1 5 i 5 k ) without crossing equals Ct’(5) (Chu [5], 1987). This problem was considered by Motzkin [25] (1948). But he could not obtain such

an explicit formulation in general. B2. + 2 side convex polygon into n,

polygons with u, + 1 sides (1 5 i 5 k ) by non-intersecting diagonals is counted by Cc)(E) (Erdelyi & Etherington, 1941; See Sands [37]).

The number of partitions of a (5,;) -

C. Takacs’ urn problem (Takacs [41], 1962; See Mohanty [23], 1966). Consider an urn which contains n cards with nonnegative integers. Cards are drawn

without replacement from the urn. Denote by y; the number on the card drawn at the ith time. Then the probability

I

1 - k/n for 0 5 k 5 n = { 0, otherwise.

In the modified form, this formula may be stated as follows. The set of ( k + 1)-ary sequence with exactly ((E,;) - [El 4- t ) O’s, n1 vi’s, 112 v2’s,

up to nk Vk’S (any pair of v, are not identical and v, 2 1 for 1 5 z 5 k ) such that the number of zeros is always greater than C,(v, - 1) * (number of v,’s) is enumerated by C$) (E) .

D. Lattice paths (Chorneyko & Mohanty [4), 1975). The number of lattice paths from the origin to coordinate point ((E,;) - I?tl +t - 1, 5)

in (k+l)-dimensional Euclidean space with unit steps along the coordinate axe directions which remain on or below hyperplane zo = CI=l(vI - 1)z, is C, (E) . k (0

E. Plane trees (Chorneyko & Mohanty [4], 1975). The number of isomorphic classes of planted plane trees such that exactly n, vertices

have degree v t + 1 (evidently, the total number of vertices is (5 ,V) + 2) equals C, (5). Corresponding to the interpretation D, we can propose the [HRI-family (multi-

dimensional [R]-family) which will be used to provide the generalized forms of the Motzkin and Schroder sequences respectively.

[HEZI-family. For some fixed integral s,, u, with s,,v, 2 1 and all z,, y, (1 5 z 5 I ; )

we take the lattice points to be (Xi”=, yI/s l ,Zl /s l , ’ . . ,z t /Sk) in EL+’ (the Euclidean

(1)

88 W. Chu

space of dimensional k + 1). In addition to the unit steps along with the coordinate axe directions, allow the diagonal steps from the lattice point yr/s2 , x1/s1 , * . . , Zk/Sh) to the point

k

k

( ( V I - 1)/s: + ~ y ~ / ~ ~ ~ x l / ~ l l " ' ~ ~ ~ - l / ~ : - I ~ ( ~ ~ + ~ ) / ~ I , Z I + ~ / S : + ~ , " ' , X ~ / ~ & ) 1=1

(1 5 i 5 k ) . Then the number of the lattice paths from the origin to point (rn + mr/s,,nl/sl,...,nk/Sk) whichremainonorbelow thehyperplane zo = cI=I(~~2-

l)z, is desired. m, = n,(v, - 1) (1 5 i 5 k ) and rn = t - 1 and

use the interpretation D, then the generalized Motzkin sequence can be obtained in the following:

k k

In the above, if we take s, = v2,

(4.2)

where the summation is over all non-negative integral vectors 7 = (j,,j2,. . . ,jk).

the generalized Schrijder numbers can be provided as follows: If we take s, = 1, nz, = n,(v, - 1) and m = t - 1 instead in the [HRI-family, then

For the lattice paths enumerated by C ~ + " ( E ) , consider the last time at which they visit the hyperplane .TO = C:=l(vi - 1)xi + s - 1. We find that (Raney [30], 1960)

(4.5)

Then from (4.4) we have

C$'(Z) = CgZ) (4.6)

&(F)= 1 +ZlcG'(?) + X z c ? ( F ) $.'*+Zkc?(F) = c$)(F),

where C:(T) satisfies the following function equation (Chu [ 5 ] , 1987):

(4.7)

Now we compute the generating functions for the sequences {&'(E)} and { r$ ) (E)} separately.

Three combinatorial sequences

From the fact that

89

= c c$'(;)z:'l "1 x . y z . . . x jk" 'c k -

j

[El + t - 1 ) f i x ; i - j i u i *? (nl - . i lv l ," ' ,nk - j k V k i=l

make the changes of ni - jiv; = ri (1 5 i 5 k) for the last sum in the above. Then it equals the following

Therefore we have

M$)(F) = (1 - lq)-~@(zy1/(1- 1F1)"','** ,z;"/(l- 1F'1)"".

M$)(F) = M i @ ) where M&f) = M$)(T). (4.9)

(4.8)

Combining this with (4.7), we find that

Hence this relation yields the combinatorial identity as follows:

(4.10)

Similarly, the following generating function for the generalized Schroder sequence and the corresponding convolution-type identity can be derived.

(4.11)

(4.12)

(4.13)

90 W. Chu

5 . SOME REMARKS IN ADDITION

The following are the standard methods for the evaluation of Catalan numbers. Andre's reflecting principle can give a simple treatment for the classical ballot prob-

lem (See Narayana [26], 1979) and it has been developed by Zeilberger [45] (1983) to give an excellent proof for the many candidate ballot problem. The method of pen- etrating analysis (or cyclic permutations) was designed by Dvoretzky & Motzkin [12] (1947) which was used to deal with the ballot problem with t time majority. Sands [37] (1978) has generalized it to the products of non-associative algebra with several Ic-ary operations. Recurrence and generating function method (or Lagrange inversion formula) was successfully used by Erdelyi and Etherington (1941) and, Chorneyko & Mohanty [4] (1975) to treat the divisions of convex polygon and the counting of higher dimensional lattice paths. The unified treatment for a number of combinatorial enumeration problems associated with lattice path counting was established by Kreweras [21] (1965), Narayana [26] (1979) and Mohanty [24] (1979) whose main means was domination correspondence. In addition, Takacs [41] (1962) has used the probability method giving a famous urn theorem which is closely related to the lattice path enumeration (See Mohanty [23], 1966).

For the details of the above mentioned methods, Narayana and Mohanty have given a systematic survey in the monographs [24] and [26]. The interested readers can refer to them.

REFERENCES

R. Alter, Some remarks and results on Catalan numbers, Proc. 2nd Louisiana Conf. on Combinatorics, Graph Theory and Computing (1971), 109-132. W.G. Brown, Historical note on a recurrent combinatorial problem, Amer. Math. Month. 72 (1965), 973-977. L. Carlitz and R.A. Scoville, Problem E2504, Amer. Math. Month. 75 (1968), 77. I.Z. Chorneyko and S.G. Mohanty, On the enumeration of certain sets of planted plane trees, J. Comhin. Theory (Ser.B) 18 (1975), 209-221. W.C. Chu, A new combinatorial interpretation for generalized Catalan number, Dis Crete Math. 65 (1987), 91-94. K.L. Chung and W. Feller, On fluctuations in coin-tossing, Proc. Nat. Acad. of Sciences (USA), 35 (1949), 605-608. J. Cigler, Some remarks on Catalan families, Europ. J. of Combin. 8 (1987), 261- 267. L. Comtet, Advanced Combinatorics, Reidel Dordrecht, 1974. T.T. Cong and M. Sato, One dimensional random walk with unequal step lenghts restricted by an absorbing barrier, Discrete Math. 40 (1982), 153-162.

10 N. Dershowitz and S. Zaks, Enumeration of ordered trees, Discrete Math. 31 (1980),

11 R. Donaghey and L.W. Shapiro, Motzkin numbers, J. Combin. Theory (Ser.A) 23

12 A. Dvoretzky and Th. Motzkin, A problem of arrangements, Duke Math. J . 14

9-28.

(1977), 291-301.

(1947), 305-313.


13 P.H. Edelman, Chain enumeration and non-crossing partitions, Discrete Math. 31

14 I.J. Good, Legendre polynomials and trinormial random walk, Proc. Cambridge

15 H.W. Gould, Research bibliograph of two special number sequences, Revised ed.,

16 I.P. Goulden and D.M. Jackson, Combinatorial Enumeration, Wiley, New York,

17 B.R. Handa and S.G. Mohanty, Enumeration of higher dimensional paths under

18 D.E. Jackson and R.C. Entringer, Enumeration of certain binary matrices, J. Com-

19 D.A. Klarner, Correspondence between plane trees and binary sequences, J . Combin.

20 D.E. Ihuth , The Art of Computer Programming I. Fundamental Algorithms,

21 G. Kreweras, Sur une classe de probleme de decombrement lies an treillis des parti-

22 F.T. Leighton and M. Newman, Positive definite matrices and Catalan numbers,

23 S.G. Mohanty, An urn problem related to the ballot theorem, Amer. Math. Month.

24 S.G. Mohanty, Lattice Path Counting and Applications, Academic Press, INC. 1979. 25 Th. Motzkin, Relations between hypersurface ratios, and a combinatorial formula

for non-associative products, and for partitions of a polygon, Bull. Amer. Math.

26 T.V. Narayana, Lattice Path Combinatorics with statistical Applications, Toronto

27 C.O. Oakley and R.J. Wisner, Flexagons, Amer. Math. Month. 64 (1957), 143-154. 28 G. Polya, On picture writting, Amer. Math. Month. 63 (1956), 689-697. 29 H. Prodinger, A correspondence between ordered trees and non-crossing partitions,

30 G.H. Raney, Funct(iona1 composition patterns and power series invertion, Trans.

31 D.G. Rogers, A Schroder triangle: Three conibinatorial problems, Combin. Math.

32 D.G. Rogers, Similarity relations on finite ordered set, J. Combin. Theory (Ser.A)

33 D.G. Rogers, Pascal triangles, Catalan numbers and renewal arrays, Discrete Math.

34 D.G. Rogers and L.W. Shapiro, Some correspondences involving the Schroder num-

35 D.G. Rogers and L.W. Shapiro, Eplett’s identity for renewal arrays, Discrete Math.

36 D.G. Rogers and L.W. Shapiro, Deques, trees, and lattice paths, Combin. Math.

37 A.D. Sands, On generalized Catalan numbers, Discrete Math. 21 (1Y78), 219-221.

(1980), 171-180.

Philos. SOC. 54 (1958), 39-42.

Morgantown; MR5315460.

1983; Ch. 5.1.

restriction, Discrete Math. 26 (1979), 119-128.

bin. Theory (Ser.A) 8 (1970), 291-298.

Theory 9 (1970), 401-411.

Addison-Wesley, New York, 1968.

tions des entiers, Cahier Bur. Univ. Recherche Oper. 6 (1965), 5-105.

Proc. Amer. Math. SOC. 79 (1980), 177-181.

73 (1966), 526-528.

SOC. 54 (1948), 352-360.

Press, 1979.

Discrete Math. 46 (1983), 205-206.

Amer. Math. SOC. 94 (1960), 441-451.

V, Springer-Verlag, Berlin (1977), 17S196.

23 (1977), 88-98.

22 (1978), 301-310.

bers, Combin. Math., Springer-Verlag Lecture Notes 686 (1978), 267-276.

36 (1981), 97-102.

VIII; Lecture Notes 884 (1981), Springer, 293-303.

92 W. Chu

38 L.W. Shapiro, Upper triangle rings, ideals and Catalan numbers, Amer. Math.

39 L.W. Shapiro, Positive definite matrices and Catalan numbers, Proc. AMS 90 (1984),

40 N.J.A. Sloane, A Handbook of Integer Sequences, Academic Press, New York, 1973. 41 L. Takacs, A generalization of ballot problem and its application in the theory of

42 L. Takacs, Combinatorial Methods in the Theory of Statistic Process, John Wiley &

43 G . Viennot, Une theorie combinatoire des polynomes orthogonaux genetaux, 1983,

44 R.L. Wine & J.E. Fkeund, On the enumeration of decision patterns involving n

45 D. Zeilberger, Andre’s reflection proof generalized to the many candidate ballot

Month. 8 (1975), 634-636.

488-496.

queues, J. Amer. Stat. Assoc. 57 (1962), 327-337.

Sons, New York, 1967.

Lecture Notes, University of Quebec at Montreal.

means, Ann. Math. Stat. 28 (1957), 256-259.

problem, Discrete Math. 44 (1983), 325-326.

Combinatorics '90 A. Barlotti et al. (Editors) 0 1992 Elscvier Science Publishers B.V. All rights reserved. 93

COMPOUND CLOSED CHAINS IN CIRCULAR PLANAR NEARRINGS

James R. Clay

Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA

1. INTRODUCTION.

In 1986, the concept of a circular planar nearring was introduced at Combinatorics '86 at Passo della Mendola [Cl]. This idea is motivated by taking the field of complex numbers (C,+,.) andconstructingtheplanarnearring(C,+,o), whereaob = (a / lu l ) .b for u # 0, and 0 o b = 0. For a # 0, C' o a + b is the circle of radius la1 and center b. Extending this to an arbitrary planar nearring ( N , +, .), we construct B' = {N'a + b I a,b E N , a # 0}, where N' = {n E N I na # 0 if a # 0). We say that ( N , +,.) is circular if for every three distinct points z ,y ,z f N , there is at most one circle N'a + b E B' with 5 , y, z E N'a + b, and that for every pair of distinct points .T, y E A', there are more than one circle N'a + b in B' containing z and y. This last property gives the circles "curvature." If ( N , +, .) is circular, then an N'a + b E B' is the circle with center b and radius a.

Also in [Cl] is a table of all the nontrivial circular planar nearrings ( N , + , . ) where ( N , + ) is the additive group of a prime field (Zp, +,*), for 13 5 p 5 1000. By studying various examples from this table, one can find properties of these circles analogous to those in the usual euclidean plane (C,D', E ) described above, and also properties in contrast to those in (C,B',E).

One source for interesting structure is to fix a circle N's of radius s with center 0. How do all the circles of radius r relate to N's? First of all, each intersects N's in exactly 0, 1, or 2 places. That is, each falls into one of three equivalence classes:

S(r; s) = {N'r + z I IN'S n (N*r + .)I = 0);

7(r;s ) = {N'r + z I IN'S n (N'r + z)1 = 1);

U ( r ; s ) = {N'r + z I (N's n (N'r + z)I = 2).

Let C(r; s) E {S(r; s), 7 ( r ; s),U(r; 9)). Then C ( r ; s) itself can be partitioned into equivalence classes

E,' = { N'r + z I z E N'c}.

That is, EL, for a circular planar nearring ( N , +, a ) , is the set of all circles with radius r whose centers are on N'c, the circle of radius c and center 0. Of course, c = 0 is possible in which case EL = E,' = {N'r}, so we shall always assume c # 0 since the E,' are of no interest to us presently. Take (C, +, 0) for an example. 7( 1; 3) consists

94 J .R . Clay

of all the circles of radius 1 which are tangent to the circle z2 + yz = 4, and 7( 1; 2) is partitioned into two equivalence classes E: and Ei .

Each Er yields a signed graph. The vertices of Er are the centers of the circles in EL. An edge exists between x and y of N, x # y, if (N'r + z) n (N'r + y) # 0. The edge is even if I(N'r + z) n (N'r + y)l = 2 and the edge is odd if ] ( N o r + 3) n (N'r + y)I = 1. In a diagram of such a graph, we indicate an even edge by a solid line segment, e.g., *-., and we indicate an odd edge by a broken line segment, e.g., . - - - -. .

An interesting example is given by the circular planar nearring ( 2 3 1 , +, 0) from the table in [Cl] where the circles have six points each, i.e., from (Z3,,Lt,', E). With N =

N' o 1 = {6,5,30,25,26,1}; "03 = {18,15,28,13,16,3}; N ' o 4 = {24,20,27,7,11,4}; N' 0 8 = {17,9,23,14,22,8}.

SO Ei = {N' o 1 + 12, N' o 1 + 10, N' o 1 + 29, N' o 1 + 19, N' o 1 + 21, N' o 1 + 2}, for example. The graphs of the E,' , c # 0, are as follows.

24 4 19 29

the circles with center 0 axe: N' 0 2 = { 12,10,29,19,21,2};

21 (y;;)lo \ 11

\ - - 2 12 27 7

Graph of E i . Graph of Ei

18 13 17 9

23 22 14 8

Graph of Ed.

16 15

Graph of Ei .

5 30

1 26

Graph of E:

Compound closed chains in circular planar nearrings 95

The graphs of Ei and E: will be the focus of our attention in this work, but one should notice something about each of the others. The graph of Ed is very analogous to the graph of E: in (273, B;, E) from the circular planar nearring (273, +, 0 ) having 8-point circles. The graph of Ei represents the circles of radius 1 whose centers are on the circle of radius 1 and center 0. In the usual plane (C, B', E), each of the circles in E! intersects each of the others at 0 as well as at an additional point, except for the circle directly opposite, and it intersects it only at 0. This is what is reflected in the diagram of the graph for E: here, also. In (2,3, Be, E), this same phenomenon also occurs for Ei .

The prism effect illustrated by the graph of Ei had never been observed in any other example by the author. Its intrinsic beauty calls for further attention. Then mother one was found in (237, Bl, E).

The graphs of E: and E; each illustrate a phenomenon which occurs frequently. For example, the graphs of Ei and El in (273,Bl,E) are analogues to that of Ei here for

here for (Z3,, Bz, E). These, and others, follow from asking the following question. (&, Bl, E), and the graphs of E5 9 and Egl in (273, Bi, E) are analogues to that of Ei

(Q) Given an E,', c # 0, for a finite circular planar nearring ( N , +, e ) , suppose there is a B E EL which intersects at most two other circles A, D E E,' \ { B } . Then what is the structure of the graph of E,'?

With this work here, we complete the answer to this question. A partial solution is in [C & Y ] , but we shall outline some of the relevant results again here because we shall need some of the tools developed there.

To answer (Q), one must consider three cases: a) B intersects no other circle in EL \ { B } ; b) B intersects exactly one other circle in Er \ { B } ; c) B intersects exactly two other circles in E,' \ {B}. Each of these three possibilities can occur, and examples will be given below together with the structure of the graphs of the relevant E,'.

2. THE TORUS PHENOMENON AND PRELIMINARIES.

For a planar nearring ( N , + , .), let N' = {n E N I na # 0 if a # 0). We shall make frequent use of the following facts whose proofs can be found in the indicated reference.

1. There is a subgroup (a,.) of the semigroup ( N O , . ) so that if a E N' , then there is a 4. E such that ax = 4.x for all x E N . We identify each 4 E 0 with an automorphism of ( N , +), which we also shall denote by 4. [C2]

if and only if b = d and N'a = N'c. [Cl] 2. If a nearring ( N , +, -) is planar, then N'a+b, N'c+d E B' satisfy N'a+b = N'c+d

3. If ( N , +, .) is circular and N'r + b, N'r + d E E,', then there is a unique 4 E 0 such that 4 b = d and #(Nor + b) = N'r +d. Consequently, if q5b = d, then + ( N o r + b ) = N'r + d. [C & Y]

4. If ( N , + , . ) is a finite circular planar nearring and there is an A E EL which is tangent to exactly m circles of E,' \ { A } , and intersects exactly n circles of EL \ { A } at

96 J.R. Clay

two distinct points, then every D E E,' has this property. [C & Y]

We now consider case a). If B E E,' intersects no other circle in EE \ {B}, then the Bame is true about each D E E,' by 4. So A, B E E,' imply either A = B or A n B = 0.

Definition 2.1. For a circular planar nearring ( N , + , - ) , suppose A, B E EL, c # 0 , imply either A = B or A n B = 0. Then

T(r; c ) = T = U D E E : D

is the torus with major radius c and minor radius r.

The following facts provide additional motivation for calling T(r; c ) a torus.

5. Let T = T(r;c) be a torus. Let x = pr + d E D = N'r + d E E: be arbitrary. Then

T = Uyr+dEDN'(rr + d).

In fact, each of these circles N'(yr + d), y E a, with center 0, is tangent to each of the circles E E EE. [C & Y]

6. If T is a torus for a finite planar nearring N , and if each circle has exactly I; points, then T has exactly k2 points. [C &. Y]

Examples of tori are the T(l,2)'s in each of (ZI~,D;,E), ( & I , & , € ) , (ZII~,S;,E), and (Z113,Bi,~). One finds a torus T ( 6 ; 3 ) in (z17,a;,E).

The graphs of the El for which T(r , c ) is a torus consists of vertices but no edges, one vertex for each I E N'c. This completes case a) with a rather uninteresting graph for EL, but with an intriguing geometric structure, the torus. In summary:

Theorem 2.2. For a finite circular planar nearring ( N , +, .), T(r ; c ) = U D E E E D i s a torus i f and only if the graph of E,' has no edges.

3. THE SIMPLE CLOSED 2-LINK CHAINS.

Now we consider case b). Let B E EE intersect A E EL. So if D E E,' \ {A, B}, then Dn(AU B) = 0. Case b) will be a consequence of this next theorem which is indicative of things to come.

Theorem 3.1. For a finite circular planar nearring (A',+,.) with 2 3 and # c E N , suppose there are A , B E El , A # B, where A n B # 0. Also suppose

D E El \ { A , B } implies D n ( A U B ) = 0. Then: 1 ) IAn BI = 2; 8) B = dA for some 4 E a, where d # 1, but d2 = 1; 3) D E E,' implies the ezistence of a unique F E E,' \ { D } such that D n F # 0, but zf G E El \ { D , F } , then G f l ( D U F ) = 0; 4 ) there is a bijection between the cosets X(4) of (4) = (1, 4} in and such pairs {D,F}

5 ) i f D , F E E l , D # F , and D n F = {d , f}, then dd = f . of 3);


Proof. Let 4 E 9 and take 4 A = B. Let A = N ' r + a and B = N'r + b , so #a = b. Also let A n B = {z,y}. Now dx,dy E 4 A n d B = B n # B , so 4 B = A since t$B # B . Hence {4zlq5y} = {s,y}. We cannot allow 42 = z and dy = y if z # y in a planar nearring, so 4s = y and 4 y = z. Hence q52 = 1. This gives us 2) and 5 ) if we but had

If A n B = {z}, say, then 42 = z and so t = 0. But then c = -r and we have 0 E X for each X E E:, which in turn implies 1E:I = 2 = 191. But 191 2 3, so we now have l), as well as 2) and 5).

From 4. we get 3). There remains to show 4). Take X E 9 \ { 1,4}. Then XA, XB E E,', and {XA,XB} n { A , B } = B. From ( A n B ( = 2, we get (XA n XB( = 2 with {Xz,Xy} = XA n XB. So each coset X(4) yields a pair {XA,XB} as described in 3) . If {XA,XB} = { p A , p B } and XA = p A , then X = p. If AA # pA, then XA = pB, and so p-*XA = B . Hence p-'X = 4, and so A(#) = p ( 4 ) . This makes X(t$) w {XA,XB} injective.

For D E E,', there is a unique X E 9 such that XA = D, so D E { XA, XB} and XA n XB # B. This makes A($) I+ { XA, XB} bijective. Now we have 4).

1).

Definition 3.2. For a pair { A , B } satisfying the hypothesis of theorem 3.1, we write and call it a simple closed 2-link chain.

The graph of El , if E,' satisfies the hypothesis of theorem 3.1, consists of [@ : (4)]

A * B a A or A2 BUA,

t Y pairs We have then

Theorem 3.3. The hypotheses of theorem 3.1 for El are satisfied if and only if the l i Yi

graph of E,' consists of [a : (4)] distinct pairs .-.. Of course, the reader is curious about the existence of such circular planar nearrings

which satisfy the hypotheses of theorem 3.1. Existence is demonstrated by looking at E: in (&,Lt ; ,E) , E$ in ( & p , & , E ) , and El in ( & p , a s , E ) .

4. THE SIMPLE CLOSED .+LINK CHAINS.

We are now ready to consider case c). This case is rather complex. First of all, we begin by breaking case c) into three subcases. We start out with B E El intersecting exactly two other circles A, D E E i \ {B}. It could be that: c1) B is tangent to each of these circles A and D; c2) B is tangent to one, say A, and intersects D in exactly two places; c3) B intersects each of A and D in exactly two places.

First, we'll see that c2) is impossible.

Proposit ion 4.1. For a finite circular planar neam'ng ( N , +, .), suppose B E El, c # 0, and that B U tangent to ezactly one circle A E E l \ { B } . Then A n B = (0) and so N'c = N'(-r ) .

Proof. Let A = N'r + a , B = N'r + b, and A n B = (3). Let 4A = B with #a = b. Then 41 E dA n $B = B n 4B and ( B n = 1. Hence I#$ = A and so 4% = 2. This makes z = 0 and since 0 E A = N'r + a, we conclude that a E Ar*(-r), hence N'c = N'(-r) .

98 J.R. Clay

Proposit ion 4.2. For a finite circular planar neam'ng ( N , +, .), suppose B E EE, c # 0 , and that B is tangent i o ezactly one circle A E EE \ { B } and intersects ezactly one D E EE \ {A, B } in ezactly two places. Then an impossibility has occurred!

Proof. If A = Nor + a , B = Nor+ b, and D = Nor+& we have p n D l = 2 and we'll work for a contradiction. Let A n B = {z} and B n D = {y,z}. If 4d = b, then 4D = B and 49, 4z E q5D n dB = B n t$B. Now q5B # B , 4B E E,', and 1B n 4BI = 2. Hence

But by 4.1, A n B = { 0 } and N * c = N*(- r ) . This puts 0 E D, and so 0 E {y, z } and we cannot have 4 y = z and $2 = y. This completes the proof of proposition 4.2 and eliminates c2) as a viable possibility.

4B = D and SO q5' = 1 with # 1. Also {4z,+y} = { y , ~ } , and so 4y = z and 4. = y.

The remainder of this section is devoted to the study of case c l ) , so assume B is tangent to each of the circles A and D. It will be convenient to modify our notation slightly. Let B1 E EE be tangent to Bo, B2 E E,'. So we are assuming each Bi E E,' is tangent to exactly two other circles in EL. Thus B2 is tangent to B1 and to a unique B3 E El \ { B1, Bz} . Similarly, Bt is tangent to B2 and to a unique Bc E EE \ { B2, B3). Since N is finite, there is a least s so that if Bs-2 and B, are the only two circles of E,' which are tangent to B,-1, then B, E {Bo, B l , . . . , Bs-3}. If B, = Bj , 1 _< j _< s - 3, then Bj-1 , Bj+l , and B,-1 are tangent to B,. Since this cannot be, we must conclude that B, = Bo. Summarizing, we get

Theorem 4.3. For a finite circular planar nearring ( N , +, .), if a B E EE, c # 0 , is tangent to ezactly two other circles in E,' \ { B } , then there is a smallest integer s 2 3 with circles Bo, B1,. . . , B,-1 E E,' with the property that B,-1 and Bi+l are the two circles in E,' which are tangent to B , . Here, i - 1, i, and a + 1 take their meaning in the integers modulo s, namely in the cyclic group (Z,, +).

Definition 4.4. For the circles Bo,Bl , . . . , B,-1 as in theorem 4.3, we write

or

B ~ ~ ~ B ~ ~ ~ ...

if Bi n Bj+l = {zj}, and we call it a Jimple closed s-link chain.

By choosing any Bi from a simple closed s-link chain Bo * B1 . . . B,-1- Bo to begin with, one would obtain the same set {Bo, B1,. . . , B.-1} of circles. Likewise, starting with B1, say, with its two tangent circles Bo and B2, one could interchange the roles of Bo and B2, i.e., one could have used B2 for Bo and BO for B2. For these reasons we identify the following simple closed s-link chains as one and the same:


Again, the indices are taken from 2, and i + 1, i + 2, i - 1, etc., all take their meaning in this group.

Theorem 4.5. For an El, c # 0, of a finite circular planar neam'ng ( N , +, suppose a B E El intersech ezactly two other circles A, D E E,' \ { B } , and suppose B U tangent to each of these. Let B = B1 and Boa B I - . . .. B,-1. Bo be the resulting simple closeds-linkchain, w i t h B i n B , + l = { t i } , B , = N ' r + b , , a n d { b o , b l , ..., b , - l } C: N'c. Then: 1) there U a cyclic subgroup (4) of @ of order s such that $2, = I ;+] , 4 b i = b;+l, and $B, = B,+l f o r each i E 2,; 2) there i s a bijection between the cosets X(4) of (4) in and distinct closed s-link chains in EL; 3) there are ezactly [a : (4)] simple closed d i n k chains in EL; 4 ) if D E E,' \ { B o , B 1 , ... ,B,-1}, then D does not intersect any B,, 0 < i < s - 1, and D yields a simple closed s-link chain D O . D1 . . . . - D,-1 - D o with D = D O and each D, E E,'.

Proof. Choose 4 E @ so that dbo = bl. Then 4B1 is tangent to q5Bo = B1, so = bo, so d a = 1 and

Now take D E EL \ {Bo, B1,. . . , B8-1} and X E 0 where XBo = D. With XBi = D , ,

@31 = Bz, q5bl = bz , and dso = 2 1 . Continuing, we have qY # 1 if 1 5 i 5 s - I, and also 1).

we have that

As0 A t 1 XZ,-Z x5,-1 Do D1 ... Da-1 DO

is a simple closed s-link chain in EL. If D intersects some B,, then D would be tangent to B, and hence D E {Bi-l, B;+1}. This gives us 4). Also X 4 ($), so we associate the coset X(4) with D ~ . D ~ . . . . . D , - ~ . D o . WithX E @\(4),weget XBo.XB1. .... XB,-i.XBo as a simple closed s-link chain in EL, so

is surjective. Finally, if XBo . XB1 + . . . - XB,-l . XBo is the same simple closed s-link chain as

pB0 - p B 1 - . . . - pB,-l p B 0 , then XBo = p B , for some i, which means that p-'XBo = B, = 4'Bo. Hence p-'X = 4i E ($), so p ( 4 ) = A(#). Now we have 2) and 3).

Corollary 4.6. With the hypotheses of theorem 4.5, suppose DO . D1 . . . . . Dt-I . DO i s a simple closed t-link chain in E l . T h e n s = t .

Theorem 4.7. Let ( N , +, a ) be a finite circular planar nearring and consider an ELl c # 0, from (N, B', E). T h e n the following two conditions on El are equivalent. I. There i s a B E E,' which intersects ezactly two other circles A, D E E,' \ { B } and B is tangent t o each of these. II. There i s a n integer 8 so that the graph of El consists of ezactly l@l/s disjoint cyclic subgraphs. Any such subgraph has vertices bo, b1 , .. . , b,-1 and the only edges are odd edges, one between each bi and b,+l.

100 J .R . Clay

5. THE COMPOUND CLOSED s-LINK CHAINS.

In the last section, we studied subcases c1) and c2) of case c). In this section, we study subcase c3). We have a B E E,', B intersects exactly two other circles A, D E E,' \ { B } , and B intersects each of A and D in exactly two places. One must consider three possibilities:

~3.2. I(An B) n ( B n D)l = 1; ~3.3. ( A n B ) n ( B n D ) = B . We proceed to show that c3.1 cannot happen, that c3.2 can only happen in very strict and limiting circumstances, and that when c3.3 happens, one gets results analogous to those of theorems 3.1 and 4.5.

~3.1. A n B = B n D;

Lemma 5.1. For a finite circular planar neam'ng (N,+,.), suppose B E El , c # 0, intersects ezactly two other circles A, D E EZ\{B}, each at two distinct points. Suppose A n B = B n D, that A = N*r +a , B = N*r + b, and D = N*r + d . Let € cf, satisfy q5a = b. Then d2 # 1.

Proof. Suppose d2 = 1. Now dA = B and 4B = A. There is also a X E cf, such that Ad = b, so AD = B . If B n D = { bl , b 2 } , then { Xbl , X b 2 } C X D n XB = B n XB. This puts XB E {A, D } . If XB = A, then Xb = a = 4 b and so X = 4 and X2 = 1. If XB = D, then X b = d and X2 = 1 in this case also.

With AnB={al,az},wehaveq5al,4a2 E d A n 4 B = B n A = { a ~ , a z } . Wecannot have q5al = a], so 4al = a2 and 4a2 = a1. Similarly, Xbl = b2 and Xb2 = b l . From A n B = B n D , we have {al,a2} = { b l , Z Q } . From Xul = 4al = a2 and Xa2 = 4a2 = a ] , we get 4 = X. So A = d B = XB = D, a contradiction.

Proposition 5.: f o r a finite circular planar neatring ( N , t,.), suppose B E El, c # 0 , intersects ezactly two other circles A, D E,' \ { B } , each at two &$tinct points. Then A n B # B n D.

Proof. Suppose A n B = B n D. Then any one of {A, B, D} has the other two as its circles in E,' which intersect it twice. Since the 4 E 9 such that 4 A = B has 4' # 1, we havet$A= B , #B = D, and 4D = A. If A = N ' r+a , B = N'r+ b, and D = N * r + d , we also have $a = b, $4 E d, and $ d = a. Hence $3 = 1. For A n B = {a], az} , we get 4a1, #a2 E 4A n #B = B n D = { b l , b} = ((11, a ~ } , since A n B = B n D. We cannot have 4al = 01 so 4al = a2 and #a2 = a]. F'rom this we get 42 = 1, a contradiction.

Having eliminated c3.1, we now attack c3.2.

Lemma 5.3. Let (N, +, .) be a finite circular planar nearring. Suppose an E,', c # 0 , has a B E E,' which intersects ezactly two other A , D E E,' \ { B } each in ezactly two places, and that I(A n B ) f l ( B n D)l = 1. If 4 E @ aatisfies #A = B, then dB = D, 4D = A, and #3 = 1. Also A n B n D = {O}, lE,'l= I @ \ = 3, and E: = EL,.

Proof. First, notice that any one of {A, B, D} intersects the other two, so each has the other two as the circles in E,' which intersect it twice. Let A = N*r + a, B = N*r + b, D = N * r + d w i t h A n B = { a l , a 2 } , B n D = { b l , b 2 } , a n d D n A = { d l , d z } . Alsolet a2 = b2 = d2 E A n B n D. Now 4al,$a2 E # A n 4B = B n $B, so 4B E { A , D } .


If QB = A, then {4a1,4az} = {a l ,a2} . If 4al = 0 1 , then 4a2 = a2 and # = 1. Hence 4al = 0 2 and 4a2 = al. With a1 = 4a2 = 4b2 E 4B n #D = A n $0, we have 4D E {B, D}, and since 4D # D, we get 4D = B. Hence 4d = b = 4a. This contradiction tells US we cannot have 4B = A.

If 4B = D, then 4 b = d and 4al ,4a2 E 4An4B = B n D = { b l , h } . So {#al ,#az} = {bl,h}. If 4a2 = b2, then 401 = i l . F'rom 4az = b = 0 2 and (b # 1, we conclude that 0 2 = = dz = 0 E A n B n D and SO A n B n D = (0). SO if X E E l , p E a, and p A = X, then 0 E X. But A intersects only two other circles in E:, so X E {A, B , D}. Hence lJ3ll = 191 = 3. Also 0 E N'r + a , with a E N*c, implies a E N * ( - r ) , so N*c = N*(-r) and E,' = ELr. Also 0 = 4bz E {$b,t$b,} c 4B n 4 0 = D n 4D. SO 4D E {A, B } . If 4D = B, then q52 = 1, 4A = B, and 4B = A, a contradiction. Hence 4D = A and d d = a , so 43 = 1.

Finally, consider what happens when 4B = D and 4a2 = i 1 . First of all, #a1 = bz = az. But then 42al = 4a2 = bl. With 4B = D, we have the same argument as above. With +bl, t#& E 4B n 4D = D n dD, we conclude that 4D E { A , B } . As above, 4D = B cannot be, so we take 4 D = A. With 4b,,4b2 E D n A = {d l ,d2} , we get {#bl,db2} = {d l ,d2} . If 4b2 = d2, then a2 = b = dz = 0, as above. So the case where #bz = dl and q5b1 = d2 must be considered.

From 4al = a2 = bz = d2 and #bz = d l , we have 4a2 = bl and 4bl = d2 = bz = az. Hence #a1 = a2 = bz and #2al = 4b2 = d l . We shall conclude that a1 = d3al = ddl = a2. WithDnA={dl ,dz},weget 4dl,#d2 ~ 4 D n t $ A = A n B = { a l , a z } . I f#dz=az , then a2 = = dz = 0 as above. So #d2 = al and 4dl = a2 , as promised. Now we have a1 = d3al = az , a contradiction.

So the only possibility is with a2 = b2 = d2 = 0, 4al = bl , and #dl = al. This puts 0 E X if X E EE, so E: = {A,B,D} and A n B n D = { 0 } , along with # B = D, 40 = A, and d3 = 1.

Suppose (N, +, a ) is a planar nearring with llpl = 3. What else is required for an ELr to satisfy c3.2? Let A = N'r - r = {O,4r - r,#2r - r } , B = N'r - Or = {r - #r,0,q52r - &}, and D = N'r - d2r = {r - d2r,4r - d2r,0}. If A n B = {O,z}, then one of the following cases must be valid: (i) #r - r = r - 4r; (ii) 4r - r = #2r - 4r; (iii) d2r - r = r - #r; (iv) d 2 r - r = 4% - 4r.

If (i) is true, then qir-r is an element of order 2. If (ii) is true, then qk-r = d [ # r - r ] . But 4 # 1, 60 dr - r = 0 and 4r = r. This means that r = 0, which cannot be. So (ii) is eliminated. If (iii) is true, then q52r - r = 43r - q5r = t$[Qr - r]. As with (ii), this would imply r = 0, and now (iii) is also eliminated. Finally, if (iv) is true, then -r = -dr, which cannot happen. So (iv) is eliminated. Our only chance is with an r so that q5r - r is of order 2.

If 4r - r is an element of order 2, then 4r - r = r - dr. Hence A n B = (0, 4r - r}. Also 42r - 4. = dr - 4zr, so B n D = (0, 42r - $r}. Finally, 43r - d2r = 42r - d3r , or r - d2r = d2r - r. Hence A n D = (0, c$2r - r). Summarizing, we have

Let N'r = { r , 4 r , 4 2 r } , so N * ( - r ) = { - r , - # r , - # 2 r } .

Theorem 5.4. Let ( N , +, .) Be a planar nearring with I@]= 3. T h e n there i s a n EL, in (N, B', E) which satisfies c3.2 if and only if 4r - r i s a n element of order 2 for some 4 E a .

Theorem 5.4 motivates the following problem.

I02 J.R. Clay

Problem 5.5. What are the planar nearrings ( N , + , - ) with ICPpI = 3 and with an element r E N such that 4r - r is an element of order 2 for some 4 E CP? We can restate this problem in the language of group theory [B & C]. What are the groups (G, +), not necessarily abelian, having a group @ of fixed point free automorphisms, with 191 = 3, having the property that -1 + X is surjective for each X E @ \ { l}, and for which there is an r E G such that $r - r has order 2? As shown in [B & C], a solution to one problem is a solution to the other. Because of theorem 5.4, a solution will also characterize all circular planar nearrings which satisfy c3.2.

We proceed now to provide only a partial solution to problem 5.5, and to indicate that we do not have the complete solution. We will provide a complete solution where (G, +) has the property that z + z = 0 for each z E G. Then we will give examples to show that this is not a complete solution by considering some nonabelian groups of order 64.

With the influence of theorem 5.4, it is natural to look at a group (G,+) whose elements satisfy z + z = 0 for each 2 E G. Of course, such a group is abelian. Since 3 must divide IGJ - 1, if [GI < 00, we must have IGI = 22". Such groups are additive groups of a field (G, +, a ) which have multiplicative subgroups = { 1, a, u2} for some a. If one constructs a planar nearring (G, +,o) from (G,CP) [C2], then for any r E G \ {0}, we have G'r = G * ( - r ) = {r,ar,a2r}. So, A = G'r + r = (0 ,ar + r,a2r + r } , B = G'r + ar = {r + ar,0,a2r + ar}, and D = G'r + a2r = { r + a2r,ar + a2r,0}. With A n B = {O,ar + r}, B n D = {0,a2r + a r } , and D n A = (0 , r + a2r}. Hence (G, +, 0 )

is a planar nearring and for each 0 # r E G, EL, satisfies c3.2. This gives us

Theorem 5.6. Let (G, +) be 4 finite group with the property that I + x = 0 for each 5 E G . Then (G,+) i3 the additive group of a planar nearring (G,+,o) with a @ of order 3 if and only if IGI = 2=" for some integer u. For such nearringcr, if 0 # r E G , then the circle3 in EL, JatiJfy condition c3.2.

Continuing with an infinite abelian group ( G , + ) with x + 5 = 0 for each x E G, we know that (G, +) is a vector space over the field 2 2 of two elements, and G has an infinite basis B = ( 1 1 ~ 2 2 , . . .} which we have indicated with a well ordered index set

extend 4 linearly, and let an arbitrary element of G be (1 < 2 < . * * I . Define 45,i+2ktl = 5CYi+2&2, and 4%+2k+2 = 5,i+2k+1 + X,,+Zk+2r

n

6, = C[aai+2k+1zoi+2k+1 + Qai+2tt2Zai+2k+21

4Y = C",i+2C+1Zai+26+2 + Qai+2k+2(5Pi+2k+1+ ZcYi+2k+2)1

= ~ [ a o i + 2 k + 2 z o i + 2 k + 1 + ( Q 0 i t l k t l + Qai+2k+2)~ai+2kt21

i=O

where the a, are limit ordinals or 0. If 4 y = y, then from n

i = O

n

i s 0 n

= c[aai+2k+1Zai+21.+l + Qai+2t+2Zoi+21+21 = I/, i=O


we have (aai+2k+l,aai+2k+2) = (aa;+2k+2,aa;+2k+l + aa,+2k+2)- Hence, each %,+j = 0, and so y = 0. This makes 4 fixed point free.

Now #3zai+2k+1 = 5a;+2k+1, and SO 4 3 z a i + 2 k + 2 = 43#Xa;+2k+~ = & $ 3 X a i + 2 ~ + ~ = 4Zai+2k+1 = Zai+2k+2. so d3 = 1 and 42 is dso fixed point free.

So that we can make a planar nearring from the pair (G, 9), where 9 = {1,4, +'}, we must show that -1 + 4 and -1 + tj2 are surjective. For this, it is sufficient to show that the image of B is a basis for each of -1 + 4 = 1 + 4 and -1 + 4' = 1 + 4. Since the proof is direct and easy, we shall not present it here. Summarizing, we get

Theorem 5.7. Let (G,+) be an infinite group with the property that t + t = 0 for each z E G. Then G i s the additive group of a planar nearring ( G , +, 0 ) with a cf, of order 9, and if 0 # r G , then the circles in ELr satisfy condition c3.2.

Towards solving problem 5.5, an easy application of theorem 5.4 tells us that a suitable finite N must have an element of order 2, and that 3 must divide IN1 - 1. Hence IN1 = 6u + 4 for some nonnegative integer u. Wen-Fong Ke and K. S. Wang, in their study of the groups of order 64, have shown the existence of planar nearrings ( N , +, .) where IN1 = 64, ( N , + ) is nonabelian with 191 = 3, and with elements q5r - r of order two for some 4 E 9 [K & W].

In fact, of the 267 groups of order 64 [H & S], exactly 7 can be the additive group of a planar nearring [K & W], and exactly 5 of these 7 have a group of automorphisms cf, of order 3 and have an element r such that dr - r is of order 2. Three of these 5 are abelian and two are nonabelian. Of course, theorem 5.6 tells us that the additive group of the field of order 64 is one of the three abelian groups. The other two abelian groups are 22 @ 2 2 @ 2 4 @ 24 and 28 @ 2 8 . The nonabelian groups are groups number 183 and number 187 on page '73 from [H & S].

Finally, we devote o w attention to case c3.3. Let's review what we are dealing with. We are concerned with finite circular planar nearrings ( N , +, .) having an E:, c # 0 , with a B E EE that intersects exactly twoother circles A, D E EL, and IAnBl = lBnDl = 2 with ( A n B ) n (B n D ) = 0. From 4., every circle G E El has the same property. It will be convenient for us to change our notation as we did for developing theorem 4.3.

Suppose B1 E E,' intersects Bo,B2 E E,' \ { B l } . Then B2 also intersects two circles B1, BJ E E,' \ {Bz } , each at two places, and (BI f l B2) n (B2 n B3) = 0. Repeating this for BJ, etc., there is a least s so that if B,-2 and B, are the two circles in EE \ { B s - l }

which intersects Be-] in exactly two places, then B, E {Bo, B1,. . . , Ba-3}. As with the proof of theorem 4.3, B, = Bo. We denote this by

Bo : Bl : B2 : - * * : B.-] : Bo,

and if B, n B,+l = {a i , c i } , we write

a0 a1 02 03-2 a,-1 Bo : B1 : B2 : 1 . . : B,-l : Bo

co c1 c2 Ca-2 c.-1

and let the subscripts have meaning in the additive cyclic group (Z,, +). In analogy to theorem 4.3, we have

Theorem 5.8. For a finite circular planar nearring ( N , +, a), suppose there is an El,. c # 0 , with a B1 which intersects ezactly two other circles Bo, Bz E EE \ { B l } , wheie

I04 J .R . Clay

IBonB1I = IB1nB21 = 2, and where (BonBl)n(Bl nB2) = 0. (That is, condition c3.3 i s satisfied.) T h e n there b a smallest integer s 2 3 with circles Bo, B1 ,.. . , B,-1 E E,' with the property that Bi-1 and Bi+l are the circles in El \ { B,} each of which intersect B, in ezactly two places, and (Bi-1 n B,) n (Bi n B,+l) = 0. Bere, the subscripts take their meaning in the additive cyclic group (Z,,+).

Definition 5.9. For a finite planar nearring (N, +, .) which meets the hypotheses of theorem 5.8, we call

Bo : B1 : B2 : ..- : Bl-l : Bo

from an Eg, c # 0, a compound closed s-link chain.

As with simple closed s-link chains, we identify the following as one and the same compound closed s-link chain:

Bi-1 : Bi : Bi+l :

Bz : B1 : Bo : B.-l : * - a : B3 : B2;

Bit1 : Bi : Bi-1 : * * * : Bi+2 : Bi+l.

- - : Bi-2 : Bi-1;

Compare the following theorem with theorems 3.1 and 4.5.

Theorem 5.10 Let a nearring (PI,+,.) satisfy the hypotheses of theorem 5.8. Let Bo : B1 : B2 : * * - : B,-1 : Bo be a resulting compound closed s-link chain i n E l . Suppose each Bi = N'r + bi, and Bi-1 n Bi = { a i l c i } , where { b o , b l , . . . , b n - i } C N'c, and suppose 4 E 1) there i s a cyclic group ($) of @ of order s 2 3 J U C ~ that qlai = a,+], dbi = bit17 4c; = c,+l, and dB, = B,+l; 2) there is a bijection between the cosets X(4) of (4) in @ and distinct closed s-link chains in El; 3) there are ezactly [@ : (4)J compound cloJed s-link chains in El; 4 ) if D E El\ {Bo, B1,. . . , L3,-1}, then D does not intersect any Bi, 0 5 i 5 s - 1, and D = DO yields a compound closed s-link chain Do : D1 : D2 : . . . : D,-] : Do with each Di E E l .

Proof. From 4bO = b l , we have ~ B O = B1. Hence q h o , q5co E q5Bo n 4B1 = B1 n 4B1. This means dB1 E {Bo,B2}. If dB1 = Bo, then weld have d2 = 1. So 4B1 = Bz,

= &, and { ~ a o , ~ c o } = {al ,c l} . Without loss of generality, we may take duo = a1

and 4c0 = c l , renaming the a1 and c1 if necessary. Having 4Bi = Bi+l, 4bi = bi+l, dai-1 = ail and 4c,-l = ci, we can use the above

argument to conclude that dBi+l = &+2, 4bi+l = b,+2, &, = a,+l, and $c, = c,+l.

This makes 4' # 1 for 1 5 i 5 s - 1 and d1 = 1, and gives us 1). If (4) = a, we are finished. If (4) # 9, then (4) has another coset X ( 4 ) in a. With

such a X E *, we define Di = M i . Since h i E XBi nXB;+I, we have, with xi = Xa, and yi = Xb,, a compound closed s-link chain

satisfies 4bO = bl and q52 # 1. Then:

5 0 z1 X I - 2 X,-1 Do : D1 : * * a : D.-1 : Do

Yo Y1 Y l - 2 Y.-1


in EL. If some D 4 {Bo, ..., B8-1}, then there is a X E @ such that XBo = D. Let DO =

D = XBo and conclude that with D, = XBi, we have Do : D1 : ... : D8-1 : DO as a compound closed s-link chain in EL associated with the coset A($) in @. Also, D cannot intersect any B;, 0 5 i 5 s - 1, for if so, then D E (Bi-1, B ,+ l } . This gives us 4).

Suppose that

XBo : XB1 : * * a : XBa-l : XBo

and

pBo : pB1 : * - * : pBa-l : pBo

are the same compound closed s-link chain in EL, with X,p E @. Then XBo = pB, for some i. Hence Xbo = pb , and p-'Xbo = b, = 4'bo. Hence p ( 4 ) = A($). 'we now have 2) and 3), and this completes the proof of theorem 5.10.

Corollary 5.11. With the same hypotheses as theorem 5.9, we have that i f

Bo : B1 : B2 : ... : Be- ] : Bo

and

Do : D1 : D2 : . . . : Dt-1 : Do

are compound closed s- and t-lrnk chains, respectively, in EL, c # 0 , then s = t

We have seen in the introduction that Ei for ( Z 3 1 , B z , E ) has two compound closed 4-link chains.

There remains to consider what happens when all the hypotheses of theorem 5.10 are satisfied except 42 # 1. Can we really have such a 4' = l ? Since Bo E EL is arbitrary, we would really need 4: = 1 for each 2 , 0 5 i 5 s - 1, where 4,B1 = B,+1. For if some 4; # 1, then one could replace Bo and B1 with B, and B,+l, respectively. Now 4, assumes the role of 4 in theorem 5.10, and with 4: # 1 we have that theorem 5.10 is valid. Hence each 4, = d,, so d2 = 1. With dZ = 1, we have each 4: = 1 where 4,B1 = B,+l , and consequently 4 ,b , = Now 4 = 4 0 . If 40 = 41, then from 4; = 4; = 1, we have dObo = bl = &b* and so bo = &b1 = b2 which implies Bo = B2 But BO # Bz, so we cannot have 40 = 41.

On the other hand, a 0 from a planar nearring ( N , +, .) cannot have two distinct elements of order 2. For if 4 E @ and 4* = 1, then since -1 + 4 is a bijection on N ,

which means 4 y = -y, and this defines a 4 E 0 \ (1) where 4z = 1, so there cannot be two such elements. In summary, we have

eachy ~ N h a s a n z ~ N w h e r e y = - s + d z . S o y + q 5 y = ( - z + 4 ~ ) + 4 ( - z+4z) = 0,

Theorem 5.12. Suppose the hypotheses of theorem 5.8 are sati3fied. Let ~ B O = B1 with 4 E @. Then dZ # 1. Bence theorem 5.10 applies.

We finish this work with a summary of case c3) in terms of the graph of EL.

Theorem 5.13. Let ( N , +, -) be a finite circular planar nearrtng with corresponding @, and consider an EL, c # 0 , from ( N , B', E). Then the following condtttons on Er

I06 J.R. Clay

are equivalent. I. There i s a B E E l which intersects ezactly two other circles A, D E E,' \ { B } , and B intersects each of these in two places. II. Either:

I) 191 = 3 and the graph of E,' U A, Or b) > 3 and there M a divisor s > 1 of 1 @ 1 and l@l/s disjoint subgraphs of the graph of EL, and any auch aubgraph has ezactly s vertices b,, b l , . . . , b,-l and the only edges are even edges, one hetween each b; and b,+].

I f I. occurs, then either condition c3.2 or c3.3 i s satisfied.

6. REFERENCES.

[B & C] BETSCH, G. & CLAY, J., Block designs from Frobenius groups and planar near-rings, Proc. Conf. Finite Groups (Park City, Utah). Academic Press, New York, 1976,473-502.

[Cl] CLAY, J., Circular block designs from planar nearrings, Annals of Discrete Mathematics, 37 (1988), 95-105.

[C2] CLAY, J., Applications of planar nearrings to geometry and combinatorics, Re- sults in Mathematics, 12 (1987), 71-85.

[C & Y] CLAY, J., & YEH, Y.-N., On some geometry of Mersenne primes.

[H & S] HALL, M., Jr. & SENIOR, J. , The Groups of Order 2" (n 5 6), The Macmillan Co., New York, 1964.

[I< & W] KE, W.-F., & WANG, K. S., On the Frobenius groups with kernel of order 64, Contributions lo General Algebra, 7 , (1991), 221-233.

[MI MODISETT, M., A characterization of the circularity of balanced incomplete block designs, Utilitas Math. 35 (1989), 83-94.


p-primitive semifield planes

M. Corder0

Texas Tech University, Department of Mathematics, Lubbock, Texas 79409-1042, USA

1. Introduction In this work we present a study of the class of semifield planes of order p" and lier-

ncl GF(p2) with the property that they admit a p-primitive Baer collineation; these are called pprimitive semifield planes . This is the class of planes obtained when the construction method of Hiramine, Matsumoto and Oyama [13] is applied to the Desar- guesian plane of order p 2 . Equivalently, this is the class of planes that admit a matrix snread set of the form

where f is an additive function in GF(p2) such that upt1 # vf (u ) for any u ,v E G F ( p 2 ) with (u,v) # (0,O).

In Section 2 somi' properties of p-primitive semifield planes are studied. First the function f that defines the matrix spread set of a p-primitive semifield plane is analyzed. The behavior of pprimitive semifield planes under several operations is considered iiex t. It is showii that if x is a p-primitive semifield plane and if rt denotes thc transpose plane then A 2 T ' . .&o we show that if a semifield plane and its dual are p-primitive then the plane is a IGiuth four-type semifield plane of all four types. Then we give the autotopism group of p-primitive semifield planes and show that this group is solvable. The autotopism group of all the known semifield planes is solvable.

In Section 3 we classify the p-piimitive semifield planes. First necessary and sufficient conditions for isomorphism of p-primitive semifield planes are given. Then the nuinher

of nonisoinorphic p-primitive semifield planes is shown to be ( p : - for every prime

p > 2. Of these, ' - + are Knuth four-type semifield planes, one is a Dickson semifield

p - l o r - are Boerner-Lantz semifield planes (according as -1 is a square 4

plane and - or a nonsquare in G F ( p ) , respectively, p > 3. For p = 3, the Boerner-Lantz semifield plane of order 81 is p-primitive.) Each of the remaining planes is either a Generalized twisted field plane or is a new plane.

In this article wc present a survey of the major results on the study of p-primitive scmifield planes which we began on [5] and continued on [6] - [lo]. In some instances we

2 f' - 3

4

I08 M. Corder0

shall give the full proof of a statement while in other cases only a sketcli is given with refercnce to a source where the proof can be found.

2. Properties of p-primitive semifield planes Let x denote a semifield plane of order q2 and kernel K Z GF(q) where q is a prime

power pr. A p-primitive Baer collineation of x is a collineation u which fixes a Baer subplane of x pointwise and whose order is a p-primitive divisor of q - l(z.e.1 0 1 1 q - 1 but I u 1 f p' - 1 for 1 5 i < r ) .

A semifield plane of ordcr p4 and kernel GF(p2) , where p is an odd prinic is called a p - p r i m i t i v e semi f ie ld p l a n e if i t admits a p-primitive Baer collineation.

We now describe the construction method of Hiramine, Matsumoto and Oyama[l3] (see also Johnson[lG]) by whirh trailslation planes of ordcr q4 and kernel GF(q) ,q = p', admitting p-primitive Baer collineations are obtained from arbitrary translation planes of order q2 and kernel GF(q) .

GF(q) with matrix spread Let x denote a translation plane of order q2 and kernel set

where g ,h are mappings from K x K into K. Let F = G F ( q Z ) 3 K. Take an elemcnt, t E F - K with t2 E K and define a mapping f : F -i F by f(x + y t ) = g(z, y) - h ( z , y ) t for x, y E K. Then

{ [ f;u) ;J : 2 1 7 2 1 E GF,,.,)

represents a matrix spread sct of a translation plane, ~ ( f ) , of order q4 and ltcrnel G F ( q 2 ) that admits a p-primitive Baer collineation. Moreover, any translation planc? of order q 4 and kernel GF(q2) with this property may be obtained with the above construction. Hiramine, et al, [13]. Notice that the function f is additive on F if and only if the functions g and h are hiadditive in K x K ; hence x(f) is a semifield planc if and only if x is a semifield plane. Thus if ~ ( f ) is a p-primitive plane theii x is a semifield plane of order p z ; hence x is Dcsarguesian. Therefore all the p-primitive semifield planes are ol)tained applying the construction method of Hiramine, h4atsumoto and Oyama to the Dcsarguesian plane of order p 2 .

Now we givc some propertics of the function f. Let x be a p-primitive semifield plane with matrix spread set

{ [ f("u, .".I Since f is an additive function in G F ( p 2 ) . f o , f l E G F ( p Z ) . (See e.g., Vaughn[2O]).

we have that f ( u ) = j U u f f l u P for some We shall denote this plane by x(f) or by

P-primitive semifield planes 109

~ ( f o , f1). In the following proposition we give conditions on the function f that give a matrix spread of a p-primitive semifield plane.

Proposi t ion 2.1 Let f : GF(p2) -+ GF(p*) be g iven by f ( u ) = fou + f lup where fo = a0 + ul t , f l = bo + b l t , ao, a l , bo, bl E GF(p) and let 0 be a nonsquare in GF(p) such that t2 = 0. T h e n f defines a m a t r i x spread se t of a p -pr imi t i ve semifield p lane T ( f ) i f and on ly if u i - (a: - bT)0 is a nonsquare in GF(p).

Proof: (See also 2.1 [S]). First we must have that the determinant of the difference of any two distinct matrices in the spread must be # 0; hence up+' # v f (v) for every u, w E GF(p2) such that (u, w) # (0,O). Since ups' E GF(p) for every u E GF(p2) we need y - vf(t7) # 0 for every y E GF(p),v E GF(p2) and ( y , v ) # (0,O). Expanding this inequality we get that 4ai - 4(al + b l ) (a l - b1)B must be a nonsquare in GF(p) . Hence we must have that ui - (a? -a:)@ is a nonsquare in GF(p). Conversely, if ao, a l , bl satisfy this condition and i f f is defined as in the proposition, then

is a matrix spread set of a p-primitive semifield planes. In the next proposition furtlier properties of the function f are studied.

Proposi t ion 2.2 Le t T( fa, f l ) be a p-pr imi t ive semifield plane. T h e n ,

(i) fo and f l cannot belong both to GF(p). In particular, if fa = 0 t h e n fl$GF(p).

( i i ) Iff, = 0 then fo is a nonsquare in GF(p) .

(iii) If fo # 0 and f l # 0 then f:+' # fp".

Proof: See 2.2 on [s].

In the cases when fo = 0 or f1 = 0 the p-primitive semifield planes T ( fa , f i ) belong to known classes of planes.

Proposi t ion 2.3 Let T ( fo, f1) be a p-pr imi t ive semifield plane.

(i) If fo = 0 t hen T is a K n u t h four- type semifield p lane of all f o u r types.

(ii) Iff, = 0 then T is a D i c h o n semifield plane.

Proof: If T = T ( fa, fl) is a p-primitive semifield plane with fo = 0 then the matrix spread set of T is of the form

110 M. Corder0

and the product in the corresponding semifield is given by ( 5 , y).(u, v) = ( z u f y f l w P , zv+ yup) for 5 , y, u, v E GF(p2) . Therefore A is a Knuth four-type semifield plane of all four types. (see [lS]).

If A = x(fo,fi) is a p-primitive semifield plane with f l = 0 then the product in the semifield is given by

This is the product in [ll,p.241] with (Y = /3 = 1 and u : z H xp and therefore x is a Diclcson semifield plane.

The next thcorem gives the nuclei of p-primitive semifield planes.

Theorem 2.4 Let x( fo f l ) be a p-primit ive plane and let N,,, NT, right and left nucleus, respectively. T h e n exactly one of the following holds:

be i ts middle,

( i) Nm = Nt = Hr 2 GF(p2) or

( i i ) Nm = Nr 2 GF(p) .

Moreover we have

( i) holds ++ fo = 0

(i) holds f o # 0.

Proof: The proof follows directly from the next two lemmas.

Lemma 2.5 Let x( f ) be a p-primit ive plane and let Nt be i ts l e f t nucle~us. Ni = {(n,O) : a E G F ( p 2 ) } .

Then

Proof: Let a E GF(p2) . By direct computation it can be shown that (a ,0 ) E Nt; so that { ( a , 0)) : a E GF(p2) & Nt. Since = p 2 we have the result.

Lemma 2.6 Let x(fo, f l ) be a p-primit ive semifield plan,e and let Nm and N, be i ts middle and right n u d e u s , respectively. T h e n N,,, = { ( n , 0 ) : fonp = n fo} = N,.

Proof: (See also [8 ] ) . Let ( n , m) E NnL. Then for every x l , z2, y l , y2 E G F ( p 2 ) we must l1ave

( ( z l , 5 2 ) ) ( n , ~ ~ ) ) ( y l , Y 2 ) = (21, 52)((n,mk/1, Yd).

Therefore the following two equations must hold for every y l , y2 E GF(p) .

P-primitive semifield planes 1 1 1

From (1) arid (2) it follows that if (n,rn) E Nm then rn = 0 a.nd fonp = fon. Conversely, if fOnP = n f o then (n,O) E N;, and therefore Nm = {(n,O) : fOnP = fon}.

By similar computations, we obtain N, = { (n , 0) : fonP = fon}.

Next we study the behavior of p-primitive planes under the operations of derivation, transposition aud dualization. The first general geometric process discovered for constructing new affine planes from given ones is the process of derivation which was invented by T. G. Ostrom. In derivation, a collection of lines in a given plane x, “the derivable net” is replaced by a suitsable collection of Baer subplanes in x to form a new affine plane.

Let x = x(f) be il p-primitive semifield plane. Then Hiramine et al, [13], showed that x is derivable. 111 particular the set of components of x contains the dcrivable net

5 = 0,y = x [ ; for every u E GF(p2) . The translation plane derived from replacing this net is not yprimitive; i t has order p4 and kernel GF(p)[l5].

Another construction technique, one which as investigated by Knuth, is the following: Let x be a translation plane with matrix spread set M = { M i } . Then taking the

transpose of each matrix M , in M , we obtain a matrix spread set M t = {Mf} which gives a translation plane, xt, called the transposed plane.

On the transpose of a p-primitive semifield plane we prove the following:

Theorem 2.7 L e t x = x(f) be a p - p r i m i t i v e semi f ie ld p l a n e a n d l e t xi d e n o t e t h e t r a n s p o s e p l a n e . T h e n d 2 x

for u,v E GF(p*). Then S = { M ( u , v ) : u,v E Proof: Let A l ( u , v ) =

G F ( p 2 ) } is the matrix spread set for ?r and St = { M ( u , w ) ~ : u , v E G F ( p 2 ) } is a matrix

spread set for n‘. Let A = [ :] . Then [ :] induces an isomorphism between

the two planes.

[ f 6 ) :p I If 7r is a semifield plane with associated semifield (S,+,.) then the dual plane x D ,

has associated semifield (S, +, *) where a t b = b . a for every a, b E S. It follows that “ ( x ) = n/,(xD) and N,(x) = Ne(rD). We use this together with (2.4) to get the following theorem.

Theorem 2.8 L e t x = r( fo , f i) be a p-pTamataVe semi f ie ld p l a n e . If t h e d u a l p h n e x D is also p - p r i m i t i v e t h e n x i s a K n , u t h f o u r - t y p e semi f ie ld p l a n e o f all f O U T types .

112 M. Corder0

9 =

Proof: Let T be p-primitive. Then N,(T) 2 GF(p2) and if 7rD is also p-primitive then Arf(rD) F G F ( p 2 ) . Since Ne(nD) = N,(r) g GF(p2) . Thus, Nf(n) g &(x) N,(r) by (2.4) and fo = 0. Therefore n is a Knuth four-type semifield plane of all four types Iiy (2.3).

0 1 -jl fl 0 !, h =

Let T = r( fo, f,) be a p-priniitive semifield plane. 111 the next theorem we give the autotopism group of K. For its proof see [7].

e = U

Theorem 2.9 Let T = n( fo, f l ) be a p-primitive .gemafield plane and let d ( n ) be its r x 0 1

1 0 O d

1 0 O d

a~utotopism QTO‘LLP. Let M ( x , y, w) = where x , y , w E G F ( p 2 ) - { O } .

e =

U

T h e n

0 1 fi” 0

1 0

O 1 1

-1 0 1

fl O J

( i i i ) if fi = 0 then d(r) = ( g , e ) . H where

r o 1

, e = U

1 0 0 1

fo 0 0 fop

P-primitive semifield planes I13

with u # 1 and H = ( M ( z , y , w ) : (z/y) '(P+') = 1 and (y/zw)P-- ' = 1). Id(n)I = 8(p2 - 1)'.

(av) if fo # 0 and f:(p+l) # 1, t hen d(T) =< g > .H where

with u = t f o and sP+l = f;+' - fop+' and H = ( M ( z , y , ~ ) : (z/y)'+' = 1 and (y/zw)P- ' = 1). Here Id(n)I = 2(p2 -

( v ) if fo # 0 and f:('--' = 1, t hen d(T) =< g , l > .H where g and H are as in ( iv) and e is given by

with 0 # 1 , v = $ and zP+' = fp-'. In this case, Id(n)I = 4 ( p 2 - 1)'.

Corollary 2.10 Let T ( fo, fl) be a p-primit ive semifield plane and let d(r) be i ts autotopism group. T h e n d(7r) is solvable.

Proof: Notice that the subgroup H of d(n) is a normal abelian subgroup of d(T) and d(a)/H is also abelian. Thus d(T) is solvable.

3. Classification of p-primitive semifield planes Let n( fo, f1) and n(F0, 8'1) be p-primitive semifield planes. The following theorem

determines necessary and sufficient conditions on the functions f = ( fo , f l ) and F = (F0,FI) for the planes ~ ( f ) and T(F) to be isomorphic. For the proof (see [GI) we studied all the bijective semilinear maps of one plane into the other (as vector spaces over their kernels), since n(f) and n ( F ) are isomorphic if and only if there exists a semilinear transformation

u[: 3 which sends a spread set of T ( f) onto a spread set of n ( F ) , where u is an automorphism of GF(p2) and A , B, C, D are 2 x 2 nonsingular matrices over GF(p2). The elation axis ( 0 , X ) is sent into ( 0 , X ) and we may assume that ( X , 0 ) is sent into (X,0) because, since T is a semifield plane, the elation group is transitive in the components not equal

I14 M. Corder0

to (0,s) ; thus D = 0 and C = 0. By analyzing the possibilities for A and 13 we obtain the result.

Theorem 3.1 Two p-primit ive semifield planes T ( f o , fl) and x ( Fo, F,) are isomorphic if and only i f one of the following is satisfied:

( i ) Fo = ucP-'fo and Fl = uf1

( i i ) Fo = ucp-l fOp and Fl = a f ;

for s o m e u E GF(p) - (0) and c E GF(pZ) - (0).

f o r some u E GF(p). In particular, x(0, f,) 2 x(FO, Fi) if and only if Fo = 0 and Fl = ufl O T Fi = uff

Using this result we call classify some of the p-primitivc semifield plaim. For the proofs of the following corollaries see [6].

Corollary 3.2 All the planes x(f0, f l ) with f l = 0 consti tute a71 isomorphism class

with - elements. T h e planes in this class are Dick3076 semifield planes. 2

Corollary 3.3 There u're p - + nonisomorphic p-primit ive semifield planes x ( f O , f l ) with fo = 0. The number of planes isomorphic to x( fo , f l ) is p - 1 if ff-' = -1 arid is 2 ( p - 1) i f fp-' # -1. A11 the planes x ( 0 , f , ) are K n u t h four-type semifield planes of

p* - 1

2

d l f O U T t y p e s .

Corollary 3.4 I f x(F0, FI) %' T ( fo, f 1 ) and F o = f o # 0, thelz Fl = 2~ f l O T PI = f f:. Conversely, if FO = f o # 0 and F1 = + f ~ or FI = +fp t hen ~ ( F o , FI) x( f o , f l ) .

Corollary 3.5 Let x = T ( fo, f l ) be a p-primit ive semifield plane with fo # 0 and fi # 0. T h e n the number of planes isomorphic to x is p z - 1 if f;('-l) = 1 a,nd is 3(p' - 1) if f:(p-l' # 1. The planes of Boerner-Lantz o order p4 are p-primit ive semzfield planes with fo # 0 and fl # 0, and f o r p > 3, f;"-"# 1.

Now we determine the number of nonisomorphic p-primitive semifield planes. The next proposition (see [5] for the proof) gives the number of possible fuiictions f such tl1at

is a matrix spread set of a p-primitive semifield plane.

Propositioii 3.6 FOT any prime p , p > 2, there are (p2 - *)' f unc t ions j such that

x ( f ) is a p-primit ive semifield plane. 2

P-primitive semifield planes 115

It follows from the proof of Proposition 3.6 that there are (p' - p ) [ "'1 p-primitive semifield planes ~ ( f o , f l ) with f i = bo + blt and bo = 0 and p - 1 with f o = 0 and bo = 0. If f1 = bo + blt # 0 and bl = 0 the condition is now ui - u:O nonsquare in GF(p)

and using Dickson [131 p. 461, we conclude that there are (p' - l)[A] p-primitive semifield planes with fl = bo E GF(p) , and consequently f o # 0. These remarks and 3.G are used in the proof of the following result, see [6].

2

2

Theorem 3.7 For a n y odd p r i m e p , there are ('q)' non i somorph ic p -pr imi t i ve

semifield p lanes of order p4.

Presently the following classes of proper (non Desarguesian) semifields are known: Dicltson semifields [ll , p. 2411, I h u t h four types [18] (these include the Hughes- Kleinfcld semifields [14]), Knuth of characteristic 2 [18], Kantor [17], Sandler [ lo] , Boerner-Lantz [3], Cohen and Ganley commutative semifields [4] and the two classcs discovered by Albert called twisted fields [l] and generalized twisted fields [2].

For p = 3 there arc four nonisoinorphic p-primitive semifield planes; two of these are I h u t h four-type semifield planes of all four types, one is a Dickson semifield plane and the other is the plane of Boerner-Lantz of order 81. For p 2 5 we say that a p-primitive scinifield plane is of tupe IV if fo # 0 and f;('-') # 0,1, and of t ype V if fo # 0 and f:('-') = 1. A ppriinitive semifield plane of type IV which is not a Bocrner-Laiitz seinifield plane and any plane of type V is either a Generalized twisted field plane or is a new plane. The distinction of these two cases is currently under investigation. The other classes of scmifield planes mentioned above could not have planes which are p-primitive semifield planes: the twisted field planes and Saiidler semifields planes are of dimension 4 over the left nucleus and the Knuth and Kantor semifields planes are of Characteristic 2. The multiplication in a semifield that coordinatizes a p-primitive semifield plane is not commutative and tlierefore it is not isoinorphic to the scmifields of Cohen and Ganley.

The following table summarizes our findings with respect to the number of non- isomorphic p-primitivt, semifield planes, the order of the autotopism group and the identification of the pliines for p 2 5 .

116 M. Cordero

pPrimitive Semifield Planes for p 2 5

Properties # of isomorphism .Order of Identification of f classes autotopism group of the class

fo = 0 1 S(p' - 1)2(p -I- 1) Knuth four-types fr-1 = -1

P A 4(p2 - 1)2(p + 1) Knuth four-types 2 fo = 0

fp-1 # -1

f i = 0 1 8(p' - 1)2 Dickson

2(p2 - 1)' Type IV ( P - l ) ( P - 3) 4 f o # 0

f;(P-l) f l (contains Boerner-Lantz)

f o # 0 P - 2 4(p2 - 1)' Type V f;(P-') = 1

References

[I] Albert, A.A. Finite Noncommutative Division Algebras. PTOC. Amer. Math. SOC. 9 (195S), 928-932.

[2] Albert, A.A. Generalized Twisted Fields. Pacif. J. Math. 11 (1961), 1-8.

[3] Boerner-Lantz, V. A Class of Semifields of Order q 4 . J . Geom. 27 (1986), 112-118.

[4] Cohen, S.D. and Ganley, M.J., Commutative Semifields, Two Dimensional over their Middle Nuclei. J. of Algebra 75 (1982), 373-385.

[5] Cordero-Brana, M. On pprimitive planes. Ph.D. dissertation, University of Iowa, 19s9.

[6] Cordero, M. Semifield Planes of Order p4 that Admit a pprimitive Baer Collineation. Osaka J. Math, 28 (1991), 305-321.

The Autotopism Group of pprimitive Semifield Planes, A R S Com- binatoria, to appear.

[71

P-primitive semifield planes I17

181 The Nuclei and Other Properties of pprimitive Semifield Planes, Inter. J . Math. 8 Math. Sci., to appear.

[91

P O I [ll] Dembowski, P. Finite Geometries. Springer, New York, NY 1968.

[12] Dickson, L.E. Linear Groups, with a n Exposition of the Galois Field Theory. Dover

A Note on the Boerner-Lantz Semifield Planes, J. Geom., to appear.

pprimitive Semifield Planes for p 5 11. (in preparation).

Reprint, New York, NY, 1958.

[13] Hiramine, Y . , Matsumoto, M. and Oyama, T. On Some Extension of 1 Spread Sets. Osaka J . Math. 24 (1987), 123-137.

[14] Hughes, D.R., and Kleinfeld, E. Seminuclear Extensions of Galois Fields. Amer . J. Math. 82 (1960), 315-31s.

[15] Johnson, N.L. Sequences of Derivable Translation Planes. Osaka J. Math. 25 (1988), 519-530.

[16] Johnson, N.L. Scmifield Planes of Characteristic p Admitting p-primitive Baer Collineations. Osaka J. Math. (to appear).

[17] Kantor, W.M. Expanded, Sliced and Spread Sets. Johnson, N.L., Kallaher, M.J. and Long, C.T. (eds.) Finite Geometries, Marcel Dekker, New York, N.Y., 1983, 251-261.

[18] Knuth, D.E. Finite Semifields and Projective Planes. J. of Algebra 2 (1965), 182- 217.

[19] Sandler, R. Autotopism Groups of Some Finite Non-Associative Algebras. Amer . J . Math. 84 (1962), 239-264.

(201 Vaughn, T.P. Polynomials and Linear Transformations Over Finite Fields. J. Reine A n g e w . Math. 262 (1974), 199-206.



The Fractal-like Steiner Triple System

Marialuisa J. de Resmini

Dipartimento di Matematica Universiti di Roma "La Sapienza" 1-00185 Rome, Italy

Abstract 19 (mod 24) there

exists an STS(v) containing a subsystem of order (v-5) /2 and (v+5)/12 STS(7)'s. More- over, these subsystems all share the same point. By repeatedly applying the construction, one can obtain STS's with interesting configurations of subsystems. In particular, one of the non-isomorphic STS(24(2" - 1) + 19), which contains 2sf2 - 1 S T S ( 7 ) Is, can be represented by a picture which reminds of a fractal, whence the title.

It is proven, in an elementary geometric way, that for every v

1 Introduction

It is well known that the existence of a subsystem T in a Steiner triple system S, of order v, affects the inner structure of S, in particular that of S \ T . This fact was used in [6] to prove the existence of an STS(e1) containing an S T S ( w ) as a subsystem for any admissible w. When T has order (v - 1)/2, S \ T is a hyperoval [2, 6, 8 , 91. If T has order (v - 3)/2, then the exterior blocks to T partition the points of S \ T and for every 2) 9 (mod 12) there exists an S T S ( v ) admitting a subsystem of order r - 1 = (v - 3)/2

Thus, it seems interesting to look at the next case in a more elementary and naive way than that in [6]. More precisely, we look for STS(v) 's containing a subsystem of order r - 2 = (v - 5)/2 and prove, by constuction, that for every v 19 (mod 24) there exists an STS( u ) containing a subsystem of the above said order. Our construction yields an ,STS(v) which also contains ( u + 5)/12 STS(7)'s. Furthermore, these subsystems all share the same point.

Provided that the arithmetic conditions are satisfied, one can repeatedly apply the construction. This results in non-isomorphic STS's with interesting configurations of subsystems. In particular, non-isomorphic STS(24(2' - 1) + 19) can be constructed which contain 2rf2 - 1 subsystems of order 7 (sect. 5). The structure of one of such systems suggested the title.

7 (mod 12)) is obtained for Steiner triple systems, but it is not a new one.

P, 61.

Obviously, as a by-product, a v + 2v + 5 construction (v

120 M . J . De Resmini

We assume that the reader is familiar with the Steiner system terminology and refer him/her to [l, 6, 71 for background and to [5] for more literature on the subject.

2 Preliminaries

Assume S is an STS(o ) containing a subsystem T of order w = T - 2 = ('u - 5)/2. Then easy calculations show that o = 7 (niod 12). Thus, 'u = 1271 + 7 implies w = 672 + 1. Next, we distinguish two cases according to n being even or odd. Obviously, 11 = 2n1

implies w = 12n7 + 1 and 'u = 24ni + 7 , whereas n = 2m + 1 yields w = 12m + 7 and 2) = 24m + 19.

In sect. 3 we give a construction which shows that for every o = 19 (mod 24) there exists an STS(o) , S, containing a subsystem T of order w = ( u - 5)/2. In [4] a similiar result is proven for 'u

Denote by t j the number of j-secant blocks to T , j = 0 , 1 , 3 . By standard counting arguments,

7 (1ri.od 24) (see also [S]). From now on, o = 241n + 19.

t o = 4(2172 + 2) , t i = 3(2n2 + 2)(121?2 + 7), t 3 = (12m + 7)(2m + 1).

Moreover, the number of exterior points to T is 'u - w = 6(2m + 2) and there are two exterior blocks on each point of S \ T .

The values for 20 and v - w suggest a nice configuration of the exterior points to T together with the exterior blocks. Namely, we split the exterior points and blocks into 2111 + 2 quasi-Fano configurations, or Pasch configurations, or quadrilaterals, i.e. four triples whose union has cardinality six. See picture:

The points of all future quadrilaterals will be numbered as in this picture. Such a configuration will be referred to as a quasi-Fano and we prefer this terminology since, by adding a suitable point, we shall complete to triples the three remaining pairs, i.e. 1 6 , 2 4, 3 5, and obtain an STS(7) , i.e. a Fano plane.

3 The Construction

In order to construct an S T S ( v ) = S, o = 24m + 19, containing a subsystem T of order w = 12n7 + 7, take as 1' any STS(UJ) on the points A l l A2, . . . , Aw. Next, partition the u - w = 6(2m + 2) points of S \ T into 21n + 2 hexads and call them 1 + 6j, 2 f 6j,

The fractal-like Steiner triple system 121

3 + 6j, 4 t 6j, 5 + 6j , 6 + 6j, j = 0,1, . . . ,2m + 1. For each j, the quasi-Fano consists of the following four triples:

1 + 6 j 2 t 6 3 3 + 6 j 1 + 6 j 4+6j 5 + 6 j

2 + 6 j 5 + 6 j 6 + 6 j 3 + 6 j 4 t 6 j 6 t 6 j ( F )

Thus we obtain 4(2m + 2) triples which exhaust all exterior blocks. To construct the tangents to T, we begin by observing that there are 3(2m + 2) of

these on each point of T. Therefore, the pairs of exterior points to T which are not covered by the exterior blocks partition into w parallel classes, each of which contains 3(2m. + 2) pairs. The quasi-Fano (F) contributes with three pairs, namely l + 6 j 6+6 j , 2 + 6 j 4+ 6j, 3 + 6 j 5+6j, .j = 0, 1, . . . ,21n+ 1. So, we form one parallel class of pairs with these 3(2m + 2) pairs and attach it to one point, A1 say, of T. This yields 2m + 2 subsystems of order 7 sitting in S and sharing the same point A1 on T.

The remaining 20 - 1 = 6(2m + 1) parallel classes of pairs consist of pairs of points from dinstinct quasi-Fanos. To construct such parallel classes, take the complete graph Ii'2,,,+? on the symbols 1 + 6j, j = 0, 1, . . . ,2m + 1, and any 1-factorization of K Z r n + 2 . There are 2m + 1 1-factors each of which yields six parallel classes of pairs and each class contains 6(m t 1) pairs.

To show how to develop, via the cyclic group C6 , a 1-factor into six parallel classes of pairs, we take the 1-factor consisting of the m + 1 pairs 1 + 12j 7 + 12j , j = 0,1, . . . , m. The next table provides the six parallel classes coming from this 1-factor, each of which -

contains 6(m t 1) pairs, i.e. the required number. Thus the s-th pardel class, s E {1,2,. . . ,6}, consists of the pairs i + 12j i E (1,. . . , 6 } , j E { O , l , . . . ,m}, with

i, + 12j ,

i + s + 5 if i t s 1 7 if i t s > 7 i. = { i t s - 1

The constuction clearly shows that every pair of exterior points appears exactly once. Finally, we attach the so obtained parallel classes of pairs to the points A2,. . . , Aw of T and this completes the blocks of S.

We observe that our constuction leaves a lot of freedom since T is any STS(ro) and A 1 is any point of T. Thus, we might obtain non-isomorphic STS's even with the same T.

In sect. 4 we provide some examples.

122

TABLE 1 7 2 8 3 9

4 10 5 11 6 12 13 19 14 20 15 21 16 22 17 23 18 24 . . . .

l + a 7 + 0 2 + a 8 + a 3 + a 9 + a

4 + U lO+a 5 + a l l + a 6 + a 1 2 + 0

1 8 2 9 3 10 4 11 5 12 6 7

13 20 14 21 15 22 16 23 17 24 18 19 . . . .

l + n 8 + a 2 + n 9 + n 3 + a 1 0 + a 4 + u l l + a 5 t n 1 2 + a 6 + a 7 + a

M.J. De Resmini

1 9 2 10 3 11 4 12 5 7 6 8

13 21 14 22 15 23 16 24 17 19 18 20 . . . .

l + a 9 + a 2 + a 1 0 + n 3 + a l l + a 4 + a 12+a , 5 + a 7 + a 6 + a 8 + a

1 10 2 11 3 12 4 7 5 8 6 9

13 22 14 23 15 24 16 19 17 20 18 21 . . . .

l + a 1 o + u 2 + a l l + n 3 + a 1 2 + a 4 + a 7 + a 5 + a 8 + a 6 + a 9 + a

1 11 2 12 3 7 4 8 5 9 6 10 13 23 14 24 15 19 16 20 17 21 18 22 . . . .

l + n l l + a 2 + a 1 2 + a 3 + a 7 + a 4 + a 8 + a 5 + a 9 + a 6 + a 1 0 + a

1 12 2 7 3 8 4 9 5 10 6 11 13 24 14 19 15 20 16 21 17 22 18 23 . . . .

l + a 1 2 + a 2 + a 7 + a 3 + a 8 + a 4 + 0 9 + a 5 + a 1 0 + a 6 + a l l + a

( For convenience, in the table we wrote u instead of 12m.)

4 Some Examples

In this section we apply our constuction to the smallest possible cases, namely m = 0 ,1 ,2 . Notation is as in sects 2 and 3.

nz = 0. We construct an STS(19) = S containing an S T S ( 7 ) = T on the points A l , A2,. . . , A7. By the results in sect. 3, S contains two other STS(7)’s . The exterior blocks to T split into two quasi-Fanos:

1 2 3 7 8 9 1 4 5 7 10 11 2 5 6 8 11 12 3 4 6 9 10 12.

These quasi-Fanos provide one parallel class of pairs. By attaching the point A1 we obtain the triples: 1 6 A l , 2 4 A l , 3 5 A l , 7 12 A l , 8 10 A l , 9 11 A l . The trivial 1- factorization of K 2 yields the remaining parallel classes of pairs to which A2, . . . , A7 are attached (and they are written at the bottom of each class):

1 7 1 8 1 9 1 1 0 1 1 1 1 1 2 2 8 2 9 2 1 0 2 1 1 2 1 2 2 7 3 9 3 1 0 3 1 1 3 1 2 3 7 3 8

4 1 0 4 1 1 4 1 2 4 7 4 8 4 9 5 1 1 5 1 2 5 7 5 8 5 9 5 1 0 6 1 2 6 7 6 8 6 9 6 1 0 6 1 1

A2 A3 A4 A5 A6 A7


m. = 1. We construct an STS(43) = S containing an STS(19) = T on the points A l , A2,. . . , A19 and 2ni + 2 = 4 systems of order 7 on A l . The 6 . 4 exterior points and the 4 - 4 exterior blocks split into four quasi-Fanos:

1 2 3 7 8 9 13 14 15 19 20 21 1 4 5 7 10 11 13 16 17 19 22 23 2 5 6 8 11 12 14 17 18 20 23 24 3 4 6 9 10 12 15 16 18 21 22 24

Therefore, the parallel class of pairs to be attached to A 1 is: 1 6 , 2 4 , 3 5 , 7 12, 8 10, 9 11, 13 18, 14 16, 15 17, 1924, 2022, 21 23. Next, take I<d on 1 7 13 19 and the following 1-factorization:

_ _ 1 7 _ _ 1 1 3 1 1 9 _ _ _ _ 13 19 _ _ 7 19 _ _ 7 13

This 1-factorization yields the remaining 18 parallel classes of pairs:

1 7 2 8 3 9

4 10 5 11 6 12 13 19 14 20 15 21 16 22 17 23 18 24

1 16

6 15 7 22 8 23

12 21

. .

. .

1 8 2 9 3 10 4 11 5 12 6 7

13 20 14 21 15 22 16 23 17 24 18 19

1 17

6 16 7 23 8 24

12 22

. .

. .

1 9 2 10 3 11 4 12 5 7 6 8

13 21 14 22 15 23 16 24 17 19 18 20

1 18

6 17 7 24 8 19

12 23

. .

. .

m = 2. In this case, I J = 67,

1 10 2 11 3 12 4 7 5 8 6 9

13 22 14 23 15 24 16 19 17 20 18 21

1 19

6 24 7 13 8 14

12 18

. .

. .

1 11 2 12 3 7 4 8 5 9

6 10 13 23 14 24 15 19 16 20 17 21 18 22

1 20

6 19 7 14 8 15

12 13

* .

* .

1 12 1 13 2 7 2 14 3 8 3 15 4 9 4 16 5 10 5 17 6 11 6 18 13 24 7 19 14 19 8 20 15 20 9 21 16 21 10 22 17 22 11 23 18 23 12 24

1 21 1 22

6 20 6 21 7 15 7 16 8 16 8 17

12 14 12 15

. . . .

. . . .

1 14 2 15 3 16 4 17 5 18 6 13 7 20 8 21 9 22 10 23 11 24 12 19

1 23

6 22 7 17 8 18

12 16

. .

. .

1 15 2 16 3 17 4 18 5 13 6 14 7 21 8 22 9 23 10 24 11 19 12 20

1 24

6 23 7 18 8 13

12 17

. .

. .

20 = 31, t o = 4 - 6; there are 6 6 exterior points and 31 parallel classes of pairs each of which contains 3 . 6 blocks. The constructed STS(67) contains six STS(7). There are six quasi-Fanos, namely the four we wrote for m = 1 and

25 26 27 31 32 33 25 28 29 31 34 35 26 29 30 32 35 36 27 28 30 33 34 36

124 M.J. De Resmini

From these quasi-Fanos we get our first parallel class of pairs: 1 6 , 2 4, 3 5 , 7 12, 8 10, 9 11, 13 18, 14 16, 15 17, 19 24, 20 22, 21 23, 25 30, 26 28, 27 29, 31 36, 32 34, 33 35 which we attach to A l . Next, take K S on 1 7 13 19 25 31 and its 1-factorization:

From the first 1-factor we obtain the following six parallel classes of 18 pairs each:

1 7 2 8 3 9

4 10 5 11 6 12 13 25 14 26 15 27 16 28 17 29 18 30 19 31 20 32 21 33 22 34 23 35 24 36

1 8 1 9 2 9 2 1 0 3 10 3 11 4 11 4 12 5 1 2 5 7 6 7 6 8

13 26 13 27 14 27 14 28 15 28 15 29 16 29 16 30 17 30 17 25 18 25 18 26 19 32 19 33 20 33 20 34 21 34 21 35 22 35 22 36 23 36 23 31 24 31 24 32

1 10 2 11 3 12 4 7 5 8 6 9

13 28 14 29 15 30 16 25 17 26 18 27 19 34 20 35 21 36 22 31 23 32 24 33

1 11 2 12 3 7 4 8 5 9 6 10 13 29 14 30 15 25 16 26 17 27 18 28 19 35 20 36 21 31 22 32 23 33 24 34

1 12 2 7 3 8 4 9 5 10 6 11 13 30 14 25 15 26 16 27 17 28 18 29 19 36 20 31 21 32 22 33 23 34 24 35

In a similiar manner one gets the remaining parallel classes of pairs from the other 1- factors.

5 The Fractal-Like STS

Of course, the construction in sect. 3 is a w --t 2w -t 5 constuction for w z 7 (mod 12). Furthermore, the obtained STS(24m +- 19) contains an STS( 12m + 7) and 2m + 2 STS(7) and all these subsystems share the same point. Obviously, one can repeatedly apply the construction. By suitably choosing at each step the common point of the constucted subsystem (A1 in sect. 3), one can obtain possibly non-isomorphic Steiner triple systems with interesting configurations of subsystem. The following table suggests some of these


and we s h d discuss two special cases.

m W 2) 2 m + 2 0 7 19 2 1 19 43 4 2 31 67 6 3 43 91 8 4 55 115 10 5 67 139 12 6 79 163 14 7 91 187 16 8 103 211 18 9 115 235 20 10 127 259 22 11 139 283 24

Take m = 2' - 1, s = 0 ,1 , . . . . Then 2m + 2 = 2'+l, 'u = 24(2" - 1) + 19. The STS(24(2# - 1) + 19) one obtains by repeatedly applying the given construction contains

+ 2"-l + 2' + 2'+l = 2'+' - 1 STS(7)'s. Since at each step we can choose in different ways the point on which we hang the STS(7)'s, we can obtain nice configurations formed by these subsystems. Two of such configurations seem worth noticing: the flower and the fractal-like STS. In the flower the 2*+' - 1 STS(7)'s share exactly the same point. In other words, the complement of this point can be partitioned into 2*+' - 1 Pash configurations. In the fractal STS , the 2'+l STS(7)'s obtained at step s , s = 0,1 , . . . , share a point on one of the STS(7)'s added at step s - 1. Obviously, s = -1 means the starting STS(7) (see table) . The following picture gives an idea of the situation and reminded the author of a fractal.

We observe that the STS(19) constructed in sect. 4, i.e. the first step of the iterated construction above, has a unique configuration of its three subsystems of order 7.

1 + 2 + 2' +

32

I26 M.J. De Resmini

References

(11 Th. Beth, D. Jungnickel and H. Lenz. "Design Theory", Bibliographisches Institut, Mannheim/Wien/Zurich, 1985.

[2] M. J. de Resmini, Sets of type (m,n) in a Steiner System S(2 ,1 ,v) , in "Finite Geometries and Designs", London Math. SOC. Lecture Note Series 49 (1981), 104- 113.

[3] M. J. de Resmini, Some subsystems of Steiner triple systems, unpublished.

[4] M. J. de Resmini, Subsystems and Pasch configurations in Steiner triple systems, unpublished.

[5] J. Doyen and A . Rosa, A bibliography and survey of Steiner systems, February 1989.

[6] J. Doyen and R.M. Wilson, Embeddings of Steiner triple systems, Discr. Math. 5 (1973), 229-239.

[7] D.R. Hughes and F.C. Piper, "Design Theory", Cambridge Univ. Press, 1985.

[8] T.P. Kirkman, On a problem in combinations, Cambridge and Dublin Math. J. 2 (1847), 191-204.

[9] H. Lenz and H. Zeitler, Arcs and ovals in Steiner triple systems, in "Combinatorial Theory", Lecture Notes in Math. 969, Springer, 229-250.


Locally partial geometries with different types of residues

A. Del Fra", D. Ghinellia and A. Pasinib

"Dipartimento di Matematica, Universiti di Roma "La Sapienza", P.le Aldo Moro 2, 1-00185 Roma, Italy. E-mail: DINAOITCASPUR.bitnet

bFacolti di Ingegneria (biennio), UniversitB di Napoli, Via Claudio 21, 1-80125 Napoli, Italy. E-mail: [email protected]

Abstract A class of locally partial geometries (L.pGs for short) is constructed where both

linear spaces and generalized quadrangles occur as point-residues. We conjecture that this class gives all possible example of L.pGs with that anomaly.

We prove this conjecture when plane-residues are projective planes (see section 5). In the general case, we are able to prove the conjecture when there is at least one point with a linear residue, satisfying an additional assumption.

1. INTRODUCTION

A locally partial geometry (L.pG for short) is a residually connected Buekhenhout geometry satisfying the Intersection Property (IP) of [2] and belonging to the rank three diagram

L PG 0 -0

points lines planes where the symbols

L PG o-------o and -0

denote the classes of linear geometries and partial geometries (see Bose [l]), respectively. We recall that a partial geometry of order ( s , t ) and parameter CY (briefly pG,(s, t ) ) is an incidence structure with s + 1 points on a line, t + 1 lines on a point, two points on at most one line, such that for all antiflags ( P , I ) the antiflag number y (P , l ) (i.e. the number of points in 1 collinear with P) is a constant CY # 0.

Clearly, linear spaces, or 2-designs with parameters 2 - ( T ( S + 1) + l , r + 1,l) are pGr+l ( T , s), hence have diagram

PGr+1 L -o=o-------o r s r S

128 A. Del Fra et al.

while when the design is symmetric (i.e.: r = s) the diagram

L PGS+l -0 = -0 = -0 S s s 9 s S

is used.

they have diagram Circle geometries, which are the trivial 2-designs for (v,2,1), are pGz(1, s). Thus

C L PG2 1 s 1 s 1 S

When a = 1 a p G l ( s , t ) is a Generalized Quadrangle (GQ) of order ( s , t ) , and we use the diagram

o - o = ~ o = ~ o

- S t

As parameters are implicit in the notion of partial geometry, all L.pGs geometries admit parameters ( r , s , t )

L PG r 9 t

and we shall speak of a L.pG geometry of order ( r , s, t ) . On the other hand, residues of points of a L.pG need not have the same index a, as it will be clear from the following example 1.1. In this example r = 1 (i.e. the first stroke denotes a circle geometry). As usual in the literature, when r = 1 we will speak of Extended partiaa Geometries (EpGs) (see [13], [12], [14], [lo], [15], [9], [8]) or, when a = 1, of Extended Generalized Quadrangles (EGQs) (see [3], [18], [4], [ S ] , [7], [ll], for instance).

EXAMPLE 1.1. Consider, for any integer t 2 1, a (t+2)-set, II* = { P I , Pz, . . . , P,+,}, anda(t+l)-set , A* = {Ll ,L2 , ..., Lt+l}, withII*nA* = 0. Callpointstheelements of II*UA*,linesthepairs{Pi,Lk},and{Pi,Pj} ( i , j = 1 ,..., t + 2 , i # j ; k = 1 , . . . , t t l ) , and blocks the triples {Pi, Pj, L k } . It is not difficult to verify that the above defined structure, say S*, is a L.pG geometry of order ( l , l , t ) (hence it is an EpG(1,t)) and residues of points in n* are Generalized quadrangles ( a = l), while residues of points in A* are Linear spaces ( a = 2) namely, since r = 1, circle geometries.

This example of EpG with two different types of residues can also be described in

Let be a Buekenhout geometry belonging to the following disconnected diagram the following more abstract way.

C -0 0

t t and let S’ be the J-shadow space of (see [2]) where J is the following pair of types

Locally partial geometries 129

The dual S* of S’ is the EpG(1,t) of the above example, and residues of points of S* are either circular spaces

C - 1 t

or d u d grids

and both cases occur. In particular, when t = 1 the geometry S* can be described as a pair of tetrahedra glued at one of their faces (that face does not appear as a block in

We conjecture that L.pGs with different types of residues are fairly rare and should be rather degenerate. We are not able to prove this conjecture in general, but we shall prove it in particular cases.

To this purpose, we establish in section 2 some result for L.pGs with at least two types of point-residues, one of which is a linear space. In particular, we also give lower and upper bounds for the diameter of an L.pG in Proposition 2.10 and theorem 2.12 (see [9] for the case where only one type of point-residue occurs).

In section 3 we prove that if there is at least one point L with a linear residue (a “linear point” for short), such that every point at distance 2 from L is linear, then r = s = 1 and our L.pG is isomorphic to the geometry S* of example 1.1. When the above assumption on L is not satisfied, but we have exactly two types of point-residues, we obtain a numerical condition (see section 4), which will be used in section 5 to prove the conjecture when r = s, that is for rank 3 geometries belonging to the diagram

S*).

PG O-------o------O S S t

These include important examples of rank 3 geometries, namely: Projective geometries:

--0 S S S

or their truncations: L

--0 S S t

or polar spaces:

or Affine dual geometries:

where

“- S S t

Af* -- S S s- 1

Af * PG8 o-------.o=-o S s-1 s s- 1


In all these examples, the parameter (Y is a costant. In section 5 , we prove that Example 1.1 is the unique example with a different from a costant.

2. FIRST RESULTS AND DIAMETER BOUNDS

From now on S will be an L.pG with order ( T , s, t ) and with at least two different types of residues, one of wich is a linear space.

We denote by P , L, and B the sets of points, lines and blocks of S, respectively. SA will be the residue of a point A E P , cr(A) will indicate the constant antiflag number (also called ‘index’) of SA, while Z = minAEp{a(A)}. The points of the two nonempty sets

are called linear and partial points, respectively. We note that T 5 s (since the first stroke of the diagram represents a linear space),

and s 5 t (because of the assumption A # 8). As the Intersection Property is assumed, the point-line system r(S) of our geometry

S is a graph without multiple edges, called the point graph of S. We say that two points A and B are collinear and we write A N B , if they are

adjacent in l?(S). A point and a line or two lines are said to be coplanar if there is a block on them.

As usual, d(A,C) will be the distance between two points A and C of P in r(S), r A , i will denote the set of points at distance i from A, while A indicates the diameter of I’(S). The distance of a line 1 or a block 2 from a point A are naturally defined by

d(A,l) = mind(A,B), d ( A , x ) = mind(A,B). BE1 B E z

As lines and blocks incident with a given point A appear in the residue SA, as points and lines, respectively, we will refer to them as ‘points’ and ‘lines’ of SA, where the quotation marks should avoid any confusion with points and lines of the geometry S itself.

With the above notation, the following lemmas hold.

LEMMA 2.1. If L is a linear point and A and B are two points of S adjacent to L, then there is a unique block x containing A, B, and L.

Considering the lines a = AL and b = B L as ‘points’ in the linear space SL (of order ( s , t ) ) , there is a unique ‘line’ x of SL on them. Therefore there is a unique block x of S containing L and the lines a and b, hence the points A, B, L.

LEMMA 2.2. If a block y contains a linear point L, then every point A collinear with L also is collinear with every other point B of y.

Proof

I

Proof For every B E y there is a unique block x on A, B , L , by lemma 2.1. Since the residue S, of x is a linear space (of order ( r , s)), A and B are collinear. I


LEMMA 2.3. Two linear points axe never collineax or equivalently: every block contains at most one linear point.

Proof By contradiction, let z be a block containing two linear points, say L1 and L2. We shall prwe that every point A collinear with L1 is linear, this will give, by residually connectedness, the contradiction ll = 8.

If y is a block on L1 but not on L2, let A, B be two points of y different from L1. By lemma 2.2, A and B are collinear with L2; thus there is a block z on A , B, Lz, by lemma 2.1.

In the residue S A the line a = AL2 and the block y are an antiflag, hence @ ( A ) is given by the antiflag number cp(a, y); now for each of the s +1 lines 1 on A contained in y choose a point B # A, L , ; by lemma 2.1 and 2.2 as above, there is a block z containing a and 1 . Thus a ( A ) = cp(a, y) = s + 1 and A is linear. By residually connectedness, this gives a contradiction. I

LEMMA 2.4. Let P be a partial point on a block x, and let A be a point with d ( A , s ) = d(A, P ) = 1 . Then x contains exactly a ( P ) lines on P which are coplanar with A .

In the residue S p the ‘point’ given by the line P A is collinear with exactly a ( P ) ‘points’ of the ‘line’ x. This gives the statement.

LEMMA 2.5.

Proof I

Let A and z be a point and a block with d ( A , z ) = 1. If we set

a(A,x) = max { a ( B ) } , BEz,d(A,B)=l

then cp(A,x) 1 a ( A , x ) r + 1. Furthermore, if A = L is a linear point, then for eveq P E 2, P - L, we have that a ( P ) = a is a constant and cp(L,x) = ar + 1.

If A is collinear with a linear point of 2 (and so A is a partial point, by lemma 2.3), lemma 2.2 implies that A is collinear with all points of 5, and there is nothing to prove. Hence, we may assume without loss of generality that A is collinear with a partial point P of z. By lemma 2.4, then A is coplanar with a ( P ) lines of x through P and so is collinear with at least the a ( P ) r + 1 points on these lines. Choosing P such that a ( P ) = a ( A , z ) we obtain the first part of the statement.

If A = L is a linear point, it remains to prove that there are no other points on z collinear with L , apart from those on the above a ( P ) lines. Suppose by contradiction that there is another point Q on z collinear with L ( Q is necessarily a partial point, by lemma 2.3). The lines LP and LQ are coplanar (since SL is linear), hence L is coplanar with the line P Q , which contradicts Lemma 2.4. This proves that the points on z collinear with L are exactly a ( P ) r + 1. If P’ is any of these points we obtain, similarly, p ( L , z) = a(P‘)r + 1 and thus a ( P ) = @(PI).

Proof

I

As an immediate consequence of lemma 2.5, we have the following two corollaries.

If P and P’ are two partial points collineax with the same linear COROLLARY 2.6. point L, then a ( P ) = a(P‘).


COROLLARY 2.7. Let L be a linear point and let x be a block with d(L, x) = 1. If a is the constant value in the statement of lemma 2.5, then the points on x collinear with L and the lines on x coplanar with L are: (i) a line of x, if a = 1,

(ii) a linear subgeometry of z of order ( r , a - l), if a > 1.

LEMMA 2.8. Every block x at distance 1 from a linear point L contains points at distance 2 from L.

Proof By lemma 2.5, there are ar + 1 points on z collinear with L, where Q!

is the constant parameter of the residues of these points ( all those points partial, by lemma 2.3). Hence (Y 5 s, whence ar + 1 < rs + s + 1. This shows that z contains points which are not at distance 1 from L and the statement follows.

LEMMA 2.9. Let P be a partial point, collinear with at least a linear point. Then a ( P ) = O , 1 rnod(r + 1). Furthermore: a(P)r 5 s.

Set a ( P ) = a. If a > 1 and x is a block on P at distance 1 from a linear point L , it follows from corollary 2.7 that there is in z a linear subgeometry, say 7, of order ( r , a - 1). Since the number

I

Proof

of lines of 7 is an integer, we have the congruence in the statement. By lemma 2.8, there is on x at least a point, say B, at distance 2 from L. The lines

of x on B and on the ar + 1 points of 7 cannot be lines of 7 (because B +! 7). Thus they contain exactly one point of 7. Clearly, the number ar + 1 of these lines must be at most the number s + 1 of lines of x on P , which completes the proof. I

We conclude this section giving lower and upper bounds for the diameter A of S. As an immediate consequence of lemma 2.8 we have the following

PROPOSITION 2.10. The diameter A of S satisfies A 2 2.

For the upper bounds and the relative lemma, we shall omit the proofs, since they can be easily obtained from the proofs in [9], where upper bounds for an L.pG geometry S are given, under the assumption &(A) = a (a constant) for all A E S. Indeed, it is enough to replace the constant a in the proofs of [9] with the minimum, say E, of a ( P ) , when P runs in the set II of partial points. In symbols:

a = min a(P) .

LEMMA 2.11. Let P be a point of S and let x be a block with d(P, z) = i 2 1.

(1) -

PEII

(i) I f E = 1, then on every point A; E r p , i n we have at least i lines consisting of points all in r p , i . Hence we have on x at least ir + 1 points at distance i from P .

(ii) If E > 1, then on every point Ai E r p , , n x we have at least E + 2(i - 1 ) lines consisting ofpoints all in r p , ; . Hence we have on z at least [E+ 2 ( i - 1]r + 1 points at distance i from P .


Proof For (i) imitate as indicated above and in [9] (see in [9] theorem 4.3), the arguments in [6] or [4] or [ll]. For (i), replace in the proof of theorem 4.1 of [9] a with a. I

THEOREM 2.12. The diameter A of an L.pG geometry with different types of residues satisfies

-

3. L.pGs WITH LINEAR POINTS AT DISTANCE 2

The main result, in this section is:

THEOREM 3.1. Let S be a L.pG of order ( r , s , t ) with at least two different types of residues. If there is a linear point L in S such that every point at distance 2 from L is linear, namely

FL,2 c A, (1)

then T = s = 1 and S is isomorphic to the L.pG S* of example 1.1.

For the proof we shall need the following results, where the notation is the same as in section 2.

PROPOSITION 3.2. If S contains a linear point L satisfying (I), then r = 1 and a ( P ) = s for every partial point P E IT. Fhrthermore, S has diameter A = 2.

Proof On a block z with d(L, z) = 1 there is at least one point of l?L,2, by lemma 2.8; this is linear by the assumption (l), hence is unique, because of lemma 2.3. On the other hand, we have ar + 1 points in z n 17L,l (see lemma 2.5), where a is the constant index of the partial points on z. Now lemma 2.2 implies 1x1 = ar + 2 = r ( s + 1) + 1,since z is a linear space of order ( r , s ) . Thus a = s + 1 - l / r . This yields r = 1, since a is an integer, hence a = s for all partial points of z. Since this holds for every z with d(L,z) = 1, we deduce that a ( P ) = s for every partial point P at distance 1 from L. From (1) and lemma 2.1, we see that II

Let E be as in 2.(1), clearly ?i = s and from 2.(2) we obtain A 5 2; since A 2 2, by 2.10, we have A 5 2.

REMARK 3.3. In a L.pG with r = 1, every line on a point A has two points one of which is A , hence IrA,11 = (s + l)(st + a(A)) /a(A) , which is the number of ‘points’ in

PROPOSITION 3.4. If S contains a linear point L satisfying (11, then n = r L , 1

and every other linear point satisfies (1). firthemore, every partial point is collinear with every point in S, and

l?L,l hence a ( P ) = s for every P E II.

I

the pGa(A)(S, t , S A .


Proof II; from (1) and Lemma 2.1, we obtain l?L,l 2 II, hence FL,1 = II.

Now r = 1 (see 3.2), a ( L ) = s + 1 so that remark 3.3 gives III = II’L,1 I = st + s + 1. If L’ # L is any other linear point, again by 2.3 FLt,1 E II; since these two sets

have the same order, we have FL,,l = II; therefore I’Lt,z c A. As L’ varies in A, this implies that every partial point is collinear with every linear point, so that two partial points are always collinear, by lemma 2.1. In particular, if we choose a partial point P , by 3.2 a(P) = s, thus IPl = II’p,l I + 1 = (s + l ) ( t + 1) + 1 (see remark 3.3). Obviously IAl = [PI - IIIl = t + 1, which completes the proof.

By lemma 2.3 FL,l

I

LEMMA 3.5. If S contains a linear point satisfying ( l ) , then every block ofS contains exactly one linear point.

Proof Let P and Q, P # Q, be partial points. By 3.4, and 2.1, for every linear point Li there is a block xi containing P, Q, L , (i = 1,. . . , 1A1) and all these blocks are distinct (see 2.3). Since [A1 = t + 1 is the number of blocks on the line PQ arid P and Q are arbitrary, the statement follows. I

LEMMA 3.6. Let S be as in 3.1. If Po, PI, Pz, P3, are four partial points such that Po,Pl,Pz, andPo,Pl,P3 areneveron asmeblock, thenalsoPo,P:!,P3 (thusPl,PZ,P3) are never on a same block.

Proof Consider the lines 1, = POP* (i = 1,2,3) , and assume, by contmdiction that there is a block 2 on Po,Pz,P3. Clearly, I 1 is not in x and is not coplanar with the lines Z2 and l3 of z; in the residue Sp,,, this would give .(Po) 5 s - 1 which contradicts the fact that a ( P ) = s for every partial point P (see Proposition 3.2). Interchanging PI and Po we have the statement. I

PROOF OF THEOREM 3.1. Consider the structure S’ = (P’, L’, E), with P’ = II, whose “lines” (i.e. elements of L‘) are the maximal sets of partial points, three by three not on the same block of S. Such a maximal set consists of any two partial points P and Q and all partial points of S not on the t + 1 blocks on P and Q. Therefore it contains exactly s‘ + 1 = IIIl - ( t + l)(s - 1) = t + 2 partial pointh (since T = 1, by 3.2, so that every block has s + 1 points, while every line has 2 points).

Clearly on any two points of P‘ we have a unique “line” of f?, and dually. The elements of C‘ on a point P are sets of t + 2 partial points meeting only in P , thus (see (2))

IIIl-1 s t + s

t + l t + l t‘ + 1 = - = - = s. (3)

Since S is finearlthis implies either t‘ = 0 , or s’ 5 t’; this last inequality would give t + 1 5 s - 1 which contradicts the assumption that S has a linear point. Therefore t’ = 0, hence s = 1 and (see (2)) IIII = t + 2, while Ihl = t + 1. It follows that a block of S consists in one linear point and 2 partial points, and S is isomorphic to the L.pG of order ( l , l , t ) of example 1.1. I


4. L.pGs WITH TWO TYPES OF RESIDUES

In this section S will be a L.pG of order ( r , s, t ) such that for every point A E P = ll U A we have a ( A ) = a, s + 1, and both cases occur (i.e.: ll # 0, A # 0).

Our purpose is to give some condition on the parameters, even when S does not satisfy the assumption in 3.1. For, we shall use the following result on pointsets in a pG,(s, t ) , obtained in [5].

PROPOSITION 4.1. Let K be a set of points in a pG,(s, t ) , and let L; be the set of lines containing at least a point of K . Then

s + 1 l q 2 ---[W(t + 1) - (s + t + 1 - a)] ,

where n k = miqEL.+{ 11 n 11'1).

Proof See [5] , theorem 2.3. I

THEOREM 4.2. Let S be an L.pG with order ( r , s , t ) where a ( P ) = a (a constant) for every partial point P. If there are a linear point L and a partial point Q with d(L, Q) = 2, then

(s - a ) ( s + 1) ( r + l ) a 5 s 5 t 5

r

Proof The assumption on L and Q is equivalent to say that 3.( 1) does not hold, by 3.4.

Let x be a block on Q. If d(L, x) = 2, then there axe no lines of x on Q with points in l?L,l. If d ( L , x ) = 1, then (see the proof of lemma 2.9) the lines of 2 on Q with a (unique) point in r L , l are a r + 1. Thus in the residue SQ the lines on Q are a pointset I( with n K = a r + 1. It follows from 4.1 that

(3) s + l

IK(1 2 -[(a. + l ) ( t + 1 ) - ( s + t + 1 - a)] . a

Since each of these line contains at most one point of I'L,1, we also have

1K1 5 I rL,1I = r ( s t + s + 1). (4)

From (3) and (4), it is easy to deduce a ( r t + s + 1) 5 s(s + l ) , which gives

(s -a)(. + 1 ) t 5 , a r or

s(s + 1) a' r t + s + 1 '

Since s 5 t , (6) yields

(5)

136

which implies S as -

r + l ’

A . Del Fra et al.

( 7 )

since a is an integer. Now, (5) and (7) give the statement.

REMARK 4.3.

1

If a > 1, lemma 2.9 implies ( r + 1 ) I a. Thus, from 2.9 and 4.2 we

REMARK 4.4. If r = (Y = 1 condition (2) becomes

2 5 s 5 t 5 2- 1.

By a well known result on GQs (see [17], 1.2.5), then 2 5 s 5 t 5 s2 -s. In particular, if theresidues haveclassical parameters, only thecases ( s , t ) = (q ,q ) , ( q 2 , q 3 ) , ( q - l , q + l ) can occur.

5. L.pGs OF ORDER ( s , s , t )

In this section we shall prove that the L.pG geometry S* of order (1,1, t ) of example 1.1 gives all possible examples of L.pG geometries with order (s, s, t ) and different types of point-residues. In other words we will prove that a rank three geometry different from S* belonging to the diagram

PG -0-------0 S S t

where, as usual in the litcrature, the first stroke denotes a projective plane of order s, has necessarily a constant a.

LEMMA 5.1. Let S be an L.pG with order ( s , s, t ) . If A and B are two points with &(A) # a(B), every point C coplanar with A and B but not on the line AB satisfies

We may assume, without loss of generality, a(A) > a(B) . By contradiction suppose there is a block 2 on AB and a point C E I - AB with a ( C ) > 1. Consider the lines 1 = C A and I‘ = CB and a block y # x on 1. Looking at the residue Sc , the assumption a(C) > 1 implies that there is a block y’ # I on 1’ meeting y in a line, say 10.

Since SA is a pG,(A)(s,t), there are a(A) blocks 11 = x , x 2 , . . . ,z,(A) on AB, meeting y on the lines l1 = 1 = CA, 1 2 . . . . , Z o ( ~ ) , respectively. Since y is a projective plane, 1, n 10 is a point C, ( z = 1,. . . ,a(A)) . Clearly I, n y‘ = 1: = BC;. Considering Sn we obtain a ( B ) 2 a(A) , which is a contradiction.

LEMMA 5.2. Let S be an L.pG with order (s,s, t ) . If a ( A ) is not a constarit for A a point of S, then every block of S contains at most one point A with a(A) > 1.

a(C) = 1.

Proof

I

Proof By contradiction suppose there is a block x with two points B # C with


a(B) > 1, a(C) > 1. Let A be any point of the projective plane I not on the line BC. By lemma 5.1, it cannot be a(A) # a(B), since a(C) > 1. Thus a ( A ) = a(B) > 1 for every point A E x - BC.

The above argument, interchanging the role of A and C, proves that all residues of points of x have the same a; this by connectedness gives the constance of a, for every point of S, which is a contradiction.

THEOREM 5.3. Let 5 be an L.pG with order ( s , s , t ) . If a ( A ) is not a constant for A E P , then s = 1, .wid S is isomorphic to the L.pG of example 1.1.

1

Proof By contradiction, assume s > 1. Since (Y is not a constant, there is at least a point A with a ( A ) > 1. We fix a point B collinear with such an A , By lemma 5.2 a(B) = 1, so that Sg is a Generalized Quadrangle. This implies that we can choose blocks 21, 5 2 containing A and B, and a block x3 on B, with q n 2 3 = 113, 2 1 n 5 2 = 0. Let 1 # 113 be an arbitrary line of 2 3 on B. In the residue Sg, the ‘point’ 1 is collinear with a unique ‘point’ of the ‘line’ 2 2 ; hence there is a unique block 2 4 on B, meeting 2 3

in 1 and x 2 in a line l z4 . Since a ( A ) > 1, there is on A a block ?f 3 B meeting z1 in a line and x2 in a line.

We set C = Tn 124, D = Z n 113, 7 = CD. Now C is collinear with A , hence a(C) = 1, by lemma 5.2; therefore there is in 2 4 a unique line coplanar with 1. The block x containing 1 and 7 meets 2 3 in the line ED, where E = 1 n 1. Now a ( E ) > 1 because there are two blocks 2, 5 4 on E , containing 1 and meeting 2 3 in lines.

Since s > 1, we may choose another line I’ # 113,l in 2 3 and using the above argument, find in I’ a point E‘ (necessarily different from E ) , with a(E‘) > 1. Then we would have in 2 3 two points E, E‘ with a ( E ) > 1, a(E‘) > 1, which contradicts lemma 5.2. This proves s = 1. Therefore a L: s + 1 = 2.

Now the assumption that a is not a constant gives a = 1,2 , the points with a = 2 are linear, while partial points have all the same index a = 1. Since T = s = a = 1, the condition in theorem 4.2 is not satisfied, hence the points at distance 2 from any given linear point are also linear and applying theorem 3.1 we complete the proof.

-

I

REFERENCES

(11 R.C. Bose, Strongly regular graphs, partial geometries and partially balanced designs, Pacific J . Math., 13 (1963), 389-419.

[2] F. Buekenhout, The basic diagram of a geometry, in Geometries and Groups Lecture notes 893, Springer (1981), 1-29.

[3] F. Buekenhout and X. Hubaut, Locally polar spaces and related rank 3 groups, J. Algebra, 45 (1977), 391-434.

[4] P.J. Cameron, D.R. Hughes and A. Pasini, Extended generalised quadrangles, Geom. Ded. (to appear).

[5] F. De Clerck, A. Del Fra and D. Ghinelli, Pointsets in partial geometries, Proc. of the 3rd Isle of Thorns Conference 1990, Fini te Geometries and Designs, (Hirschfeld et al. ed.s),93-109.


[6] A. Del Fra and D. Ghinelli, Classification of Extended Generalized Quadrangles

[7] A. Del F'ra and D. Ghinelli, Large extended Generalized Quadrangles, Ars

[8] A. Del Fra and D. Ghinelli, Tkuncated V,, Coxeter complexes as EpGs with

[9] A. Del F'ra and D. Ghinelli, Diameter bounds for Extended partial Geometries,

[lo] A. Del h a , D. Ghinelli and D.R. Hughes, Extended partial geometries with

[ll] A . Del Fra, D. Ghinelli and A. Pasini, Diameter bounds for an EGQ, J. C.I.S.S.

[12] W.H. Haemers, Regular two-graphs and extensions of partial geometries, (preprint ). [13] S.A. Hobart and D.R. Hughes, Extended partial Geometries: nets and dual

[14] D.R. Hughes, Extended partial Geometries: dual 2-designs, E U T O ~ . J. Comb.

I151 D.R. Hughes, Partial geometries of rank n, Proc. of the conference Combina-

[16] D.R. Hughes and F.C. Piper, Design Theory, Cambridge University Press

[17] S.E. Payne and J.A. Thas, Finite Generalized Quadrangles, Pitman Boston

[18] J.A. Thas, Extensions of finite generalized quadrangles, Symp. Math. 28

with maximum diameter, Disc. Math. (to appear).

Comb., 29 A (1990), 75-89.

maximum diameter, J . Geom. (to appear).

Europ. J. Comb., 12 (1991), 293-307.

minimal p, Geom. Ded. (to appear).

Proc. of the 2nd Catania Comb. Cod., 15 (1989), 256-270.

nets, Europ. J. Comb. (to appear).

( to appear).

torics '90, Gaeta (May 1990).

(1985).

(1984).

(1986), Academic Press, London, 127-143.

Combinatorics '90 A. Barlotti et al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. I39

(0,n)-Sets in a generalized quadrangle

A. Del Ra', D. Ghinelli' and S.E.Payneb

'Dipartimento di Matematica, Universiti di Roma "La Sapienza", P.le Aldo Moro 2, 1-00185 Roma, Italy. E-mail: DINAQITCASPUR.bitnet

*Department of Mathematics, Box 170, University of Colorado at Denver, Denver, CO 80204, USA. E-mail: [email protected]

Dedicated to Prof. G. Tallini On the Occasion of his 60th Birthday

Abstract We study subsets of points in a generalized quadrangle (GQ), in particular, those

meeting every line in just n points or in none at all, (i.e: sets of class (O,n)), simply called here (0, n)-sets.

1. INTRODUCTION

Subsets of the pointset P of an incidence structure S = ( P , 0 ,2) , subjected to certain intersection conditions with blocks, have been widely studied when S is a projective or &ne geometry, a Steiner System or a 2-design (see for instance: Tallini [32]-[39], Tallini Scafati [40]-[42], Hirschfeld [18], or the various papers in the Proceedings [3]-[6]).

In this paper, S will be a generalized quadrangle (GQ) of order (s, t ) and as usual, the blocks will be called 'lines' (see Payne and Thas [28] for the definition and all assumed results).

For the size k of an arbitrary subset I< of points, there are lower bounds depending on s , t and on the minimum number n 2 1 of points of K which are on a secant line of S (i.e. a line with a non-empty intersection with K ) (see section 2, or [13]). These bounds have already been applied by the first two authors to the study of extensions of GQs, which was the original motivation for the present paper (see [12] and [13]). The first bound

k 2 TI[(. - l ) t + 11 = bl (1) is reached if and only if Ii' is a subquadrangle of order ( n - 1, t ) .

Equality holds in the second bound bz

k L (s + l ) [ n ( t + 1) - (s + t ) ] = bz (2)

if and only if Ii' is a se t of class (0,n) (briefly, (0,n)-set) in S (i.e., on every secant line, there are exactly n points) with the property that on every point P not in I< the

I40 A. Del Fra et al.

number of esternal lines (i.e. lines not meeting I<) is a constant e (see 2.2, 3.6 and 3.10).

When there are no external lines, (0, n)-sets in a GQ are called n-ovoids. The dual notion of an n-ovoid was first considered by Segre (see [31]) in H(3,q2), the GQ(q2, q ) arising from the nonsingular hermitian variety of the projective space PG(3, q 2 ) . Segre proved that if K* is a lineset of H(3,q2), such that any point of the surface lies on a constant number n of lines, then either

(i) n = q + 1 and K' consists of all the lines in the surface; or (ii) n = ( q + 1)/2, and I(* contains exactly half of the lines and is called a hemisys-

tern of H(3, q'). An equivalent dual statement of Segre's result is the following one. A proper n-

ovoid (i.e., q 2 n 2 1) in the GQ(q,q2) arising from an elliptic quadric Q = Q-(5,q) in PG(5, q ) has n = ( q + 1)/2 and contains half the points of Q. Following Segre, this is also called a heniysistem (see Thas [45], and Cameron, Delsarte and Goethals [8] for further results in this direction).

From section 3 on, A' will be a (0, n)-set. Now k = 11-1 also has an upper bound

k I n(st + l), (3) which is reached if and only if K is a n-ovoid.

A (0, 1)-set is simply called a k-arc. A k-arc with no external lines (i.e. a 1-ovoid) is simply an ovoid. Clearly, for a k-arc k 5 s t + 1 and equality holds if and only if I< is an ovoid.

For k-arcs and ovoids we refer to [28]. We recall, in particular (see [28], 2.7.1) that if the GQ has no ovoids, then any k-arc has necessarily k 5 s t - t /s . Since if t > s2 - s , then S has no ovoids (see [28], 1.8.3), when t = s2, any k-arc has

k 5 s3 - s. (4)

Recently in the classical case this upper bound has been improved by Thas (cf. [46] and [47]). The reader is referred to Thas [45], for other new results on ovoids, k-arcs and hemisystems of GQs and, more generally of partial geometries.

In this paper, after giving the above mentioned bounds on k , we start to build some general theory of (0, n)-sets in GQs (section 3).

In section 4 we observe that (0, s)-sets with s > 1 are complements either of ovoids or of subquadrangles of order (s, t / s ) or of 'geometric hyperplanes' (i.e. subsets of the form R I , R a point).

Thas gave in [45] a characterization of n-ovoids with strongly regular point-graphs (see 5.2 for the statement, [45] section 3 for the proof). In section 5 we give an upper bound for the size k of a (0, n)-set with strongly regular point-graph, which is better than (3) for certain values of n.

In section 6 we exhibit a general method of constructing (0,n)-sets, using ovoids with common intersection. A special case of this construction gives examples when the GQ(s,t) has a subquadrangle of order ( s , t ' ) , (1 5 t' < t ) (see 6.2). In the rest of the paper we give examples in most the known GQs: in the last part of section 6 we consider the classical GQs, in section 7 we concentrate on the GQs associated with an oval.

(0,n)-sets in a generalized quadrangle 141

2. BOUNDS FOR THE SIZE OF A POINTSET IN A GQ

Let IC be a nonempty set of points of a finite GQ(s, t ) , say S. We denote by L+ the set of lines meeting K (secant lines) and by L- the set of lines disjoint from K (external lines).

We consider the positive integer

l + E L + n = min [ I + nKI. (1)

Bounds for the size k of IC can be expressed in terms of s, t and n.

THEOREM 2.1 With the above notation,

k 2 n[(n - 1)t + 11, (2) where if n # 1, equality holds if and only if Ii is a subquadrangle of S of order ( n - 1, t ) .

If L- = 0 the bound in (2) can be improved. Namely:

k 1 n(st + I), and equality holds if and only if Ii' is an n-ovoid.

(3)

Proof Let 1 be a line of S meeting K in precisely n points. On the lines through these n points there are at least n[(n - 1)t + I] points of IC and we have (2). The statement on equality is trivial.

If L- = 0 each of the lines other than 1, through the s + l -n points of 1-ICcontains at least n points. Hence

k 2 n[(n - l ) t + 11 + ( S + 1 - n)tn = n(st + l), where equality holds if and only if IC is an n-ovoid. I

The lower bound given by (2) is not always the best possible.

THEOREM 2.2 minimum defined by (1). Then

Let IC # 0 be a subset of points in a GQ(s,t) and let n be the

k 1 ( S + l)[n(t + 1) - (S + t ) ] , (4)

and equality holds if and only if IC is a (0, n)-set and for every point P , the set P l of points collineax with P satisfies

k s + - for all P E Ii,

for all P @ I<.

(s + 1) '

(s + 1) (5)

Proof By definition of n, for any point P of K we have

1pL n h'l 2 ( n - l ) ( t + 1) + 1 = n(t + 1) - t , (6)

and equality holds if and only if every line on P meets IC in precisely n points. Hence

c I P L I{' 1 n(t + 1) - t . k

PEK (7)


By 1.10.1 of Payne and Thas [28] we have

where equality holds if and only if ( 5 ) holds, in which case K is said to be i-tight where i ( s + 1) = (Kl (see [26]). By (7) and (8) we get (4). Now the last part of the statement is trivial. 1

REMARK 2.3 better than that of (4) if and only if

Comparing (2) and (4) we see that the lower bound given by (2) is

S n < 1 + - .

t (9)

The two bounds are equal when n = s + 1 or n = 1 + s / t . If n = s + 1, then (2) implies k = JSI = (s + l)(st + 1). Hence n = s+ 1 if and only if K = S , which is an uninteresting case. As a byproduct, we get the following well known result: a finite GQ(s,t) admits a subquadrangle of order ( s ' , t ) only when s 2 t . Moreover, we have s' = 1 if s = t .

REMARK 2.4 Theorems 2.1 and 2.2 have important applications in the study of extended GQs (i.e. connected structures such that the residue in each point is a GQ). The first two authors have used inequalities (2)-(4) to find upper bounds for the number

of points in an extended GQ, say I? (see Del Fra and Ghinelli [12]) and to give bounds for the diameter of I7 (see Del Fra, Ghinelli and Pasini [13]).

In the particular case of a (0, n)-set a different proof of (4) is in Cameron, Hughes and Pasini [lo], lemma 3.11. However, this proof cannot be generalized to the case when / I + n 1il is not a constant and n is the minimum defined in (l), which yields the above mentioned results for Extended GQs. We note that our (0, n)-sets are called 'saturated n-sets' in [lo].

3. ARITHMETIC CONDITIONS FOR (0,n)-SETS

From now on S will be a GQ(s,t) and I< a (0,n)-set in S with k points, and 1 5 n 5 s (to avoid the trivial case I< = S, corresponding to n = s + 1). As in section 2, L+ and L- will be the sets of secant lines and external lines, respectively. Clearly:

IL+I + IL-1 = ( s t + l)(t + 1). (1)

Counting flags (Q, 1 ) with Q in K , we see that

k=- nlL+ I ( t + 1)'

From (2) and (1) we obtain

We denote by 0 the integer

(0,n)-sets in a generalized quadrangle I43

Obviously 6 2 0 and we have equality if and only if I< is an n-ovoid. Then (3) implies

k I n(s l+ I), ( 5 )

where equality holds if and only if I< is an n-ovoid. Since k = n(st + 1) - 0, it is natural to call 6 the deficiency of I<.

Condition ( 5 ) and Theorem 2.1 are equivalent to the following proposition.

THEOREM 3.1 satisfies

The deficiency 6 = nlL-l/( t + 1) = n(st + 1) - k of a (0, n)-set K

0 5 6 5 n t ( s - n + 1). (6)

Furthermore, 6 = 0 if and only if K is an n-ovoid, 6 = nt(s - n + 1) if and only if h' is a subquadrangle of order ( n - 1, t ) .

If P is a point of S not on A', we denote by e(P) the number of external lanes on P (so t + 1 - e(P) is the number of secant lines on P ) .

THEOREM 3.2 external lines meeting 1+ is a constant h which does not depend on I + , namely

For every secant 1+ of a (0,n)-set I< in a GQ(s, t ) , the number of

firthermore h satisfies

0 I h 1. t ( s - 72 + I), (8)

andh = O i f f l i ' isanu-ovoid, h = t(s-n+1) iffICisasubquadrangleoforder(n-1,t).

Let 1+ = { Q l , . . . ,&*,PI, . . . ,Ps+l-n} with Q; E K and Pj $ K ( i = 1,. . . , n ; j = 1 , . . . ,s + 1 - n). Since S is a GQ(s, t ) , the points of Ii' are partitioned by the pencils of lines on the points of I+. Thus

Proof

S+l-n

k = n[(n - 1)t + 11 + 11 (t - e(Pj)]. (9) j=1

s+1-n Hence the number h = e(Pj) does not depend on I + , and

j=1

k = n(st + 1 - h).

This, by (3) and (4) yields

6 = nh,

thus, in particular

IL-1 = ( t + l ) h , IL+I = ( t + l ) ( s t + 1 - h) ,

144 A. Del Fra e t al.

which proves (7). Clearly by (6) we have (8). I

As an immediate consequence (see also (3)) we have the following:

COROLLARY 3.3 If there is a (0, n)-set K in a GQ(s, t ) , then

n must divide k and 8, (13)

( t + 1) must divide IL-1 and IL+I. (14)

REMARK 3.4 In a similar way, if L- # @,we choose a line 1- = { P I , . . . ,Ps+l}. Since S is a GQ, the s + 1 pencils of secant lines on the P,!s partition the points of K. Thus

S+l

k = n(t + 1 - e(P,)). i=l

(15) k

This implies that the number h' =

does not depend on the fixed 1 - . Since 1- is counted in the CPE,- e ( P ) exactly once for each Pi E I - , we see that h' - (s + 1) is t he total number of ex ternal l ines dif ferent from a chosen 1- meet ing tha t I - . Obviously

e(P) = (s + l ) ( t + 1) - ;, PEI-

k = .[(s + l ) ( t + 1) - h']. (16)

h'= h + s + t , (17)

h ' > h > O and h ' > s + t . (18)

(18). I

THEOREM 3.5 With the above notation, if L- # 0, then

so that

Proof Since L- # 8, h > 0. Comparing (10) and (16) we get (17) and obviously

As a consequence of (17), we have for the deficiency 6 = nh, IL-1 = ( t + l ) h , and IL+I the following expressions as functions of h'

e = n(h' - - t ) , (19) p-1 = ( t + l)(h' - s - t ) , (20) IL+I = ( t + l)[(t + l ) (s + 1) - h'].

Furthermore, since now h 2 1 we have from (8) and (17)

s + t + 15 h' 5 t ( S - n + 2) + s, and h' = t ( s - n + 2) + s if and only if I< is a subquadrangle of order ( n - 1, t ) .

THEOREM 3.6 If I< is a (0, n)-set, then

k 2 (3 + l)[n(t + 1) - (s + t ) ] = bz , (22)

and equality holds if and only if e ( P ) is a constant, say e , for every P of S not in I<, then we have L- # 0 (i.e. # 0) and

(0.n)-sets in a generalized quadrangle 145

k -- - t + l - - h'

n s + l - n s + l n(s + 1)' - - h

- s + t e = - -

In particular, if equality holds, then n divides s + t . Proof Inequality (22) was proved in 2.2 in a more general situation. If equality

holds in (22), then each point P $ K is collinear with exactly k / ( s + 1) points of K (see 2.(5)). Now, on each of the t + 1 - e(P) secants on P there are exactly n points of K , hence

k - = ( t + 1 - e(P))n , s + l

which proves that e(P) must be a constant for all P $ I<, namely k

n(s + 1) e ( P ) = t + 1 - ~ = e . (24)

Now, by definition of h' and h we have h = (s + 1 - n)e, h' = (s + 1)e, (25)

(26)

so that by (17)

h' - h = s + t = n e ,

and e = (s + t ) / n . This with (24) and (25) gives (23). Conversely, if e (P) is a constant e for all P $ I<, from (25) and (26) we get

h' = (s + l)(s + t ) / n . Substituting in (16) we obtain k = (s + l )[n( t + 1) - (s + t)], so that in (22) we have equality.

REMARK 3.7 bound

I

From Theorem 2.1 we also have for the size k of a (0,n)-set the

k 2 n[(n - 1)t + 11 = b1, (27) and as already noted bl is better than the bound bz given by (22) if and only if

S n < 1 + - .

t If n = 1 + s / t , then bl = bz = (1 + s ) ( l + s / t ) . It is easy to prove that I< is a subquadrangle, with constant e (P) if and only if n = 1 + s / t and k = bl = bz; we note that in this case K is a subquadrangle with order ( s / t , t ) .

EXAMPLES 3.8 following.

K = {Q,R}'U {Q,R} lL .

Here n = 2,

Known examples with n = 1 + s / t and k = bl = bz are the

(i) s = t , and S has some regular pair (Q, R) of non collinear points. Then

k = 61 = bz = 2(1+ t ) .

(ii) s = t2 = qz and K is the pointset of W(q) (see (28]), chapter 3) as a subquadrangle of H(3, q'). Here

= l + s / t = 1 + q , k = bl = bz = (1+q)(q2 + 1).


(iii) (9, t ) = ( q 3 , q z ) and K is the pointset of the point-line dual of H(3, q z ) contained as a subquadrangle in the point-line dual of H(4, q 2 ) . Here

n = 1 + s i t = 1 + q ,

Theorem 3.6 is clearly equivalent to

THEOREM 3.9 satisfies

k = bl = b2 = (1 +q)( l + q 3 ) .

The deficiency 6 = nlL-l/(t + 1) = n(st + 1) - k of a (0,n)-set K

6 I ( s + 1 - n) ( s + t ) , (29)

and equality holds if and only if e(P) = e is a constant (see (23)). This upper bound for 6 is better than that of (6) when n > 1 + s i t .

Similarly, we have for h = 6/n and h' = h + s + t the obvious upper bounds, improving (8) and (21) when n > 1 + s i t .

From 2.2 and 3.6 we have immediately the following corollary.

COROLLARY 3.10 Let I( # 0 be any subset of points in a GQ(s,t) and let L+ be the set of secant lines. If

n = minltELt IZ+ n Kl and k = IK(1 = (s + l)[n(t + 1) - (s + t ) ] ,

then K is a (0, n)-set and for every point P 9 Ii the integer e(P) is the constant e given by (23).

4. (O,s)-SETS IN A GQ(s,t)

REMARK 4.1 Let S be a GQ(s, t ) , s 2 1, t 2 1. If s = 1, then S is a dual grid, i.e., a complete bipartite graph with parts PI and Pz. In this case a pointset Ir' c PI U Pz is a (0,2)-set precisely when K c PI or K c P2. So we may suppose that s 2 2. Then the complement A of a (0,s)-set I( has the property that each line having at least two points of A must lie entirely in A. It follows from 2.3.1 of [28] that there are precisely the three possibilities given in the next result.

THEOREM 4.2 if and only if one of the following occurs:

Let S be a GQ(s, t ) with s 2 2. A k-subset Ir' is a (0, s)-set in S

(i) K is the complement of an ovoid. Here

L- = 0, e ( P ) = h = 0 VP E S - I<, k = s (s t + 1).

(ii) IC is the complement of R I , where R is any point of S. Here

e(P) = h = 1 VP # R, P E S - K , k = s 2 t .

(iii) K is the complement of a subquadrangle of order ( s , t ' ) with t' = t / s . Here

e (P) = h = 1 + $ VP E S - I<, k = (s2 - 1)t.

(0.n)-sets in a generalized quadrangle I47

5. (0, n)-SETS WITH STRONGLY REGULAR POINT-GRAPH

Let S be a GQ(s, t ) and let r be the point-graph of S. If H is any subset of points in S, then the graph induced by I' on H will be denoted by l?(H).

THEOREM 5.1 regular then

Let I( be a (0,n)-set in a G Q ( s , t ) say S. If r(K) is strongly

k = IK() I n + t (n - 1)[1+ (t + l ) ( n - I ) ] = ,& ( 1 ) The upper bound pz is strictly better than the bound satisfies

= n(s t + 1) given in 3.(5) if n

2t + s + 1 - J4ts + ( s + 1)Z

2(t + 1) 2t + s + 1 + J4tS + (s + 1)2

2( t + 1 ) < n <

Proof If r(K) is strongly regular, then it has parameters (V,k,X,p), with - - v = k,

Thus n 2 2 and

c1= -

Since 5 -

k - ( n - l ) ( t + 1) - 1 5 ( n - l ) ( t + I ) [ ( . - I ) ( t + 1) - ( n - I ) ] .

Hence

k I (n - l ) ( t + 1 ) + 1 + ( n - 1)'(t + l ) t ,

which gives (1 ) . If

p1 - P Z = n(s t + 1) - n - t ( n - 1)[1 + ( t + l ) (n - I ) ] > 0,

k = ( n - l ) ( t + I ) , X = n - 2.

- - k(k - X - 1 )

v - x - 1 . - 1 > 0 and p 1 1, this implies

p2 is strictly better than the upper bound pl. It is elementary to see that ( 5 ) holds if and only if

( t + l)n2 - (2t + s + 1)n + t < 0,

which yields ( 2 ) , since the discriminant 4ts + (s + 1)2 is always non negative. I

Obviously, if K is an n-ovoid (i.e. k = P I ) , then n does not satisfy (2 ) . In this case, Thas proved in [45] the following theorem, using a matrix technique which was developed in a slightly less general form by Cameron in [S] .

THEOREM 5.2 then one of the following cases occurs:

(i) n = (s + 1 ) / 2 and t = s2. (ii) n < (s + 1) /2 ; if n = 1 then t I s2 - s; if n > 1 then t 5 s2 - 2s. (iii) n > (s + 1) /2 and t E {s2 - s,s2 - s - 1); if t = sz - s - 1 then n # s. The graphs r(K) and I'(S - K ) are both strongly regular iff one of the following

Let Kbe an n-ovoid of the GQ(s , t ) , S . If r ( K ) is strongly regular,

occurs.

148 A. Del Fru et ul.

(a) n = (s -+ 1)/2 and t = s2, (b) n E (1 , s ) and t = s2 -s.

Proof See Thas [45], section 3.

6. A CONSTRUCTION OF (0,n)-SETS FROM OVOIDS

In the remainder of this paper we shall give several examples of (0,n)-sets in the known generalized quadrangles. Most of these examples are special cases of the following general construction.

THEOREM 6.1 Let S be a GQ(s , t ) with $2 1) distinct ovoids 81,. .. ,Qy, pairwise intersecting in the same set X with x = 1x1 points. Then, for every n, 1 < n 5 y the set

n

I< = upi -X), (1) i= l

is a (0, n)-set with k = n(st -+ 1 - x) (2)

points.

Proof It follows immediately from the definitions. I

We note that when y = 1 then X = 8 and x = 0. In the particular case x = 1 we obtain:

THEOREM 6.2 For each n = 1,. . . , s there is a (0, n)-set in S' with k = nst'

points.

By 2.2.1 of [28] each point of S not in S' is collinear with the 1 +st ' points of an ovoid of S' (see Thas [44]). Suppose I is a line of S - S' on a point PO of S'. If PI,. . , , P, are the other points of S on 1, each Pi, i = 1, . . . , s, determines an ovoid 0; of S', so that each two of the ovoids have only Po in common.

On these s ovoids we have exactly s (s t ' ) = s2t' points of S' different from PO. Since there are exactly (s -+ l ) ( s t ' -+ 1) - [(t' -+ 1)s -+ 11 = s2t' points of S' not collinear with PO

Let S' be a subquadrangle of order ( s , t ' ) in a GQ( s, t ) , (1 5 t' < t ) .

(3)

Proof

s

U ( U i - { P O } ) = S' - P,I (4) i= 1

For each n = 1,. . . , s, the set n

I< = up: - {Po}) i=l

is a (0,n)-set of S' with nst' points. 1

(5)


EXAMPLES 6.3 With the notation of 6.2 and the point-line duals of the examples given in 3.8 (Q(4, q ) is the dual of W(q) and Q(5, q ) is the dual of H(3,q2); see Section 3.2 of [28]), we have the following three types of examples:

( s , t ) = (q ,q ) or (q ,q2) . Both Q(4,q) and Q(5,q) have regular pairs of non concurrent lines. Here

(i) ( s , t ' ) = (q , 1);

k = n q , l s n s q (6) so n < 1 + s / t ' = 1 + q and the best possible bound for k is bl = n2 < k = nq (see 3.7).

(4 (3, t ' ) = ( Q , q ) ; ( $ 7 4 = (q , 8). Here S

k = nq2, 1 + ~ = 2 I n I q , (7) but the better bound b2 is smaller than k . For example, Q(4, q ) is contained in Q(5, q) , but other examples occur even in the non classical cases (see Kantor [20],[21], Payne and Maneri [27], Payne [23]-[26]).

(iii)(slt') = ( q 2 , q ) ; ( s , t ) = ( q 2 , q 3 ) . Here

k = nq3, 1 + disp lays ty le5 = 1 + q1 1 5 n 5 q 2 . (8) In all cases k is larger than bl or b2.

REMARK 6.4 Obviously we cannot use the construction in 6.1 when S has no ovoids. However, if S has a spread (i.e. a set of lines such that each point of S is incident with a unique line) then the point-line dual ST of S is a GQ(t , s) with an ovoid and we can use our construction in ST.

First we examine the classical cases.

REMARK 6.5 The generalized quadrangle Q(5, q ) of order (q, q 2 ) arising from an elliptic quadric in PG(5, q ) has no ovoids, since t > s2 -s (see [28], 1.8.3). The following theorem proves that H(3, q 2 ) has ovoids with common intersection (and so (0,n)-sets constructed as in 6.1).

THEOREM 6.6 Let H=H(3,q2) be the GQ(q2,q) given by the points and lines of a nonsingular hermitian variety of the projective space PG(3, q 2 ) . For every point P E PG(3, q 2 ) - H the polar plane P" of P meets H in an ovoid say O p = P" n H. For each n = 2,. . . , q(q - 1) there is a (0, n)-set K in H with

k = nq(q2 - I ) , (9) obtained as in 6.1 from n of the above ovoids (here the common intersection X is 1 n H , where 1 is a line of PG(3, q 2 ) meeting H in q + 1 points).

Proof Let T be the polarity of PG(3,q2) determined by H. If P $ H the intersection P" n H is a nonsingular hermitian curve C, hence has q3 + 1 points. Clearly the q + 1 lines of H on a point Q of C are the q+l components of the singular hermitian curve Q" n H.

For every non tangent line 1 of PG(3,q2) (i.e. I I n HI = q + l ) , let P I , . .. ,Py (y = q2 - q ) be the points of 1 external to H. Th e y ovoids

Opi = Pi" n H , (10)

I50 A. Del Fra et al.

pairwise meet in the polar line 1" = OPE, P". Since z = 11" n HI = q + 1, the set

IC = U(oi- (1" n H I ) ,

is a (0,n)-set with k = n(q3 - q ) points.

n

2 I n I y i=l

I

REMARK 6.7 In the examples of Theorem 6.6 b1 = n[(n- l ) t+ l ] is the best bound for q = 2 and when q > 2, 2 5 n < q + 1; when q > 2 and q + 1 < n 5 q2 - q then bz is the best bound; while b l = bz = (4' + l ) ( q + 1) for n = q + 1, but Ic is greater than its lower bound.

EXAMPLES 6.8 The GQ(q2,q3) of points and lines on a nonsingular hermitian variety H = H(4,qZ) of PG(4,q') has no ovoid (see [28], 3.4.1). For q = 2 it has no spread (Brouwer ["I). For q > 2, whether or not it has a spread seems to be an open problem. Therefore we cannot use the construction in 6.1.

However we have examples of (0, q2)-sets in H(4, q 2 ) , taking complements of hyperplane sections. The complement in H of a tangent hyperplane is a (0,q2)-set with k = s 2 t = q7 points (see 4.3 (ii)). While a complement of a non tangent hyperplane is a (0,q2)-set with Ic = (s2 - l ) t = (q4 - l ) q 3 points (see 4.3 (iii)), we note that here k attains the lower boud bz of 3.6.

REMARK 6.9 In the next section we shall give examples of (0,n)-sets in the GQ of order q TZ(R2) discovered by Tits (see 7.3 and 7.4 below). The quadrangle Tz(R) is isomorphic to Q(4, q ) iff 52 is an irreducible conic, it is isomorphic to W(q) iff q is even and R is a conic (see [28], 3.2.2). Using these isomorphisms the examples in 7.4 will yield examples in Q(4, q ) and, for q even, in W(q).

7. EXAMPLES IN THE GQs ASSOCIATED WITH A N OVAL

DEFINITION 7.1 Let x = PG(2, q ) be embedded as a plane of C = PG(3, q) , q odd or even. Let fi = {Po,. . . , P,,) be an oval of 7r and let ti be the line of x tangent to R at Pi (0 5 i 5 4). We consider the structure S = ( P , L , T ) where the set of points is P = PI U Pz, with PI = C - x the pointset of the 3-dimensional f i n e space, and Pz the set of planes in PG(3,q) containing at least one ti (in particular x is a point of Pz). Now L is the set of lines of PG(3,q) meeting R in exactly one point. The incidence Z is induced by the incidence in C (for example the "point" T is incident with the q + 1 lines to , . . . , t , , ) . One can prove that the structure S is a GQ(q, q) .

In fact, it is easy to see that Definition 7.1 is just a slightly revised description of the GQ of order q Tz(C2) first given by Tits (see Dembowski [ll], pag. 304, or [28], 3.1.2). Since each of t o , . . . , t,, is a regular line, we may expand about t i , as in the point-line dual of 3.1.4 of [28], to obtain a GQ P(T2(R),ti) of order ( q + l , q - l), defined below.

DEFINITION 7.2 Let 7r = PG(2, q ) c C = PG( 3, q ) and let R- = {PI,. . . , P,,) = R - PO),^ an oval of x . Each point P, of R- (j = 1,. .. , q ) , has two tangents: the line I , = POP, and the tangent t , to R at P,. Put


P; =PI = C-A,

p ; = { p ( p # ~ i s a p l a n e o f C; p n ~ = t j , j = l , ..., q } ,

P; = { p I p # A is a planeof C; p n x = ljl j = l , . . . , q } .

Now P- = PF u PT U P;, L- is the set of lines in PG(3,q) not in A and meeting A in a point of R-. With the incidence 2- induced by the incidence in PG(3, q ) , the structure S - = (P-,,C-,I-) is a GQ of order ( q + l , q - 1 ) .

When q is odd S - = P(Tz(R), t o ) is isomorphic to the point-line dual of the GQ(q- 1 , q + 1) discovered by Ahrens and Szekeres [ l ] (see [28], 3.1.5 and 3.2.6).

When q is even this construction is more general than the construction of S+ to be given below, due to Ahrens and Szekeres and to Hall ([l] and [17], respectively).

When q is even, there is a unique point P, of A , called the nucleus of R, for which R+ = R U {P,} is a hyperoval (i.e. a q + 2)-arc). The point A of S = Tz(R) is regular, so we may expand about A as in 3.1.4 of [28] to obtain a GQ(g - l , q + l), defined as follows.

DEFINITION 7.3 Let R+ = 52 U {P,} = {Po, . . . , Pq, P,}be a hyperoval of the projective plane A = PG(2, q ) , q = 2 h , and let A be embedded as a plane in PG(3, q ) = C. The incidence structure S+ = (P+,L+,Z+) with pointset P+ = PI = PF = C - A,

lineset

L+ = {I I 1 is a line of C meeting A at a unique point of R+},

and incidence Z+ induced by that of C, is a GQ(g - l , q + l), and S+ = P(Tz(R),A). (See 3.1.4 of [28].)

Using ovoids with common intersection (see (6 ,1)) , we have examples of (0, n)-sets in the above defined GQs associated with an oval.

EXAMPLES 7.4 Let S = T2(R) be the GQ(q,q) associated with an oval R of x = PG(2,q) c C = PG(3,q) . Let Q be a point of A not in R (and different from the nucleus P, of R, if q is even). We choose a line r of C not in 7r and meeting A at Q. For each line I of A external to R and passing through Q, the plane 0 1 = IT determines an ovoid 8 of T(R) whose points are the point A E Pz and the g2 points of 0 1 - 1.

By Segre’s famous theorem (see [29]), if q is odd R is an irreducible conic, so Q can be exterior or interior to 0. There a.re on Q 7 exterior lines 1 1 , . . . ,1, to R with

if q is even, -

if q is odd and Q is exterior to R, . if q is odd and Q is interior to R -

The point A and the g points of T different from Q form the pairwise intersection X = 81i nolj, i # j of the ovoids O,l,...,Ql,. Hence z = JlXl = g + 1 and if 7 2 2 the set


n

I< = U ( U l i - X ) ,

k = n(q2 - q )

i= 1

is a (0, n)-set with

(2)

points for every n = 2,. . . , y (see (1)).

EXAMPLES 7.5 In the previous example let q be odd, so R is a conic, and S = T2(i2) is isomorphic to Q(4,q). In S let A be a set of q + 1 pairwise noncollinear points with the property that no three points of A are collinear with a same point of S. Such a set of points in Q(4, q ) will be called a BLT-set (following a suggestion by W. M. Kantor) because L. Bader, G. Lunardon and J. A. Thas [2] have pointed out the important role they play in other geometric constructions. For example, in the model of S as Q(4, q ) , any plane conic of PG(4, q ) lying on Q(4, q ) is such an A , and several other examples are known to exist (see [2]). Let I( be the set of points of S not collinear with any point of A. Then K is a (q(q2 - 1)/2)-set which is a (0,n)-set with n = ( q + 1)/2. This is essentially the content of Remark 2(c) of [2].

EXAMPLES 7.6 Let S- = T(S2-) be the GQ(q + 1,q - 1) defined in 7.2. PT and PT are both ovoids. If a is a plane of C different from x and meeting x in t o , then the points of cy - x form an ovoid of S-. These q + 2 ovoids partition the points of S-. It follows that the union of n of them forms a (0, n)-set with

k = nq2 (3)

points for every n = 1,. . . , q + 2 (see 6.1). Let Q E x - R- (Q # P, if q is even), and let r be a line of C not in x meeting 7~ at

Q. For each line 1 on Q external to R- the plane a/ = lr determines an ovoid as above. The q points of r not in x form the pairwise intersection of the ovoids 01,, . . . , Q 7 - ,

hence we have a (0, n)-set in S- with

Ic = " ( 4 2 - q ) (4)

points for every n = 2 , . . . , 7 - . Here

if Q $ ! t o , y + 1 if Q E t o . ( 5 )

One may check that in all the examples in 7.4 and 7.6 k > b2 2 bl (see section 2), and b2 = bl in the examples 7.4 if n = 2.

EXAMPLES 7.7 When q is even, the GQ(2h - 1,2h + 1) S+ defined in 7.3 has ovoids of the same type: all points of cr - x , where 0 is a plane meeting x in a line external to the hyperoval R+.

Let Q be a point of x - R+ and r be a line of C not in A , r n x = Q. Through r there are q/2 = 2h-1 planes ali meeting x in a line 1, exterior to R+ (i = 1,. . . , 2h-1). Since the q points of r not in x form the pairwise intersection of the ovoids Oli, we have

(0,n)-sets in a generalized quadrangle

a (0, n)-set in S+ with

k = n2h(2h - 1)

L 53

points for every n = 2 , . . . , P-' ( h 2 2).

EXAMPLE 7.8 We now construct in S+ a set K* of k* lines such that each point of S+ is in 0 or n = q/2 = 2h-1 lines of S+. So K* will be a (0,2'+')-set an the point-line dual of S+.

Let A' be a plane different from A and meeting K in a line I exterior to R+. Let R' be a hyperoval of A' for which I is also an exterior line. Let P o , . . . ,rq+i be the lines joining P, to the points of 0'. As above, ti = P-P,, i = 0,. . . , q. Let ti meet 1 at Q,. Through Qi there are q/2 - 1 lines rij, 1 5 j 5 q/2 - 1, which lie in A', are different from I and are exterior to R'.

For each i , j there are q lines mij,,, 1 5 p _< q, 0 5 i 5 q, 1 5 j 1. 4/2 - 1 which are incident with P, and lie in the plane P,r,j but not in A. Put

K* = {rni j , , : 0 5 i 5 q, 1 5 j 5 q / 2 - 1, 1 5 p 5 q } . (7)

By construction the plane P,r,j meets each of ro, . . . , rq+l only at the point P,. Let R be a point of C - A. Then if R is on one of ro, . . . , rq+l it is on no line of K*.

Suppose R is on none of ro , . . . , rq+l, and let R' be the point of P,R on A'. Then R' is exterior to 0' and is not in A. So there are q/2 lines r of A' through R' and exterior to R'. Of course each such r is one of the lines r i j , and the plane Pirij = P,r,j contains one line mijp through R. So R is on q/2 lines of K*. Hence I(* has

lines and each point R of S+ is on 0 or n = q/2 lines of K * . Again (the point-line dual of) bl = ( q + 2)(q - 4)q/2 is strictly less than k*.

For n = 1 + s / t we have examples in 3.8 of (0, n)-sets with k equal to the common

For n = 1, a single point gives k = 1 = b l .

For n = s we have examples with k = b2 taking complements of subquadrangles of order (s, t / s ) (see 6.8 and 4.2).

For 2 5 n 5 s - 1 and n # 1 + s / t , we have yet to construct examples that show bl and b2 are the best lower bounds we can expect in general. In fact we now accomplish this goal with n = 2.

THEOREM 7.9 with

lower bound bl = b2.

With ( s , t ) = ( q + 1, q - l), q any prime power, there is a (0,2)-set

k = b1 = 2q. (9)

Proof We use the GQ S- with the same notation as in 7.2. Let r be a line of C not in A meeting A at Po. The q planes Pir, i = 1 , . . . ,q are each points of Pq collinear with the q points of r not in A . This gives a q x q dual grid whose k = 2q points form a (0,2)-set. 1


THEOREM 7.10 With ( s , t ) = ( q - l , q + 1) = (3,5) there is a (0,2)-set I< with

k = bz = 16

points. Exactly half of the 96 lines are secant to IC and half miss K. As a pointset Ii’ is 4-tight (see [26]). The 48 secants form an 8-tight set of lines meeting each of the 4608 spreads in exactly 8 lines

(10)

Proof We list the points in I<, starting with the hyperoval.

R+ = {(0,0,1,0), (0, L0,O)) u {(1,C,CZ,O> I c E GF(4)). (11)

The points of the GQ(q - 1 , q + 1) S+ defined in 7.3 are the points of C - A where A = [O ,O,O, 1IT. The lines of S+ are the lines of C meeting T in a point of R+. Then the points of K are:

Here GF(4) = {0,1, I, t2}, with 1 + E = E2. We have to show that each line of S+ incident with a point of Ii‘ is incident with

a unique second point. Let P = ( u , v , u + w , l), with v = 0,1, w = 0 , l be a point of I(. It is elementary to verify that the line on P and Q = ( l , c ,c2 ,0) with c # 0 , l is incident with the unique point P‘ = (u + c + 1, v , u + c + w , 1) E IC; the line on P and Q = (1,1,1,0) is incident with the unique point P” = (u+ 1, v, u + w + l , 1) E I<; the line on P and Q = (1 ,0 ,0,0) is incident with the unique point P”’ = (u, TI + 1, u + w , 1) E I<.

It follows that K is 4-tight. Then there must be (1 + t ) k / n = 6 . 16/2 = 48 sccant lines. Each secant line must meet 12 other secant lines. Finally, since each spread covers the points of I<, it must do so in exactly 8 secants. 1

REFERENCES

[l] R.W.Ahrens and G. Szekeres, On a combinatorial generalization of 27 lines

[2] L. Bader, G. Lunardon and J . A. Thas, Denvation of flocks of quadratic cones,

[3] A. Barlotti (ed.), Combinator ia l and geometric s t ruc tures and the i r applications,

[4] A. Barlotti, M. Marchi and G. Tallini (eds.), Combina tor i c s ’81 in honour of B.

associated with a cubic surface, J.Austral. Math. Sac. 10 (1969), 485-492.

F o r u m Math. 2 (1990), 163-174.

Ann. Discr. Math. 14 (1982).

Segre, Ann. Discr. Math. 18 (1983).

(0.n)-sets in a generalized quadrangle 155

[5] A. Barlotti, M. Marchi and G. Tallini (eds.), Combinatoraal incidence geometry: principles and applications, (La Mendola 1982), Rend. Sem. Mat. Brescia, 7, Vita e Pensiero, Milano (1984).

[6] A. Barlotti, M. Marchi and G. Tallini (eds.), COmbinatOTiCS '86, Ann. Discr. Math. 37 (1988).

[7] A.E. Brouwer, Private communication 1981. [8] P. J. Cameron, P. Delsarte and J.M. Goethals, Hemisystems, orthogonal config-

[9] P.J. Cameron, J.W. Hirschfeld and D.R. Hughes (eds), Finite geometries and

[lo] P.J. Cameron, D.R. Hughes and A. Pasini, Extended generalised quadrangles,

[ll] P. Dembowski, Finite geometries, Springer Verlag (1968). [12] A. Del Fka and D. Ghinelli, Large extended generalized quadrangles, ATS.

[13] A. Del Fra, D. Ghinelli and A. Pasini, Diameter bounds for an EGQ, J . C.I.S.S.

[14] D. Ghinelli, Varieth hermitiane e strutture finite, Rend. d i Mat., (1-2), 2

[15] D. (Smit) Ghinelli, Varieth hermitiane e strutture finite, 11, Rend. di Mat., (l) ,

[16] D. (Smit) Ghinelli, On the applications of hermitian geometries for the con-

[17] M. Hall Jr., Affine generalized quadrilaterals, Studies in Pure Math. (ed. L.

[18] J.W.P. Hirschfeld, Projective geometries over finite fields, Clarendon Press

[19] D.R. Hughes and F.C. Piper, Design theory, Cambridge University Press

[20] W.M. Kantor, Generalized quadrangles associated with Gz(q), J . Comb. Th.,

[21] W.M. Kantor, Some generalized quadrangles with parameters (q', q) , Math. 2.

[22] S.E. Payne, An inequality for generalized quadrangles, PTOC. Amer. Math.

[23] S.E. Payne, Generalized quadrangles as group coset geometries, Congress. Nu-

[24] S.E. Payne, A garden of generalized quadrangles, Algebras, Groups and Ge-

[25] S.E. Payne, A new infinite family of generalized quadrangles, Congress. Numer.

urations and dissipative conference matrices, Philips J . Res. 34, 3/4 (1979), 147-162.

designs, LMS Lecture note ser. 49, (1980).

Geom. Ded. (to appear).

Comb., 29 A (1990), 75-89.

Proc. of the 2nd Catania Comb. conf.,l5 (1989),256-270.

(1969), 23-62.

6 (1973), 13-29.

struction of association schemes and designs, Convegni Lincei 11.17 (1976), 213-228.

Mirsky) Academic Press (1971), 113-116.

Oxford (1979).

(1985).

A 29 (1980), 212-219.

192 (1986), 45-50.

SOC., 71 (1978), 147-152.

mer. 29, Proc. 12th S.E. Conf. Comb. Graph. Th. and Comp. (1980) 717-734.

ometriea 3, 1985, 323-354.

49, (1985), 115-128.


126) S.E. Payne, Tight pointsets in finite generalized quadrangles, Congress. Numer.

[27] S.E. Payne and C.C. Maneri, A family of skew-translation generalized quad-

[28] S.E. Payne and J.A. Thas, Finite generalized quadrangles, Pitman Boston

[29] B. Segre, Ovals in finite projective planes, Canad. J. Math. 7 (1955), 414-416.

[30] B. Segre, Lectures on modern geometry (with an appendix by L. Lombard0

[31] B. Segre, Fonne e geometrie hermitiane con particolare riguardo a1 caso finito,

[32] G. Tallini, On caps of kind s in a Galois r-dimensional space, Acta Arith. 7

[33] G. Tallini, Problemi e risultati sulle geometrie di Galois, Rel. n. 30 1st. Mat. Univ. Napoli (1973), 1-30.

[34] G. Tallini, Graphic characterization of algebraic varieties in a Galois space, Teorie combinatorie, Atti dei Convegni Lincei 17, Itoma (1976), 153-165.

[35] G . Tallini, Problemi di immersione nei sistemi di Steiner, Sem. Geom. Comb. 21, Roma (1979).

[36] G. Tallini, Finite line spaces and k-sets of PG(r,q), Conf. Sem. Mat. Univ. Bari 192, (1984), 1-24.

[37] G. Tallini, On sets of given type in a Steiner System, Finite Geometries (Win- nipeg, Man., 1984), Lectures notes in pure and appl. Math. 103, Dekker, New York

[38] G. Tallini, Teoria dei k-insiemi in uno spazio di Galois. Teoria dei codici correttori, Sem. Geom. Comb. 64 (1985), 1-139.

[39] G. Tallini, Some new results on sets of type (rn,n) in projective planes, J. Geom. 29 (1987), 191-199.

(401 M. Tallini Scafati, Sui {k,n}-archi di un piano grafico finito, Rend. Accad. Naz. Lincei (8) 40 (1966), 1-6.

[41] M. Tallini Scafati, { k, n}-archi di un piano grafico finito con particolare riguardo a quelli con due caratteri, Nota I, 11, Rend. Accad. Naz . Lincei (8) 40 (1966), 812-818,

[42] M. Tallini Scafati, I k-insiemi di tip0 (m, n ) di uno spazio &ne A(r,q), Rend.

[43] M. Tallini Scafati, Recent results on (m,n)-type k-sets in an A n e plane a*,

[44] J.A. Thas, A remark concerning the restriction on the parameters of a 4-gonal

[45] J.A. Thas, Interesting pointsets in generalized quadrangles and partial geome-

60 (1987), 243-260.

rangles of even order. Congress. Numer. 36 (1982), 127-135.

(1984).

Radice), Cremonese, Roma (1961).

Ann. Mat. Pura A p p l . (4) 70 (1965), 1-202.

(1961/62), 19-28.

(1985), 307-319.

1020-1 025.

Mat. Roma VII (1) (198l), 63-80.

J. Geom. 29 (1987), 94-100.

subconfiguration, Simon Stevin 48 (1974-75), 65-68.

tries, Lin. Alg. and A p p l . 114/115 (1989), 103-131.


[46] J.A. Thas, A note on spreads and partial spreads of hermitian varieties, Simon Stevin, 63 (1989), 101-105.

[47] J. A. Thas, Old and new results on spreads and ovoids of finite classical polar spaces, Combinatorics '90, Abstracts and texts, (eds. A. Barlotti, A. Bichara, P. V. Ceccherini, G. Tallini), GAETA, Hotel Serapo, May 20 - 27, 1990.


Combinatorics '90 A. Barlotti et al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved.

A

0

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159

A B C D E F O H I J K L .UN 0 P Q R 8 T U V W X Y Z A

6 D E F G E 1 J K L Y N O P Q R S T U T W X Y Z A B Y

E F Q H I J K L M N O P Q R B T U V W X Y Z A B ~ ~ D W 6 O H I J K L Y N 0 P Q R 8 T U V W X Y Z A B V D E V

H I J K L M N O P Q R 8 T U V W X Y Z A B ~ ; D E ~ a T I J K L Y N O P Q R B T U V W X Y Z A B V D E E Q H B J K L Y N 0 P Q R B T U V W S Y Z A B U D E F O 8 I R K L Y N O P Q R B T U V W X Y Z A B V D E P Q H I J Q

M N O P Q R 8 T U V W X Y Z A B C D E E Q E I J K L 0 N O P Q R B T U V W X Y Z A B U D E F a H I J U L Y N O P Q R B T U V W X Y Z A B V D E P O HI J K L Y N II P Q R S T U V W X Y Z A B C D E F O H I J K L Y N O L

B C ~ E E ~ H I J K L Y N O P Q R ~ T U V W X Y Z A z

D E F ~ H I J K L Y N O P Q R B T U ~ W X Y Z A B V x

an I J K L V N O P Q R ~ T W V W X Y Z A B C D E F u

L Y N O P Q R ~ T U V W X Y Z A B V D E P O H I J K P

O R B T U V W X Y Z A B v D E F an I J K L M N o P K R B T U V W X Y Z A B U D B E G H I J K L Y N O P Q I

T u v wx Y ZA B c D E P o n I J E L M N OP Q R 8 e I V V W X Y Z A B C D E F R H I J K L Y N O P Q R B T a

8 T U V W X Y Z A B V D E F O R I J K L Y N O P Q R I

V W X Y Z A B C D E E R H I J K L M N O P Q R B T U P W'X Y Z A B V D E F O H I J K L M N 0 P Q R 8 T U V E X Y 2 A 0 V D E F a H I J K L U N 0 P O R 8 T U V W D Y E A B C D E I Q H I J K L Y N O P Q R B T U V W X C Z A B C D E F a H I J K L M N O P Q R B T U V W X Y B

COMBINATORICS AND CRYPTOGRAPHY

Franco EUGENI Diprtmjmto di Inqeqneria E l e t i i'icd

Chiversit.' I.+ L'Apila - I ta ly

The purpse of t h i s lec t -ure is not t u give a survey aLwut (?riptiiluqy, but. t.o int.rcdiice t h e basic i d e a s of t - h i s Science. I b e l i e v e that the

I t a l i a n h i s t o r v (and nm-e g e n e r a l l y " E u r o p n 1~ist.ot-y) of C q p t o l o g y

from Caesar to World War 11 c a n be red as a Chapter of H i s t o r y of

Combinat-orics. The nlat:hematical p r i n c i p l e s i n v o l v t d are of c d i n a t o r i a l t > p . For exanple , the square used i n t.he Viqenei,e cocle (Viqenere 1522-1.596) is p r o b d > l y t.he f i r s t algebrair: s t r u c t u r e ! p r c ~ i s e l y the r i n g of r e s i d u e classes nndulus 26 1 used i n t.he h i s t o r y of Phthenmt.ics. Such

s t r u c t u r e t h a t appa r s i n Tri t .henuus ( 1462-1516) is older t.han Vigenere aqe, t .hat is t h e arr:.ient. z i ruph . The Viyenere square w a s r e d i s c o w r e d by

many cr i p t i q r a p h e r s

Fi9.1 Vig e n e r e s q u a r e

160 F. Eugeni

After 1900 neny new d i r e c t i o n s of r e s e a r c h b v e k e n propsed. I h i ~ p ? t h a t some young M a t . h m t . i c i a n l wnrkinq i n C o n h i n a t o r i c s , c a n feel t h e i-*hanli of t h i s C;c'iencce. W e dre goiny t o s h o w s o n e i:onnect.ions k t - w e e n i:onhinat.r-ltics and t.he real w o r l d c f Elect-ronic Cixmunicat ions.

Suippse t.bt J t . ransnut t .er 'T wants to send a " t o p secret" n e s s a g e to a

wceivei- R. .q nOn-dUthc:JYlZd tnd c ~ i v can i n t e r f e r e i n two ways.

1. Me c a n j u s t read t h e nrssaqe without. pnnissiixi.

I n o r d e r to prevent. t h i s , we c a n p r o t e c t the mssage bv us ing a

c r y p t c q r a p h i c svstem.

I f ' someone i t a h l r to r c ~ d a ~csret. message without key, w e say that the cot i c has hcen d?crittcd or broken. During World Wars I and 1 1 many

cpnsat ional code-breaking occurred. T h e cryptosnalysis o f Zimmerman telcgramm in 1917, was read hy Woodrow Wilson a t American Congress. ZimnitTrrnan promtscd !sccretuly) a p'irt o f USA t o Mexico, D O United States drclared w a r on Germany. 'The Llnitcd States Navy's great victory at Midway

lolsnda (1'342) w a a a consequence o f srrrft knowledge of Japan's PURPLE machine code. Particrllnr rmphasis m u s t be placed in t h e studies ahout

ENIGMA, the german f a m o u s chipher machine, one of causes of the german milit.ary failure in W a r 1 1 . We rememher a!:o the m o s t famous incident

about Pearl Arhour. The decritted message did not send 3 n the J U S ~

tahlr!

yLg.2 Enigma machine

2. H e c a n change t h e mssage so that "pay one thousand d o l l a r s to Mr. C" , f o r example, becomes " p y one thousand dollars t o m. X" and

.Mr. Y 1s t h e saw hid quy.

Combinatorics and cryptography 161

I n order t o prevent t h i s , w e can use an au then t i ca t ion system.

T h e i d e a o f a n air!henticat:cn s y c t o m l c d s to p r o j e c t i * ~ c s p a c e s a n d

d e s i g n s in a n a t u r a l w a y a n d in c o n n e c t i o n w i t h G i l b e r t - M c W i l l i a m - a n d

S l o a n e T h c o r c n i .

3. I n some cases t h e mssacie g ives an order, for example of nulit.ar type , or a payment order or t h e acces to a syst.em or ot.her. Scmlrtims t.he order can be r ea l i zed i f s eve ra l pople agree. In order t o ob ta in t .his w e can use a threshold schem.

The following sc*t.ions are devoted t o t h e s e t h r e e problems.

2. CRIPTOGEUWHIC SY-: THE ROMANTIC PERIOD.

Now w e are going to t a l k about. t h e f i r s t . problem. Criptology w a s born when m n ' s evolu t ion leads t h e politic and t h e

wars. The European scenar io of CrlTpt.ography from 1400 to 1900

includes nunv nanes w e l l kncm i n Matlemat.ic-.s and o the r f i e l d s as t h e following Crypt.ologist.s 1500: Leon Ratt ista A l b e r t i , Giovan httista Pellaso, Giovan ht.t'ist.a Dr l l a Porta, Gerolm Cardano - who st.udied t h e

t h i r d degree equation and invented t.he universa l j o i n t said cartlanic joint. - B l a k e de Vigenere, Jdmnnes Trithemius, Francois V i e t e 1600 Francis Bacon, Jaques Ozanam, John Wallis 1800: J u l e s Verne, ELlgar A1 lan Pce , F r i t d e r i c h Kasiski , Lewis Carrol , AucJuste lierckohof f s , Lyon

Playfa i rChar les Whetstone, Sanuel Morse, Thomas Je f f e r son ( t h e t h i r d Presi-dent- of USA). The writers J u l e s Verne and Lewis Carrol. alias t h e

Professor C h a r l e s Dodson from Oxford t h e writer o f Alice's advent.ures i n Wunderland. A l s o Martin Gardrier w r o t e a didactic book about. Cryptc-raphhy. W r e recent.ly we have t-wo g r e a t nanles: Alan Turing and C l a u d e Shannon.

We can d iv ide t h e Cr ip to locy ' s h i s t o r y i n t.wo big pxids:

a ) The romantic {=rid (before Shannon p p r i n 1949)

b ) The modern age ( a f t e r Shannon pqxr).

Now w e p resent t h e tradit . iona1 mxlel of a connumicat ion system. Prec ise ly w e can de f ine a c iphe r system as follows.

I62 F. Eugetii


A B C U E F O H I J K L M '1 / W X Y Z X O P Q R S T U V

A B C D E F O H I J K L N I V W X Y Z N O P Q E B T U

A B C D E F O H I J H L C U V W X Y Z N O P Q R S T

o p l T U V W X Y Z N O I ' Q R B

I A B C D E F O H I J K L M qr 1 S T U V W X Y Z N O I ' Q R

A B C U E F O H I J H L Y R S T U V W X Y Z N O P Q

A B C D E F Q H I J K L Y u v ~ Q 1 8 B T U V W X Y Z N O P - ____

A B C D E F O H I J K L M P Q B B T U V W X Y Z N O 1 A B C D E F O H I J K L M-1 O P Q R B T U V W X Y Z N I

-~

____

A B c 1) E F o H I J K - L ~ - ~ _ _ .

_____-

i ____

w= I

consti-uctai bv k o n R3t.tist.d A l b e r t i , t h e fannus it-alian Architext. In fourteen sixt-y SIX (1466) A lhe r t i wl-0t.e a bwk that earned lmn the t i t l e of "Father of m:&rri Crypt.ol1 yiy" . Sou can see i n t h e f iqulure t. he A1txrt.i d isk .

Combinawrics and cryptography 163

m n s t n r t e d by Leon Rat.tish A l b e r t i , the fanruus i t d i a n W h i t e c t . In fourteen s i x t y s i x (1466, A l h r t i w r o t e a book that earned hi in the t.it.le

of "Father of rn:ldr?rn Ci-ypt.oll.n;f. You can see i n the f i g w e t.he A l b e r t i disk.

5ig.3 B e l l s Fortil rqunre and Albert( dI rk

Many plialphdbetic oxbs wre cs.utstnu=ted. We render iXwanni Battista cklla Rsrta fnm Naples (15581, italian writer, acientist a d

magican. His fane is linked to the invention of the dark ram. Now we

m l d like to talk about the mre used wlial@mbetic code: the Viqmere

code. This cale w u redmxwemd by a fwndman, B i a i w de Vii@nem. The

philosophy of V i g t m e m ade is the following. Fix an algebraic finite

structw W,*) such that v a, h c: G the awtios x*d 5 b &g

unirme g?lutbn, for etraaqde a finibe quasi-gnxlp or a gimp. bt UB

mndder an alphaat A C G and a Wmrd X of elements of G. It is easy

enciphring and kipherinq. We &in the Vigenere COQ when the

st.nu!ture G is exactly the g m of mi&ea ao&.lus 26. In other YID& 26

Caesar codes. The Vigenere txde was cXw18tmctd i n 138t 4 was

cumridered unbwkable for 300 w, it. via8 broken by Ehsisky in tN3 by

twiny a irtatiatical way : so called =&4 M s .

164 F. Eugeni

N c m w e present. a pe r fec t s e m r i t y system: t h e one t h pd. The

c l e a r t e x t a ,a ,..., a is a sequence of 1et. ters of lenght n. The keys are t.he scrpiences k l , k -,,..., k of letters of lenght ti. The enciphering

func t ion is t h e Vigenere systeni with a key word k l ,k2 , . . .,h. In o t h e r words t h e syst.en1 is a Vigenere sys t .m wit-h a lcey of s i z e as t.he t e x t . This system has k e n invent-ed by Vernani i n nineteen thii-t.y. Usually

t.he one t.uw pic1 is not used with 1ett.el-s but wi th b i t s and t h e sequences

arc binary sequences and t h e enciphering func-.t.ion is t h e sum nrr-lulus two.

1 : n

L n

The Shannon theory (1949) i n p l i e s that the one t i n e p d o f f e r s perfs& s e c u r i t y when all sequences of lenqht. n occur with t h e sane probab i l i t y : i n o the r words t h e key sequences nust. he chosen a t random. W e can do t h i s by f l i pp ing n t . in rs a f a i r co in (hed= l , taile=O).

I t has been reported that. t . l i e ho t 1 ine between t h e Wi1it.e House and the k;renll has been secured by a one tirw pad.

For real a w l i c a t i o n w e must produce au tonn t i ca l ly t h e random b i t s . I t

is indeed a se r ious problem to t ransmi t a l m y random key sei-.ret.ely. X

prac t i ca l so lu t ion is to substi tut .e t h e random key sequence by a

pseudorandom seyllence. Such sequences look a t f i r s t glance as random sequences, bu t they are determined by very few dab. So, t h e t.ranmutt.er

has t.u send cmly these f e w data!

Thi s p-~x:*ed~u-e dtx?s not, provide a solut-ion c€ t.hc key change prrhleni,

hut it is rmly a p r a c t i c a l so lu t ion . We pay f o r t h i s advantage: such

system are not perfect.ly secure. Today w e have pseudo-random sequences by using a technica l device sai.d

s h i f t - r e g i s t e r (cf. f i g 4 atid 5). They can k e r ea l i zed inhardware i n a

very e f f i c i en t . way and they are very f a s t . Moreover, t h e mathenutical theory of sh i f t . r e g i s t e r has been developed i n t.he last twenty years i n nBny d i r ec t ions . But. it. is too long to speak ahwt t h i s !

A s h i f t - r e g i s t e r corrsists of a svstem of n-ce l les i n series, each of which can conta in one b i t . I n a s h i f t r e g i s t e r t h e bit cont~airietrl i n t.he cell i a t tint? t is shift-ed i n cell i - 1 at. t.im t.+l. hlien a l l t . 1 ~ rells are f u l 1 w e can st.art.. So, t h e content. x o f cell c: on t.he r i g h t is t.he oritput c->€ t h e system, a l l t h e o the r values are s h i f t e d and t.he last cell c would be enpty arid w e have t.o f i l l it.

I1 0

n


FG.4 S h i f t r f q i ' ; ! c r

To ob ta in t h i s w e use t h e so-called f e d b a c k func t ion

x = € ( x , s ,..., x ) .

Suppose f u l l t-he state t , at. t i n = t +1 a f t e r t h e sh i f t . w have an OUTPUT arid an INPUT by using t h e feedback funcst,ion. So, when the st.at.e 1

is known and a l s o t h e feedback funct.ion is known the s h i f t r e q i s t e r can

work!

n 0 1 n - ?

When t h e feedlack funct ion is l inea r w e have t h e so-called Linear Feedback S h i f t Register of lenght n (LFSR) given by:

x = d s + a x - . . . + a Y n 0 0 1 1 n - 1 n - 1

W e de f ine t h e associated p lynomia l i n G F ( 2 ) as follows

If a se:juenc:e r e p t - s it.self a€t.er a c e r t a i n tmr? w e call it pxidic.

The sequences of A l i nea r f e d b a c k s h i f t - r e y i s t e r are pricxt-lic. It is

clear that a long period is better. W e need t.o o tka in l i n e a r sh i f t . r e g i s t e r which achieve t h e i r nmxirmun period 2"- I.

W e have a connect-ion w i t h pulinonw-a1 theory i n Galois f i e l d s . A l i n e a r

shift r e g i s t e r of lenght n has nmxineun p r i o c l 2"- 1 if and o n l v i f t h e asstsiat.ed plirionyal is p r i n u t i v r i n GF( 2 " ) . This means that. t h e [mlinomial is i r r educ ib le i n G F ( 2 ) has one of h i s zeros i n t h e set. of genera tors i n GF( 2" ) .

Tmhy LFSR are easv to cons t ruc t t.o produce periodic pseudo random sequences but t h e i r corrispunding secret. ccdes are easy to break. For

esmple i f w e known t-wo t m s n bit . of cleartest. and cor r i sp indinq c iphe r t ex t . and we kncxn t h e lenght. of t h e LFSR w e are able ti:, reconstiiict. the.

key.

166 F. Eugeni

%BIT SrATlC SWIFT REGISTER

Tho MC14014B ml MCI4OZlB &bil sIa11c shill rqislrrc #I* constructcd with MOS P-chwful a d N.chmnul onhaiimmonl mod4 dcvicrs in a single mondilkir ilructurr. Thaso shill rqistrrs lwd primary use in pw i l l t l .ww iJ drta comrrsion. svnchronow md asynchronous pirallol inw, uU oulpul data quauring; wd 0 t k . r pcnaral purmsr rqistrr rpo4isrlions requiring low power and/or high nois4 immunity.

Ouiasctnl Currrnr = 5.0 nA/prk* lvpiul Q 5 Vdc 0 Synchrrnoui Pardlrl Iiiwt/Ssrial Oulpul lMC140148l 0 Asynchronous Parallel IniwtlSvrirl Oulpul IMCI402lBl

Svnclironouc Serial InwtISoiial Output r Full Static Operalion lrom DC 10 svp. 6.0 MHz (11 IOV V m 0 “0“ Outputs IIM Sash. Sevmih. end Liphlli S11p0s

Ooulila Dioda Iiipul Piolrclion r Supplv Voliaga Rrnpo - 3.0 VJc 10 1 1 Vdc

Schoilky TTL Lomi la Two HTL Lords Oror Ihc Rated Tcmirv aiura Rmy. MC140140 Pin.lor-Pin Rr tdronenl (of C04014B MC140711 Pin.lor.Pin Replrrmunl In CD402IB

CWlblO 01 Ofivinp TWO Lmwpower TTL Loads. One Lw.lW*f

CMOS MSI tLow.mwEn COMPLEMENTARV MOW

&BIT STZTIC SHIFT REGISTER -

aig.5 C o m m e r c i a l shift r e g i - t e r

l1sua11y t h e pwple u s e s h i f t register w i t h m.ire t h a n one hundred h i t . W e have also (crmrirrrial p l a s t i c black case (:ontAuninq t-lw DES .

4. AUl’HEXTICATION SCHEMES.

Ttrlav i i L + n y o r i q i n a l idtt3s have keen rleveelopl i n ct-~~>t.ol iqy. One of t h e mxst-. lxrvasive pmblems i n nu1 i t a r y and i n c-.ormrrcial connumicat ions

system is t h e nee1 t-o a u t . h e n t i c a t e d i g i t a l messages [ 4 1 , [ 5 1 , [7 1 , 181,

[ 14 1 . Elany s w ~ ~ t ’ r Elect . ronic Rind T r a n s f e r n e t s ncml [ ~ ~ ~ ~ ~ t . ~ ~ ~ ~ r ~ ~ ~ J t i i , *

syst-enis. They need p r o t e c t i o n and security against. unauthor ized int . rusi tm

of a LHCI qiy . T h i s c x x u r s , for exanlple, i n t h e Snlart. Card’s syst.enls. I n a

Combinatorics and crypwgrapplty 167

!hut Card's system the receiver has to be sure that t h e received message is the original t.ransmitted nEssaqct. .% the use of a n authentication procedure has i n t h i s area its natural application. hk deal with these problems i n several Data systems or sen7ices such as: e lec t ronic transacticsns, Point of Sale (RB)-banking, m n f i t h n t i a l data i n medicine. etc. I n I t a l y there are not mmy of these Smart Cards. Recently two

I t a l i a n Clniversity, namely t h e University of R a m "La Sapienza" and the University of Bologna introdwed a %rt Card system f o r confident ia l data of their student.s. The design and the hardware has been studied by "ENILXTA" .

We know also t h e so-called magnetic stripe card (for exarple the classical Bancornat). According w i t h the 1nt.ernational 0iyanizat.ion of

Standard, a Smart Card is a card, similar to a magnet.ic strifli, card. but. containing a nucrcjprocessor . Clsually connect ion with t.he c i r c u i t is insured ttuwugh a round printed goldered p t c h divided in e ight zones. One of these zones, the so-called secret zone, is inpossible to read from outside the ch ip and its physical protection is assured by t e c n o l q . So, we have to design a procedure called authentication schens

(cer t i f ica t ion et.c.1 i n order tr, nuke the bad guy's life d i f f i c u l t . Many of these procedures be designed using geon&ric f i n i t e st.mt.ures.

The procedure. of an authenticdt.ion s!-st.m runs as follows. The and the receiver B f i x an a l ~ L i k ~ r i h n f and :I secret key ti. tranmutter A

Then - A sends t.he messxje M together with the aut.henticator a = f ( K . M )

- B receives a message M' and an authenticlator a'. - B Lwtes a*:= f (K,M' 1.

- Only if a* = a', B accepts the messaqe M' as the or iginal . Sun.~,= t h a t w e have t o cilaarye M and a i n an illegal way. hk want ti>

delek M and to inscxt anot.her msyaye M'. S i w e w e do t u A know t.te secret key K and a' = f (K,?l' 1 is different. from a we can only t.q! cur chances of success are not as b d as it nuy seem. In fact., there is the following

THMRElrl ((;ilhert.+jac Williams-Sloane l.111). Ytqqauce LAat any dA.enk& h mtg olne mmage and blak acC -6 amd at4 &44 OC.CW y i t A cRe urn. ~ e ~ e . l . & i ~ . !&m.de tig t &. t&iP. W ~ P A 4 4.e-p- %?em, &n. ( s q aiLt.$.wt,.tt.cati.m .v_.4i.me U e c$.cmcm upl gue.&ag 4.h. cuwect ut~&l,,c.r uw3 UL h a t . 1 !It

168 F. Eugeni

An au then t i ca t ion system i n which the above chances are p rec i se ly IiJt is called a p e r f e c t aut.hentication system. Sone s c h e m s are not [-wrfect but. t h e i r chances are only O ( l . ' J t . ) . TheSi? system5 are r:allrd e s s e n t i a l l y

perfec-t.. A per fec t q e o m t r i c aut.hent.icat.ion scheme is t h e f o l l m i n y . F i s a

f i n i t e project.ive plane, f o r ex,mple a p ro jec t ive plane P over a Galu is f i e l d GF(q). Fix a l i n e L. The nessages are t h e p i i n t s of L, t.he keys are t h e p i n t s of P off L ( i n number of 13'). F o r a nrssage M a key

ti t he au then t i ca to r is t h e l i n e through M and ti. I t is easy to prove that one can t r y with a p i n t of set PK\,;?( M}. S o , t.he bad [ply has t.o do 13

proves.

and

The use of snwrt. cards f o r payrwnt. app l i ca t~ ions o f f e r s a very high leve l of securit.y. There is a previous au then t i ca t ion of t h e card-holder using a Persnnal 1dent.ific:at.ion Number (so-called PIN) arid t h e

ident i f icaticm card-syst.em. via a cryptographic: alqorithm. PIN is d i f f e r e n t from s y s t e m used i n Bancornat, it is checked i n s i d e

t.he card so t h e r e is not a c e n t r a l P I N - f i l e , t h e lenqth of PIN is not

f ixed and PIN can e a s i l y be (:hang4 by the card-holder. The authent.icat.ion card-syst.em has t o qive 1 qcarar1t.e tha t it is

deal i r q with an untnanipul;rt.ed card. The b n l i or "cmt ral s t . a t ion"

iises a g l ~ h i l secrrat. key h: .md an alqorithni F (usua l ly F is t h e Dat.a Encript-ion St.andard, i .e. DES) .

Any c ~ r d has a n ind iv idua l C a r d IDentit-y, said CID, itnil an individual

secret key kiC criwn by KC = F(C1D.K)

Evbreover, t h e card conta ins a second alclorittun f ( it m y he f =F) . The card-holder who does n o t how K , cannot. comput.e KC. ,"in au then t i ca t ion syst .al card-syst.m rms as follijws: - The card sends the C I D t o "cent ra l" stdtirm.

- The s t -a t ion conpute KC = F(CID,K) . I n a d d i t i o n , it sends random ninnber R to the card.

- The ca rd ccmput-es an aut.hent.iL:atrjr .A=f (R,K;C) md sends A to t.he stat. il .m .

- The c e n t r a l st.atiori conpit-es A ' = f(R,KC), with its own KC.

- Only i f A = A' is t~he card regarded as authent.ic:. - In a sinulai- way t h e card checks t h e s t a t i o n .


In a t r ansa t i im prcxess t h e card and t h e system au then t i ca t e ear-h o t h e r , t h e PIN is v e r i f i r c l ( i r k n t i f Lr” , i t ion owner-rard) , t h e t r ansac t ions are c e r t i f i e d (a1 1 messacJes have an au then t i cu to r ) and a l l t r ansac t ion ihta art? s tu red i n the retailer’s tievice and i n the card.

5. DIVISIBLE DESIGN AND AUTHENTICATION SCHFMES.

A d i v i s i b l e des i cp D is a p i r ( S , R ) where ( c f . [ G l , [141) :

a ) S is a set. of element-s, called w i n t s , parti t icined i n t o m

n-subsets (2 ,G,, , . . . ,G , said yenerat.ors. 1 , rn

b) R 1s a f a i l l 7 e->f k-stbset.s of S.

c) Any t w o p) in t . s x and y- of S are cont-airiel i n p rec i se ly x blccks i f and o n l y i f they ,ire i n t-wo d i f f e r e n t genc-rators.

P a r t i c u l a r cases of such a s t r u c t u r e are t h e €011 iwinq:

- I f n= l t h e genera tors are s ing le ton and t h e strut-ture is a

so-called 2- (rn , l c ,~ ) design ( c f . [ 6 1 ) . Sup~xxe k=n and A = l . Then t h e new f m l v containiny blocks and g m e r a t v r s is a S te ine r s i s tem 2- (mn,k , l ) , (cf . [61, [151).

-

If b is t h e number of hlccks of R and L- LS t h e nunhei of b l ~ k 5 containinq a f i x e d p i n t x , then:

(5.1) r s n , ni k

Note t h a t r = I n i f and only i f ni = k. In t h i s case D is said t o he a t r ansve r sa l clesiyn. If we count t h e p i r ( s , R ) and ( T , E ) whew T

is a ?-set contained i n t h e h l m k Ei and s is a p i n t of B we obt.ain:

The dual s t r u c t u r e of t h e pair ( S , R ) is obtained bv i n t e r - cliarqing pwint-s wi th bloclcs. A diL-isihle dcsiyn D is c a l l e d synnx?ti-ic i f 1t.s dual

st .ructure is a d i v i s i b l e cjesicjn with the s a w pwxwt-ers. E x a n p l e s of

d i \ , i s ib l r rlesiqns c a r i k found i n [ 61 , [ l f l l , (111, [131. Son= of them ! r f . 1161) are t he folliwiing, which are embedded i n p ro jec t ive spaces.

170 F. Eugeni 170 F. Engeni

In a pmiect.ive space HXd.q) over Gabis field GF(q) choose a point x

a d a hyperplane H. Remvi.ncJ H with al l its points and p i n t x

tmpther w i t h a l l the hyperplanes containing it, we obtain tw e q l e s of s -mtr ic divisible desiqns, according with x lies i n €1 or not., Khii -h

are of course square. The parameters are the follLwinci ( i f x lies i n ti) n=q, n p d - l . k=qd-l, I=&

n-q-1, m.:qd*. . . ? I , k-qd-1, a=cjd-2. 0 1-

so, i f d.2, we have spare divishle desiqns with s-1. In the next sect ion we s h a l l prwve that wing m h des igm 8om3 new p r f e c t authentication sclemea can be c:onstnr;td. Nuw we L w n S t n r t . a nf?w

authent imt.ion systeni.

DEFINITION.- I& D be a divisible desim with , ~ = l . tet the nrmR'iqr?h

he the pints :>f '3 fixed hlcxck M csf D.

€'or each message m 1 . d M let G(m) be the generator through m. We

define the set of keys of m as SM U Otrn)). Firlally EN each msstiip m and for each related key c the

autJw?nticator a(m.1:) is ttE un- blcxlk containing m aid I : ! .

-.- A and B f i x the key h. !K'tris is a p i n t of design D c3ut.side of the line M of the mssages. If 9=C(ni 1 is t.he generator containing K,

we i-emrk that A cannot mnd the mssage m . So, i f A a d B use the key K, then the pssible messages are the p i n t s of M \.( m }.

Wfi we pi-ove tlw following

u

a

0

mm3RFM.- Y l l e y p . . M M W U l qph?an #d a.&m k p&?eck

it a d mP.9 D i a a Llc)uti.w. dimidR& &Ayn ? , = l a

-.- Let Ij tx! a d i v i s i b l e design with a = l . Thr total nwtlher of related to a fixed mewage m. is:

t.(nr) = run - (k-n-1)

adiry to the Cillwi*+c Wil l ias-s lvane ttmtmi w suppxsc-

that a l l messages and all keys occur with ttle s.uw piu tsb i l i ty . So, the had guy's chatu:!cs of .success is a t least. I '4 t ( n i ) .

On the otter hand, suppse that. t h e p i r ( m , a ( m . c ) 1 ha5 heen alt.erd in (m*.a*(m*.c)). TIM? hd quv's chance of sLILyJess i n t h i s case is l/(k-1). It follukx that. 1/ t ( m ) i U t k - l ) , fnxn which

t ( m ) . (k-l):! .

Combinatorics and cryptography 171 171

This inplies n(nr1) -% k(k-1)

where equalit.\- holds if and only if the authentication system is perfect. N o t e that by (5.1) can be writ ten

b/nm = n(m-l)/k(k-l) Then the auttwnticxitiuti system is perfect. if and only if b= m, i.e.

the desiqn is squre. So, t he asser t ion is pnwed.

I t remits to r.mstiwt. a sy~temtic crrllection of exanplen.

Finally we m u l d l ike to speak about threshold scheme.

Threshold schemes have been i n t . d u c e d by Blackley and shmir i n 1978. i n mi l i ta ry mrld: f o r exanple a

n rhey have their traditional appl i ra t ions r?wlear missile may be released only i f at least two or mre o u t of pereons aqree.

The idea urderlying a threshold scheme is to create I'shadcws'' of rmr

secret -sage so that the p la in mssaye rannot. be r-etraived w i t h o u t the lamwledqe of a cei-tAm nunher said t.he threslwld.

A (t ,v)-threshold xlmle consists of v people p ,p ,..., p sharing a cer ta in n i h r b of secrets S ,S ,..., S tirat we call blocks of inhrmatione i n swh a way that the following properties holds

1 2 V

1 2 b

1.- t 5 p 1 5 v ;

2.- each p

3.- I . m ~ l t 4 q a of s w t l y t of n inforni t ions cmntained i n S

&tennines s ; 4.- hwledye of less than t infornration does not detemune any

block.

has one information or shadaw of infommt.ion I ; I i

i

i

It is well kxwn that.

Theorem (Shanur 1983). Y w - b e q u h 1 and v qua& n, 4 . h !ma d.t'!. t.

r-mi n ulehe PA& a ( t . . n ) - t h A . d d wAeme.

The proof is a comequeme of existewe of c e r t a i n i i r r a x i n g SeqlEW*S

m c m . ....... m 1 2 n

172 F. Eugeni

such that ml my. . .nik n\ ni . . . m

and t h e f m u s CHINESE FJNUWER TFIUIREIY.

:I n - 1 n - k t l

In g m r w t r i c lanqllage a niiiltiple threshold schmie is a p i r (P,%P) where

F is a v-set of e1enent.s c a l l e d "poin ts or shadows" and 3 is a family of subse ts s a i d blocks (of informations) such t.hat:

- t. e1enent.s of P determine e i t h e r zero or one h1~rr.k R.

- through t - 1 or fewer p i n t , s t h e r e are m n y blocks. A con t ro l syst.eni of a nultiple thrshold schenE is a set of p i n t s C

such t.hat. a.- C conta ins nu block b.- each hlmls i n t e r s e c t C.

i n o the r words: L' is a blocking set Lri ( P , " $ ) .

Suppxe that. w e have a t.hreshold sr-.hemm+ (F ,%) . A noti au thor ized bad

guy ML-. X tries t.o e n t e r t h e system, YO Mr. X choise t p i n t s . The systeni tries t k ) cunst ruict. a block through these t point.s:

I- I f there are 110 bloc:k, then the process s tops and t h e access is din ied .

2- I f t h e r e is a block B through them t h e system i n k r s e c t s i t wi th

C and ob ta in a set ( C i n t e r sec t ion R ) . Only i f t.he bad y y knows t h i s set t h e acces is f r e e . W e can c o n s t r w t 2-t.hreshold scherws using pro ja-tive or a f f i n e f i n i t e

spaces, s ince i n t h i s case w e have algorithms to operat.e. Sonetuies w e can obtx in geonet r ic indicat.es f rwi qroliEtry.

ror axanple s u p p s e to have a threshold s c h e m cons t ruc ted us ing

the l i n e s of a PROJECTIVE PLANE as blocks of infoi-mtios. Then i E we f i x a blorkinq set it can be r i ~ ~ t o q i - a l h i [ - ~ ~ l dangerous t o

use snlall b lwkinq sets. In f a c t G.Tal l in i has r e c e n t . 1 ~ proved t h a t :

All t hese o f f e r s new ideas for t h e deve lopwnt of b l w k i n g set theory .


I h o p that t h e ideas imclerlvining ny t-alk are pleased t.o yrxnc~

nathenlaticians. I shal 1 verl- qlad i f sonwne of t h m mul (1 f e e l sme

in teres t . in this research area.

ACENMLECGE4E". Fina l ly I would l i k e to thank a l l t h e cwcianizers t - h i s In te rna t iona 1 Conference "Combinat.orics ' ?O "and i n p r t . i c u l a r t.le

Frofessors Phria T i l l l i r i i Ycafa t i . Adridno Barlc-1t.t.i , AZessandro Eichara, Pier Vit . t .c>rio Ceccherini and Giuseppe T a l l i n i .

REFERENCES

[ 1 1

[ 2 1

[31

141

[ 5 1

(61

171

[81

191

L.EERrWI, Constructing -?-de.siq~s fi-oni spreads and l i n e s , Discrete Phth., 71 (1988) , 1-2.

T.BETH-D.JtiWNICkEL-H.LEN2, Dt?sqm Theory, B.1 . h i s sensha f t s Verlctci, Yknnhem Wien Zuric.11, 1982.

I101 R.C.B)SE. Synetric group d i v i s i b l e desiqn with the dual p m p r ' t j - , J. S t a t 1st. Plarin. Inference , 1 ( 1977) , 87-101.

174 F. Eugeni

Franco EUGENI Diparthato d i Ingegneria Elettricia Universita' de L'kpila Italy

I 74 F. Engad

1111 R.BDSE-W.S.Cx"Xt, -torial w r t y of ynxps dir-isible

1121 W.S.aXMlR. -&ne relatian the blocks of symetric yrmy

1131 H.CIW+SU-F.EU;ENf-M.~MI, E l a m t i di EIRtenuticd Disceta,

incaylete block design. AM. Math. Stat. 23 (1952). 367-383.

divisible design8 AIM. Math. stat. 23 (1952). 602-609.

Zaiuchelli. Bologna, 1988.

I141 E . N . G I ~ - F . J . M l ~ , ~ ~ . ~ . ~ ~ . Ct&$ which &tr?Ct &?€$rtlW, Bell Syst. Tech. J. 53 (1974). 405-424.

1151 H.GLONERIDDO, St#i Siisteni di Steiner. Sun. Gean. Oombin. Univ.

I161 D . J C I N h t f C K E L r K . V . divisible design w i t h k = l r i * l k . Arch.

L'Plquila, Quad. 9 (19861. 1-18.

mth. 43 ( 1 9 8 4 ) s 275-284.

I171 R.L.Rn~EST-A.StWMIR-L.ADEUUW, .9 nethod €or obtaining d i # . a l siqrwur?? 6 public key c-riptwsystern, taboratory for ooapltet. Sciemx?. t4ITlfiCS1Ri-82, April 1977.

Rend. kc. Naz. t.irrui 20 (13Sb). 311-317 ad 442-446. 1181 G.TALLIN1, Ssrlle k-10th clegli qar i lim-i € i t t i t i . We 1-11 .

Frana, EtEBJI Oieert immto di I-ia E l a t t r i c a Univetraitrr' de L'kquila Itrrly

Combinatorics '90 A. Barlotti et al. (Editors) 0 1992 Elsevicr Science Publishers B.V. All rights reserved. 175

Recent intrinsic characterizations of ovoids and elliptic quadrics in PG(3,K)

Giorgio Faim

Dipurtimento di Matematica. lJniversit8, Via Vanvitelli 1 , 06100 Perugia, Italy

A bsttact Let PG(3.K) be a pappian projective space. There has been much study

of ovoids and elliptic quadrics in PG(3,K) as well as subset with various properties which characterize them. A more difficult question is to characterize such subsets in an intrinsic way, i.e. starting with an ahstract set which is not iiecessarily embeddable in any PCr(3.K). Starting with the celebrated Buekenhout's article [ 71, this paper surveys some important known approaches and results, ohtained after 1973, on this intrinsic characterizations of ovoids and elliptic quadrics to bringing up to date references and results.

I . INTRODUCTION

In PG(3 ,K), the 34imcnsional projective space on some commutative field K, ovoids and quadrics have inspired many theories. One purpose among others is to characterize ovoids and quadrics. In this regard in 1973 F. Buekenhouc [7] surveys all main known approdches and results from a geometric point of view.

A first general approach is the rmWdkf one in which, following the pioneering studies by Kustaanheimo [21] and Segre [36], we consider ovoids and elliptic quadrics as well as subsets S of PG(3,K) and ask for conditions on S in order that S be an ovoid or an elliptic quadric in PG(3.K). This has Itad to developments in two directions:

(1) the m q m r &prpkne itpprmach where finiteness is not assumed all the time and where the fact that all tangents to an ovoid through R given point constitute a hyperplane played an important role (see Tits [48], [49], Segre [36]. [37]. Panella [33]. Barlotti [Z], [3]. Buekenhout [7]. Maurer

I76 G . Faina

[31], Kantor 1241, Thas [45]; (2) the fhmwcf.bn numEcr#pprmch in which finiteness is essential and

which is discussed exhaustively in Tallini's papers [41]. [42] and [43].

The intersection number approach as well as the tangent hyperplane approach has its origin in the study by Bose [S] of arcs and caps which are characterized by the fact that their 1-intersection numbers are 22.

A second general approach is the 12trhsre one: starting with an abstract set S which is not necessarily emheddable in a PG(3,K) we determine the properties which have to be satisfied by a family of subsets of S in order for it to induce on S the structure of an ovoid or an elliptic quadric of a PG(3,K). In this area we have no general answers to our main questions and the situation is f a r from being as satisfactory as in the embedded approach. Among the classical results on our subject special mention is deserved by those of Bern [4], Buekenhout [8], [9], Mlurer [32], Dembowski [ l l ] , Liineburg 130) and Hering [19].

This paper surveys some importanc approaches on intrinsic characterizations of ovoids and elliptic quadrics in order to up date references and results. Recause of the wealth of matererial involved, we do not attempt a complete listing of topics, and we restrict our survey to some results obtained after 1973. The results obtained in this area up to 1973 are discussed admirably in Buekenhout's paper [7] where the author has succeded both in surveylng the general area of szw&quzhnc sts and in stimulating further research. Among the approaches to our subject which have not been discussed in this paper, we especially mention those based on topological inversive planes. For some indications on this problem see [35], [40] and [Sl].

Many open questions are implicitely or explicitely mentioned in the text and the author hopes that these items may generate new interest in their., even partial, solutions.

2. OVOIDS AND ELLIPTIC QUADRICS OF PG(3.K)

A quadric Q of a d-dimensional projective space PG(d,K) on some commutative field K is the set of all points on which a given quadratic form vanishes. In other words> Q is the set of points (XI ,x2,...,xbtl)€PG(drK) such that

Recent intrinsic characterizations 177

where [qij] is a matrix of order d+l defined on K. Let f2 be a set of points of PG(d,K). A fa-nf to fl at X E f l is a

line r of PG(d,K) such that rnn={ X}. A subset R of PG(d,K) is called an o v U a (or o ~ d i f d=2) if

(1) for all X E f l , the union OX of all tangents to f2 at X is a hyperplane;

(2) each non tangent line intersects fl in 0 or 2 points. An ovoid in PG(d,K) which is also a quadric in PG(d,K) is called an efhp3C

It is well known that there are no ovoids in fijui-r projective spaces of dimension 24 (see [ 121, p. 48).

qZa?hC.

3. THE POINT OF VIEW OF INVERSIVE PLANES

An hm?tw'ct=pf#m is an incidence structure I=( 0.C) where the elements of 0 are called puhfs and elements of C are called crdes, which satisfies:

(3) circles are non empty;

(4) for each PE 0, IP is an affine plane (where IP is the &md structure whose points are the points of 0 other than P, whose blocks are the circles of Cwhich contain P, and whose incidence is that inherited from

The am'rr of /is the (common) order of the affine planes Zp. Inversive planes arise quite naturally in geometry in the following way. Let K be a skew field and 0 an ovoid in PG(3,K). Then the following incidence structure Z{D) is an inversive plane:

4-

points of Z(0) are the points of U; Circles of Z(D) are those planes which meet 0 in more than one point; incidence is inclusion.

An inversive plane is said to be ~ggb2~ if it is isomorphic to some I(i7). Van der Waerden and Smid [SO] have shown that tha inversive planes TO),

178 G. Faina

where 0 is an elliptic quadric of PG(3,K), are precisely those in which the rhfeaz ofhfiqueJhofds (see [ 123, p. 257). These inversive planes are also called aziqwJ!n. An immediate consequence is that miquelian inversive planes are egglike. Egglike inversive planes play a role in the study of inversive planes similar to that of the desarguesian planes in the theory of projective planes. The role of the Pappian planes is taken by the subclass of miquelian inversive p h s . In our context are fundamental the following three classical theorems are fundamental.

THEOREM 1 (Dembowski-Hughes [13]). Let fq0,C) be an egglike (miquelian) inversive plane. Than the incidence structure 1 may be embedded in a three-dimensional projective space PG(3,K), where the set 0 is an ovoid (elliptic quadric).

The points and planes of our PG(3,K) are as follows: points are (a) the points of 0 (real points), the circles of C (ideal points)i planes are: (a) circles of C (real planes) and (b) points of 0 (ideal planes). Such a PG(3,K) is called suifiioble (for 0.

In a fliuirr projective space PG(3 ,q) of odd order q, the quadrics are the only ovoids (see Barlotti [2]), hence we have

THEOREM 2 . Let /(Q,C) be a egglike inversive plane of odd order q. Then 0 is an elliptic quadric in a suitable PG(3,q).

On the other hand in a space over GF(2"), n odd >l,there exist ovoids which are not quadrics (Tits [49])1 the corresponding egglike inversive planes are the only non miquelian finite inversive planes known at the present time. To this purpose we can reformulate a classical result by Dembowslri [ 111 as follows

THEOREM 3. Let 1(;O,Cc) be a inversive plane of even order q. Then 0 is an ovoid in a suitable PG(3,q).

From theorem 3 we have that the following important question remains

Must all inversive planes of odd order also be egglike? open:

Another old, but fundamental question is the following: If each internal structure JP of the inversive plane /(Q,C) of odd order q

is Desarguesian, then prove that ]is Miquelian (i.e. that 0 is an elliptic


quadric in a suitable PG(3,q)).

Recently Thas [46] gave an answer to that question fo r q = l (mod 4) under the even weaker assumption that Zp is Desarguesian for at least one point P. In [47], Thas himself gave an answer for all odd q with qE{ 11,23,53}. Starting from those results and from theorems 1, 2 and 3, we can easily prove that

THEOREM 4. Let Z=(Q,C) be a finite inversive plane of order q r l (mod 4) such that Ip is Desarguesian for at least one point P. Then 0 is an elliptic quadric in a suitable PG(3,q).

THEOREM 5. Let I=(Q,Q be a finite inversive plane of odd order q, q&{ 11,23,53}, such that Zp is Desarguesian for at least one point P. Then 0 is an elliptic quadric in a suitable PG(3,q).

Another interesting partial result concerning the case of odd order appears in a different context (see Kroll [29]) and may be also rephrased as

THEOREM 6. Let /(i?.C) be an inversive plane of order q z 3 (mod 4)

(1) Im is a Desarguesian affine planer (2) let A,B.CE Cwith IAfU31=lBnCi=l and -g{AVS}, ~ E C , then there

exists a DE Cwith DnC={ =} and IDM(=l . Then 1 is a Miquelian inversive plane and 0 is an elliptic quadric of a suitable PG(3,q).

and let -E 0 be a point such that the following two conditions hold:

Following Dembowski [12, p. 2551, we define a 4-dmk. of an inversive plane Z{Q,C), to be any quadruple Ci,C2,C3,C4 of circles in C such that no three of the Ci contain a common point, but CinCi+, tfl, with subscripts taken mod 4. Define, for such a 4chain, pencils or bundles or~=[C~nCl+,]. We say that Z satisfies the 6 u d r rlrrurrrn (or BT, for short) if the following condition holds:

For any 4-chain with fxi,ctZ,fx3,c?t4 as above, f x l and a2 are compatible if an only if 013 and a4 are compatible (i.e. if they have disjoint supports and contain a common circle).

It is known (see [ 121, p. 257) that every egglike inversive plane satisfies

180 G . Faina

the Bundle Theorem but the BT does not hold in all inversive planes. Hesselbach [20] was able to prove that all topological inversive planes which satisfy the BT are egglike. In [22], Kahn showed that all finite inversive planes wich satisfy the BT are egglike. Finally, it is interesting to note the beautiful improvement by the same author [23] of previous results even under the assumption that the inversive plane is infinite. So that we can give the next result.

THEOREM 7 . If I=('Q,C) is a (finite or infinite) inversive plane which satisfies the bundle theorem, then Q is an ovoid of a suitable PG(3,K).

Given an inversive plane /(O, C, of order q, we define a p p h r(;r) as follows: the vertices of f(0 are the circles of /and two circles are adjacent if they intersect in precisely 1 point. Specializing to the case q odd. in [16] it is shown that the graph of the known inversive planes have 2 connected compenents. The main result obtained in [ 161 provides a partial converse to this result and hence R characterization of miquelinn inversive planes of odd order. As a consequence we have

THEOREM 8. Let I(0,CI be an inversive plane of odd order q. If f(Y) has 2 connected components and there is some point P such that IP is a Desarguesian affine plane, then 0 is an elliptic quadric of a suitable pG(3 A>-

Up to isomorphism, only two series of inversive planes of even order q, herein called the &.ssixZ inr.euldw p h e s , are known. Namely, the Miquelian inversive plane A?@,) corresponding to an elliptic quadric (see [12], p.104) and the Suzuki inversive plane .%(q.J corresponding to Tits ovoid (see [12], p.43). We shall also say that an ovoid in PG(3,q) is classical if it is either an elliptic quadric or a Tits ovoid. All the known finite ovoids are classical. The full automorphism group of a classical inversive plane of order q is transitive on its point-set. In [ 11, Ragchi and Sastry improve a theorem of Liineburg [30] by proving a result which (in view of Theorem 1) may be rephrased as

THEOREM 9. If f(Q,C) is a finite inversive plane of even order such that the full authomorphism group Aut( I> is transitive on the pointset of 1:

then 0 is an elliptic quadric or a Tits ovoid in a suitable PG(3,q).

It is interesting to note that the beautiful proof of theorem 9 involves coding-theory.


4. VON STAUDT'S POINT OF VIEW

In his Fuwakmenml Theam of Proy&cn.w L2mcneh-y-? of 1847 Ch. von Staudt [39] showed clearly that it is important to consider the manner in which the blocks are embedded in order to get information on the surrounding geometrical structure. According to von Staudt's point of view, we discuss in this section some intrinsic characterizations of elliptic quadrics. Many incidence structures allow an adequate definition of the group of projectivities of a block. so that using von Staudt's insights a kind of Fundamental Theorem can be proved. For all those geometries of the same type the group of projectivities operates n-fold transitively on the points of a block, where n is characteristic for the respective type. The Fundamental Theorem then characterizes the classical model by sharply n-- transitive action.

Let /{C?,C) be an inversive plane. In 1 there are several possibilities to define the concept of a bmcpmqxrfzw<y. For a definition of this concept see Kanel [ 2 6 ] . With these basic perspectivities we can form the composition of perspectivities from one circle C, to another circle Cz. Here the following type is studied (for their definition see [ 2 6 ] , p. 78):

with centets {P,,P2). The set of all proper perspectivities will be denoted by n Every product of

basic perspctivities is called proy&fi'w<y-. Let B be a fixed circle of f(U, ci Let n@) be the set of all projectivities which map B onto itself. It is not

difficult to check tmt n(B) is a permutation group on the set B, and that if B,,B2 are two distinct circles, then n(Bi) is isomorphic to I'I(B2). Hence, n(B) is also denoted by n. The subgroup -:n > of proper projectivities is

defined in the same way.

p o p r ~ ~ l ~ ~ ~ ~ Y [C,,(P,,PZ),CJ: x ----f X'

P'

P

An inversive plane is said to be rn-qyul' or to have the prop~~<y (Pm,,fl), if the stabilizer of any rn points on a circle within the group of projectivities consists only of the identity.

By means of this property we can to do some intrinsic characterizations of eIliptic quadrics. The first result is due to Freudenthal and Strambach [ 171 and, in our context, it can be f ormubated as follows

182 G. Fuina

THEOREM 10. Let KQ,C) be an inversive plane. The pointset 0 is an ovoidal quadric of a suitable PG(3,K) if, and only if, Z ( U , O satisfies the property fP3 ..Ill.

In [ 2 8 ] , Kroll generalizes this result as follows

THEOREM 1 1. Let Z(Q, C) be an inversive plane. The pointset 0 is an ovoidal quadric of a suitable PG(3 ,K) i f , and only i f , Z(0, C) satisfies the property (P3..<-l7 ..). P

From a result of Funk [ 181, one can also obtain the following

THEOREM 12. Let f(0,C) be an inversive plane. The pointset 0 is an ovoidal quadric of a suitable PG(3 ,K) if, and only i f , &a, GI satisfies the property (P' ..<I7 .>). P

5. THE POINT OF VIEW OF FINITE LINEAR SPACES

A f ! r spm (q,L) is a non empty set S whose elements are called pihts, provided with a family L of parts in S, whose elements we call lines, such that any two distinct points are in exactly one line. A finite linear space in which each line has the same number k of points will be called umYwm. A subspace in (3,L) is a subset 5' in S such that for any x , y E S ' , xty, the line joining them belongs to S: Suppose a family P of subspaces in (;S;L) exists such that [Plz2, every .n€!f' contains three non collinear points and through three non collinear points there is only one element of P. Then, the triple (;5;L.,P) is called a p.mr s p e , the elements of P are called phme5.

A subset n of a linear (or planar) space is a mp (or h-ap, where h=lnl) if no three of its points are on the same line. A line is a se-aurf, a &agent or an rxtwmf line to a cap n if it meets n in two, one or no points, respectively.

From now on (S,L) (or (2+5,dP) ) will be supposed to be a finite uniform linear (phnar) space and we suppose that its lines have size kr3.

In order to reach very interesting intrinsic characterizations of ovoids in PG(3,q), in [44] G. Tallini extended the concept of an ovoid to planar spaces as follows:

let (S.L.:.dp) be a planar space. We call ovoiil in ($,L,,P) a cap r? such


that given any point F F a the set theoretical union of the tangent lines to (?

P at P is ti subspnce t p in f.5:L) such that every plane through P nor in T

meets tP ina line.

The space tP is called the tangenc space to n a t P. In theorem 13 we

give a first. intrinsic characterization of ovoids and elliptic quadrics i n PG(3 4).

THEOREM 13 (Tallini [44]). Let (SL,,:?) a uniform planar space where the planes have the same size v. If in fiYL,<Pl an ovoid fl exists. then (-Y.L,p,J -PG(3.q) for some q . and n is an elliptic quadric if q=k-l is odd. an ovoid in PG( 3 ,y) if y is even.

A more difficult question is to characterize oL-oids by startiizg from h- caps of a p h m r spuce. In Theorem 14 we gather some parlial results towards the answer of this quesrion. I n order to do this, let R be the riumhrr of lines through a point of (SL,.?, let r be the number of lines through a point of p,L ) and let s be the number of planes through a line in rr’ (S L.. brl.

THEOREM 14 (Tallini [44]). Let I I be an h-cap of a pls-mr space (&,.L,f‘/ . I€ r is even, or if r is odd and (nnHf==r+l for any N E ~ , we have hsR-s+ l , the equality holds i f , and only i f , (sL,p,J =PG(3,y) and FI is an ovoid in PG(3 ,q ).

In [38]. Sherman attempts to reach sirniltar results for linear spaces. of course with a different definiiion of an ovoid, since Tallini‘s definition can only be applied LO a plnnar spuce.

Let (s,L,j he a finite uni€orrn linear space. A 4jprpkuze is a proper subspace which meets every line, so [hat any line i g either a subset of a hyperplane or meets it in exactly one point. A cap is an O W I ~ (in the sense of Sherman) of (S,L./ if all its tztngmt spnces are hyperplanes.

THEOREM 15 (Sherman [38]). T h e following LWU atatctrnctm\ are equivalent. (1) (S,L,J is a fimtc uniform linetr space of even order with an ovod n, and not a projective plane; (ii) S i s PG(3.2”) for some h , and R an ovoid in PCr(3,2”).

That there is a5 yet no definirive result for the odd case is a direct

184 G. Faina

parallel to the statement that it is not yet known whether all inversive planes of odd order have been determined.

Using the classification of finite doubly transitive permutations groups, Kantor [25] has proved that the finite linear spaces containing a line of size >2 and admitting an automorphism group which is 2-transitive on points are exactly the Desarguesian projective or affine spaces, the affine plane over the exceptional nearfield of order 9, Hering's affine plane of order 27, two spaces with 3 points and lines of size 9 due to Hering. the Hermitian unitals and the Ree unitals. As a consequence of this result, Delandtsheer [lo] has implicitely proved the next result on finite planar spaces. We recall that if (S,L,p) is a planar space where the planes have the same size v, then (S,L..f) may be viewed as a Steiner system S(3,v,l5$.

6

THEOREM 16. Let (sL.,p) be a finite planar space such that the corresponding Steiner system is S(3 ,q+ 1 ,q2+ 1) whose blocks are the images of {-}UGF(q) under PSL(2,q2) with qr3. (SL.,dD, admits an automorphism group which is transitive on the pairs consisting of a plane and a line intersecting this plane if and only if S is an elliptic quadric in a suitable PG(3,q).

6. BUEKENHOUT'S POINT OF VIEW

In the course of the proof of his by now famous theorem which provides the converge of the classical Theorem of Pascal, Buekenhout [6] generalizes, in an algebraic direction, the idea of an oval in a projective plane by introducing the concept of an abstract oval which is essentially based on the theory of n-transitive sets of involutorial permutations. Buelcenhout has shown how to define the structure of an oval without resference to the plane containing it starting with a set of points together with a family of permutations and he asks for characterizations of this abstract structures. According to Buekenhout's point of view, ovoids and elliptic quadrics are characterized both combinatorially and in terms of automorphism. In this section, we outline those results that have been developed in this context.

A family 3 of involutorial permutations, called involutions. can be associated to any ovoid f2 of PG(3,q) in the following way. For any point P of PG(3,q)\R, welet f p : f 2 --+ R denote the bijection which maps a point XEf2 to the point X'Efl. Here X'=X if the line (PX) joining P and X is tangent to f2 while X=(PX)n(f2 \ { X}) If (PX) is secant. The permutation


fp is an involution for any P€PG(3,q)\R and the family

(1) every f p E 3 has h fixed points, with l s h sq+ l ; 3={fp I PEPG(3,q)\n} has the following properties:

f P f P (2) any three distinct points of f2 are fixed by a unique involution; (3) for any three distinct points A,B,CEf2, there is a unique involution f

Moreover, if n is odd, 3 verifies the property: (4)any involution mapping a point A to a point B commutes with

any involution fixing both A and B. Finally, we racall that /nl=q2+1 (see [Z]).

At this point, the following question naturally arises: what properties must a family 3 of involutorial permutations of an abstract set E of n2+ 1 elements have in order to define a J-dimensional projective space PG(3,K) containing 8 as ovoid?

such that f(A)=A and f(B)=C.

In theorem 17 we gather an answer to our question.

THEOREM 17 (Korchmaros-Olanda [27]). Let E be an abstract set with I81=q2+1, q22 and 5 a family of involutorial permutations of E having properties (l) , (2), (3). If q is even, then E is an ovoid in a PG(3,q) where 3 is the family

If q is odd and 3 has also property (4), then E is an elliptic quadric of a PG(3,q) where 3 is the family

vp I P E P G ( ~ , ~ ) \ E I .

I p ~ p ~ ( 3 , w . e 1-

Another interesting intrinsic characterization of ovoids is obtained by starting from some structures which are a generalization of inversive planes.

Let E be a set containing q2+ 1 points (qr2) and 3 a non empty family of subsets of E called circles. The structure ( C , 3 ) is said to be a cf?zY+pAzm of order q if the following properties hold: (5) each circle contains q+ 1 points5 (6)for every pair of points P,Q and for every circle c with PE c and QE c , there exists an unique circle d such that P,QE d and cn d={ P }.

THEOREM 18 (Quattrocchi-Rinaldi [34]). If (E,3) is a circle plane of order q, q even, then E is an ovoid in PG(3,q) where 3 is the family of p~ane sections nnE, such that InnEl+-i.

186 G . Faina

Another question is to see whether any intrinsic characterization like the one by Korchmaros-Olanda [27] can be found for ovoids starting from a family of partitions of an abstract set. We first of all discuss some properties of ovoids in PG(3,q).

Let €2 be an ovoid of PG(3,q). Lf r is a line having no points in common with n, then the q+l planes through r intersect fl in the elements of a partition of 0. It is well known that if TC is a plane of PG(3,q), then Innnla{ i ,q+i) .

The following question naturally arise: Let E be any abstract set and ’9 be a family of partitions of &. What

properties must ’9 have in order to define on & an ovoid structure in a PG(3,K)?

A partial result due to the author [14] characterizes ovoids in the following way.

Let E be a finite set and ’9 a family of partitions of P, such that (7) for all PE’9 and for all BEP, IBI€{ l,q+l}, with qr2.

We shall use the following symbols:

C

Cx

3,

the family {B I BEP, P E ~ } the family of all elements in C through the point XEE the family of all partitions in 3 which contain the set { X}, where XEP,.

A pair (&,3) is called an ucIsrrucr oFIoid of order q if it satisfies the following properties: (8) any three distinct elements of E lie on exactly one element of

(9) any two disjoint elements of Cl ie on exactly one partition P E ’ ~ . C.

THEOREM 19 (Faina [ 141). Let ( E , 3 ) be an abstract ovoid of order q such that there exists at least one BE C with IBl=l. If I QI=lzB(+q+ 1, then

(i) IEl=q2+lj (ii)there exists a family r of involutory prmutations of E such that I 1’9(q2+ 1 >i (iii) if q is even, then E is an ovoid of pG(3,q)~ (iv) if n is odd and any involution gEr mapping a point A to a point B commutes with any involution g’Er fixing both, then P, is an elliptic quadric in PG(3 ,q).


7. FINAL REMARKS

None of the other outstanding results in this area go beyond those that have been stated in this paper.This field of inquiry remains open to numerous and interesting refinements.

Because of the wealth of material involved, we shall not discuss results on the intrinsic characterizations of hyperbolic quadrics nor parabolic quadrics. A survey of the attempts up to date to characterize these interesting geometric structures in an intrinsic way will be published elsewhere (see [ 151).

REFERENCES

1.

2.

3.

4.

5.

6.

7.

8.

9.

BAGCHI, B. and SASTRY, N.S.N., Even order inversive planes, generalized quadrangles and codes, @om. A&k#m 22 (1987), 137- 147. BARLOTTI, A. , Un’estensione del teorema di Segre-Kuswanheimo, B&. Un. Afar. Zrd. 10 (1955). 498-506. BARLOTTI, A., Some topics in finite geometrical structures, L.N

BECNZ, W., Uber Mobiusebenen, a h & w a r . &hes&r. 0.M I.: 63

BOSE, R.C., Mathematical theory of the symmetrical factorial design,

BUEKENHOUT, F., Etude intrinsque des ovales, Read Mir. -4ppf.

BUEKENHOUT, F., Characterizations of semi quadrics. A survey, AffY Cmqwz’Lbzrz . 17 (1976), 393421. BUEKENHOUT, F., An axiomatic of inversive spaces, L Com6. Z-9.

BUEKENHOUT, F., Inversion in locally affine circular spaces I, 11,

UniK N d GwI2.k N f a M p J HW, 1965.

(1960), 1-27.

5 h M r ~ l 8 (1947), 107-166.

25 (1966), 333-393.

11 (1971), 80-212.

A&/&. 2212. 119 (1971), 189-202; 120 (1971), 165-177. 10. DELANDTSHEER, A., A geometric consequence of the classification

of finite doubly transitive groups, &-om. M i x f a 21 (1986). 145-156. 11. DEMBOWSKI, P., Mijbiusebenen gerader Ordnung, Mrh. &. 157

(1964), 179-205. 12. DEMBOWSKI, P., Finite geometries, Springer Berlin, 1968.

188 G. Faina

13. DEMBOWSKI, P. and HUGHES, D.R., On finite inversive

14. FAINA, G., Ovoidi generalizzati, R d a m . hhf. UnkPoMx. Tmno

15. FAINA, G., I pkni di Mobius, Laguerre e Minkowski, Atti Convegno

16. FISHER, P.H., PENTILLA, T., PRAEGER, C.E., ROYLE, G.F.,

planes, J London h4vfh. Sa. 40 (1969, 171-182.

46 (1988), 247-257.

Giornate di Geometria, Univ. L'Aquila 1991 (submitted).

hrws'r~,dutesaf odd &e, Eurp. J. Combinatorics 10 (1989),331- 336.

17. FREXJDENTHAL, H. and STRAMBACH, K., Schliepungssatze und Projektivitaten in der MSbius- und Laguerregeometrie, A&fh Z 143 (1975)) 213-234.

18. FUNK, M., Regularitat in Ben-Ebenen, D?kswf&ffofl UN'r: fihqgeff-

19. HEiRING, C., Endliche zweifach transitive Mlibiusebenen ungeraden

20. HESSELBACH, B., Uber zwei Vierecksatze in der Kreisgeometrie,

21. JARNEFELT, G. and KUSTAANHEIMO, P., An observation on finite

22. KAHN, J., Finite inversive planes satisfying the bundle theorem (to

23. KAHN, J., Inversive planes satisfying the bundle theorem, L Corn&.

24. KANTOR, W., On unitary polarities of finite projective planes, Ghmd

25. KANTOR, W., Homogeneous designs and geometric lattices, L Cum&.

26. KARZEL, H. and KROLL, H.J., Perspectivities in circle geometries, Gwneqv - F ' O ~ SaLdrrspoinr of w&w (BE. NA TO Adrmx.-& .9uc$v hsf.d2ufeJ Pfaurrw/M P. aVnd.Sh?.m&& K. &.) Reidel Pub. Co., Series C - Math. and Physical Sciences 70 (19%1), 51-99.

NurffLqy 1980.

Ordnung, A&. Ahfh. 18 (1967), 212-216.

A6h. AZafh. SPm. Urn'$: Ham6urg 9 (1933), 265-271.

geometries, Shm hfifh. KO%. TmmA%??h 1 1 (1949), 166-182.

appear).

72. (A) 29 (1980), 1-19.

d h&r&. 23 (1971), 1060-1077.

Th. (A) 38 (1985). 66-74.

27. KORCHMAROS, G. and OLANDA, D., On egglike inversive planes,

28. KROLL, H.J., Die Gruppe der eigentlichen Projektivitaten in Benz- 1 OfGFOLPI. 21 (1983), 53-58.

Ebenen, &om. Miam 6 (1977), 407413.


29. KROLL, H.J., On the characterization of the finite Miquelian Mobius planes, ANxrrls Dfkmre 12fifh. 18 ( 1983), 549-552.

30. LmEBURG, H., On Mobius planes of even order, A&... Z 92

3 1. MAURER , J. , Spiegelungen an Halbovoiden. A m . h&fh. 2 1 ( 1970), 41 1415.

32. MAURER, J., Eine axiomatischer Aufbau der mindesteiis 3 dimensionalen Mobius-Geometrie, Izfitrh- 2. 103 (1 968), 282-305.

33. PANELLA, G., Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito, Boff. c/n. A3r. hi1 10

34. QUATTROCCHI, P. and RINALDI, G., Firute tangent-circle structures

(1966), 187-193.

(1955), 507-513.

Proc. International Conference "Cornbinatorics '88, Mediterranean Press 1991.

35. SALZhlL4", H., Topologische projektive Ebenen, Itfifh. 2. 67 (1957), 436466.

36. SEGRE, B., Ovals in a finite projective plane, r3mtd X Mtfh. 7 (1955), 414416.

37. SEGRE, B., On complete caps and ovaloids in three dimen=iional cialois spaces of characteristic two, .4carArifhm. 5 (1959), 325-352.

38. SHERMAN, B.F. , Ovoids in finite umform linear spaces, J; of'&um.

39. STAUDT, G.K.CH. v., Geometrie der Lage, Bauer & Raspe, Nurnberg

40. STEINKE, G.F., Locally Miquelian Benz planes, A6h- 12hrh. Sm.

41. TALLINI, G., Sulle k-calotte di uno spazio lineare finito, Am. Mw.

42. TALLINI, G., Graphic characterization of algebraic varieties in a

43. TALLLNI, G., Problemi e risultati sulle geometrie di Galois. Zg&ufu d

44. TALLINI, G., Ovoids and caps in planar spaces, Amtfs D i s e f e Ai5if.h- 30 (1986), 347-354.

45. THAS, J.A., Ovoidal translation planes, '4mh- Athrh. 23 (2972), 110- 112.

46. THAS, J.A., Solution of a classical problem on finite inversive planes Proc. Conf . Fibre BUI~~I&S, Rekjfd L;irome%c+s and .4ppfi~if1ons , Colorado State Umv., Pingree Park 1988.

47. THAS, J.A., Flocks, maximal exterior sets and inversive planes, Comempomty Mtrkm&zcs (A. M . S . j 1 1 1 ( 1 990 j , 1 87-2 1 8.

37 (1990), 159-170.

1847.

Un'r-. Biim6urg 54 (1984), 141-161.

A+YW AppL 42 (1956), 119-164.

Galois space, A f t i Cb~r.~gm'C'hx~' 17 (1976), 153-165.

hfarmafh?i L~N'K Napofi Relazione n. 30, 1973.

190 G. Faina

48. TITS, J., Ovoides a translation, Re.d Mtf. AppA 21 (1962), 37-S9. 49. TITS, J., Ovoides et groupes de Suzuki. A d - 12rUrh 13 (1962), 187-

198. 50. VAN DER WAERDEN. B.L. and SMID, L.J.. Eine axiornatik der

Kreisgeometrie und der Laguerre-Geometrie, hfifi5. ALW. 1 10 ( 1935). 753 -7 76.

333. 51. WOLK, D., Topologische Mobiusebenen, Ma&. Z 93 (1966), 311-

Combinatorics '90 A. Barlotti ei al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. 191

A NATURAL ASSOCIATION OF PGL(V)-ORBITS IN THE SEGRE VARIETY

(P(V))" WITH FLAGS AND YOUNG TABLEAUX

FEDERICO GAETA. Universidad Complutense, Madrid

ABSTRACT

The usual motivations to introduce the YOUNG tableaux were: first the representation theory of the symmetric groups (cf [B], [L], v c " 9 ] , ( ; W ] , LY]) and then the (closely related) of GL(V) in n. K , K commutative ground field of characteristic zero). The projectivization P ( 1 ) of any GL(V) -invariant irreducible I if) V"",) is the natural ambient space of a flag manifold F ( V a V 2...2 V"') (cf DEF. 3 , 5 . 2 ) whose dimensions di give the shape 3c (or empty frame, or

FERRER's diagram) characterizing the isomorphism type of I by means of a vertical (or llcolumnll) partition of m

(0.1) m = d 1 + d 2 +...+ dc dl ~r d2 2 ... 2 dc > 0

(satisfying n 2 dl). It is well known that although the v"@ representation of GL(V) is fully reducible (cf. loc. cit.) the single 1's are not uniquely determined: only the isotypic components are unique. Moreover there is a natural choice of such components parametrized by the set of all standard YOUNG tableaux (cf. DEF.1,$2) without repetitions with entries in the label set L = (1,2,...,rn) (cf.51) used to denote the m I1copies1l of V.

In this paper we show (cf. $1) that: Every non zero monomial el ement x o x o . . . o xm, xi E V,

~=1,2,...,m defines in a unique way a YOUNG tableau Y(Z) with entries the (unrepeated) m factors x in such a way that each

one of the c columns B1,B2,. . . ,B are basis of the components of a flag F(Vd a Vd a . . . a Vd ).

F characterizes the frame 3. (satisfying (0.1)) and the labels form a standard YOUNG tableau without repetitions with entries in L (cf. Example at the end of $1).

The construction is a simple combinatorial and characteristic free sophistication of LITTLEWOOD'S abstract llregular application of nodes" [L] (cf. $1,3) making every

1 2

J

C

1 2 C

192 F. Gaeta

step of the construction fully well determined, and since it depends essentially on L and on the relations of linear independence (or dependence) of the x the algorithm is

invariant by the GL(V); on the other hand we can interpret-mutatis mutandis-all the constructions in terms of the projective group PGL(V) acting on tQe projective space P(V) and on the SEGRE variety (P(V)) representing the ordered rn-tuples of points of P(V). In fact (x , )x (x , )x . . .x(x,) is represented by (xl@. . .axn) (cf. §1,2

for almost standard notations), ie. the monomial (e fully decomposable) tensors def i",e (up to a # 0 proportionality factor) the points of (P(V)) .

The full construction is given informally in a self-contained way in §l (for a reader familiar with the notions involved). Otherwise, perhaps such a sketchy presentation might appear as llunmotivatedll. Since the Author is not a combinatorist (but an algebraic geometer) we need to take into account the choice of reasonable prerrequisites! §2,3 give of all of them with a more formal description of the construction and geometrical examples.

My combinatorial friends, in particular BRINI, ROTA, SENATO, VENEZIA asked me in Gaeta about the possible relationship of this intrinsic algorithm (dealing with domino

J

pieces .! E L , one M(!) E V-(0)) with the universally 1 - I

fascinating abstract one due to SCHENSTED). I do believe indeed that there is one! In §5 I give a sketchy tentative geometrical interpretation of SCHENSTED *lburnping1l procedure (corresponding to an unfinished research). Let me concretize the SCHENSTED construction [S], [GI, [R], [Sch] assuming that the given map (between two ordered sets) is

S : L --+ r r with r={lI2,...,n} a coordinate label set (V = K , L n r=0).

S is not necessarily injective! (repetitions of the r indices are allowed as in commutative monomials happen...).

SCHENSTED deals with numerical Itdomino piecest1 pq .!EL, S ( l ) E r. Our M --) YM insertion algorithm deals with a

sequence F"' c F'2' c F'3' c ... c F(m'

# # # #

of flags (where G means V(s-l'') S V(*") for every pair of corresponding components of F'"'' , F'" but the strict inclusion c holds because we added M ( s ) to FS-') in order

to construct F"' . The introduction of coordinates leads to products of

monomials of type

#

A natural association of PGL(V)-orbits 193

J

computed in the dl=r vectors e l l e2 , . . . , ed of a canonical

basis of F('-l) The tentative insertion of a numerical

domino piece wl keeping the ordering in F") is not

possible using only M + Y,, but the commutative property

enables to replace the standard tableaux of the same shape Y(F), Y r ( F ) (with entries in L, r) by another similar pair

1

obtained inserting and then disrupting according to

Schensted rule. A few more details of this work in progress is given in

95. I welcomed the opportunity of meeting the above mentioned colleagues for calling my attention to this problem.

1. SUMMARY AND INTRODUCTION.

Let M ( L ) be the image of the map

(1.1) M:L + v-{o) M E vL (=P, =mth Cartesian

power)

of the label set L = {1,2,.. . ,m} in the n-dimensional

K-vector space V (with 0 excluded to avoid trivialities) (cf.

9 3 )

We shall insert M ( L ) in a well defined YOUNG tableau Y,,

(cf. DEF. 2) depending only on two structural ingredients:

1) The natural order of the labels: 1 < 2 <...< m

2) The relation of linear independence (or dependence)

of finite families of non zero vectors of V .

The construction is recursive on Ll = {lI2,...,i}

(L = L,) and it is a further sophistication of LITTLEWOOD'S

abstract construction named "regular aplication of nodes1' (s

[L] Ch V 95, p. 67,68). LITTLEWOOD uses only 1) , ( V does not

194 F. Gaeta

appear) and only the first step putting 1 in the upper left

corner is uniquely well defined in [L]. For any further step

s (>1) there are finitely many ways of adding the label s to

a standard tableau Ys-lr in order to obtain again a standard

Ys (very often there is more than one way ! ) (See example

below). In our construction (cf. $ 3 ) the entries of YH are

not labels .?. E L; they are vectors M ( l ) E V ( t ) of the t-th

copy V(.?.) of V (EEL) . The vectors of the rth column form a

basis of a vector space VS-'" component of a flag Fs-l (s.

DEF. 4 in $2)

6-1

These properties (using condition (1.2) above) complete

the [L]-construction acting on the entries of each M ( t ) . They

are sufficient to find the right place to insert M ( s ) in Fs-'

"i.e to construct Yn from Yn adding M ( s ) " . Namely: s s-1

M ( s ) must be at the bottom of the first available column

r of Ys-in order to keep the two previous properties (linear

independence in any column of Ys and (1.2) for s instead of

s-1) ; more precisely:

The construction of Yn is naturally translated to @M

usually written in a factorial way:

( 1 . 3 ) @M = M ( l ) @ ... @ M ( m ) z O (e=, M ( t ) * O , V t E L)

L L because @M is the canonical projection of M by KV + KV /N

since the submodule N ("where all multilinear maps vanish1#,


cf. [Bou]) is invariant by the symmetric group Sm = S(L) I ie.

we consider defined @M + Y e n .

Let (sM) be the point of the projective space P(sVL)

canonical projection of @ M E @V . (sM) is the point of the

SEGRE variety (IP(V) ) " representing the m-tuple of points

(M(l))x(M(2))x ... x(M(m)) (M(t)) E P ( V ( l ) ) V t E L, Cf. [H-PI.

L

The M 1-9 Y, , - inser t ion a lgor i thm of t h e M ( t ) i n

s u i t a b l e YOUNG t ab leaux i s a l s o ex tended to (@M)

EXAMPLES :

I-> Y ( @ n , .

1 )

L = {1,2,3,4,5,6} M:L I-) V is defined by x ( l ) , y ( 2 ) ,

z ( 3 ) I u(4) , v(5) I w ( 6 ) under the assumptions:

XAY#O , XAYAZ = 0 , XAYAU = 0 ,

ZAU+o ZAUAV = 0 , XAYAW+Q

x ,y ,z ,u ,v ,w E V-{O}, dim V > 3

Let us separate in ( 1 . 4 ) the label YOUNG tableau and

the vector tableau

1 3 5 x z v

( 1 . 4 ) l 2 4 Standard y u 6 W

( 1 . 4 ) " Flag F3,2,1: V3 3 V2 2 v1

Insertion order:

196 F. Gaeta

1 1 3 1 3 1 3 5 1 3 5 l 2 2 2 4 2 4 2 4

6

are (s.[L]): 1 3 The regular insertions of 4 to

1 3 2 4

1 3 2 4

1 3 4 2

but o n l y 1 3 is allowed to us because we need to insert

u ( 4 ) and the first one is not correct because we assumed

XAYAII = 0. The last one is not good because Z A U ~ O . In the

final flag we have

2 4

v‘” = ( X A y A W ) 3 vC2) = ( Z A u ) 3 v ( ~ ) = (v )

where xlhx2h. . .hxd ( * O ) defines a vector subspace of

dimension d represented by the point ( x ~ A x ~ A . . .hxd) E I P ( d ~ V ) ,

of the Grassmann manifold G ( V d C - - , V) canonically embedded in

! P ( d A v ) (Cf. $ 2 ) .

2)

and Sm (for m s n): l,l,. . . , 1

The extreme cases: 3;

9 (m = c) (corresponding to the natural Western

way of writing) arises in our algorithm iff any two of the m

given vectors x(t ) are proportional (dl = 1, i = lr2,...,m):

l , l , . . . ,1

X A X = O l s i < j s m 1 J

x(1 ) x ( 2 ) . . . x ( m )

3 )

tn (c = 1) is possible only iff m s n and X ~ A X ~ A . . . A X ~ * O

x ( 1 )

is the natural tableau iff X ~ A X ~ A . . .AX,+O

REMARK.

We shall denote the constructions by


where Y(Y) is a standard YOUNG tableau with LI entries of

shape Yi such that YM = MYi:21 - V denotes a tableau of

shape F1 with values in V satisfying the flag conditions for

every i.

1

The t w o algorithms M I-) YMl @M I d YaM are

invariant by the corresponding actions: GL(V)xVL - VL and GL(V)x(@VL) + (@VL) (since N is also GL(V)-invariant) and

the algorithm (@M) I-> Y(@M, for the SEGRE variety is a l s o

invariant by the corresponding action of the projective group

PGL(V) on (P(V))", thus it is well suited for the study of

PGL(V)-orbits. Since the condition (1.2) indicates that the

subspaces V"') are the components of a flag we see that

these algorithms not only attach to every M (or @MI or (@M))

a well defined YOUNG tableau 2d . . . d YL (without 1 2 c

repetitions, with values in L) ; but also a flag Fd . . . d of 1 2 c

the same shape whose dimensions equal the column partition

(0,l) of m defining the shape 2 d.d2.. . d of the empty frame

(M FERRER's diagram) of Ym.

C

DEF. 1. A YOUNG tableau Y of shape Y with values in a

set S (L,V, etc,) ( s . 92) will be denoted as before by a map

(1.4)' ' ' Y:2 - s

whose image Y(2) is visualized inserting Y(i,j) E S as a

(ij)-entry in the "empty square" (or dot) of 9 (cf the

EXAMPLE of 92).

198 F . Gaeta

In our insertion algorithm we associate to M E VL a

frame S in such a way that the diagram

Y M (1.5) 3 + L __j v-(0)

means that Y(S) (with values in L) is a numerical standard

YOUNG tableau and YM = MY is a tableau with values in V-(0)

whose columns are bases of the spaces V") of a flag

~"'2 ~ ' ~ ' 2 . . .2 v', of type Fdl.dZ....dc (s. DEF.4 in 92).

If M contains a canonical-basis B (cf. 92,3) the

recursive construction can be simplified using the BOERNER'S

footnote lemma (cf 92): then the degrees A l of M (or @ M I

( s M ) ) ) = number of occurences of the basis vector el of Bi

define the row shape 3 A l A 2 . . . A

r = 3d i.e. . . . d 1 2 r

(1.6) m = A l + A 2 + . . . + A r A l 2 A2 2 ... A r > 0

is the horizontal partition of m conjugate with (dl,d2,.. . ,dc) Then a simple column ordering of a row ordered

Y:S --+ L produces the desired standard tableau (cf. 93).

2. NOTATIONS AND PRERREQUISITES.

The power set notation VL for Vm is standard in set

theory. Our use here is justified because we need to

emphasize the r6le of the I1labelst1 ! E L = (1,2,. . . ,m). The llsimplefl, llnaturalll notation x I ~ x z l . . . ,xm is just a

caligraphic western llconveniencell I similar to the semitic or

Chinese way of writing but it can be geometrically misleading

because it is not always l1naturall8 (cf. the Extreme cases

examples 2,3 in 0 1) .


Accordingly an element M E VL is a map

(2.1) M:L - V

The left (and right) actions GL(V)xVL(SmxVL - VL) are

naturally defined by the compositions shown in the diagram

- 1 M ( 2 . 2 ) V & V + L 5 L OL E GL(V) s E S m , s : L + L

The definitions of aoM, som are:

(2.3) (OL~M) (i)=a(M(i)) S~M=MS-' o (S~M) (i)=M(s-'(i))

As usual we have

( 2 . 4 ) aos = S O O L OL E GL(V) s E Sm

Alternatively we can consider m copies V ( t ) of V

labelled by the E E L with a copy map v v ( t ) I Vv E V for

each l .

I noticed many sophistications on the approach to

FERRER's diagram (H empty frames) and YOUNG tableaux. I

adopted here the informal approach of [ L ] and [B], which

enables the reader to choose his own improvement. Thus a

YOUNG tableau with frame 2 with values in the set S is a map

( 1 . 4 )

Accordingly the symmetric group Sm = S ( 2 ) (m = #2) acts

on Y on the right:

yuxtaposition is used only for map compositions. In

particular it always makes sense to speak on the horizontal p

or vertical permutations q ( E P I Q) where the subgroups P ( Q )

200 F . Gaeta

of Sm are characterized by leaving invariant all the rows

(columns) of Y (9) . When S is a subset of Z (for instance L) it makes sense

to introduce row (column) ordered YOUNG tableaux (R(C)).

( 2 . 6 ) R(i,j) s R(i,j+l) Vi C(i,j) 5 C(i+l,j) V j

When S = L Y ( L ) will always be bijective (w there are no

repeated integers in Y(3)).

'3,2,2,1 EXAMPLE of Y:L = {1,2,3,4,5,6,7,8} & 3;4'3''=

( 2 . 7 )

Our preference for the vertical partition to describe 9

is founded in the fact that we have in mind the flag with

dimensions dl . . ,d (cf. 41) :

DEF. 2. A YOUNG tableau Y:3 + L row and column ordered

simultaneously is called a standard YOUNG tableau.

For instance in the previous example Y(34'3'1 ) is a

standard YOUNG tableau.

We shall use another label set r (with r n L = 0) (i*i).r

labels the basis (M coordinate systems in V : r = {lr2r...,n}.

The basic vectors e (j = 112,...ln) actually are inverse

images of the unit vectors in a bijective linear map V - Kn=K ) . The YOUNG tableaux with values in r might have

repeated indices (remind that meL is the basis of fl@

functorially attached to the basis e of V; however the

frequent use of vertical antisymmetrizations (by the action

of Q cf. (2.6)) eliminates the generalized standard tableaux

J

r

A natural association of PGL(V)-orbits 20 I

(keeping the 5 sign for rows) with equal integers in the same

column (the strict > holds then in (2.6) for any column).

+ Z be any numerical YOUNG tableau.

There exists a horizontal (vertical) permutation p(q) such

that pY is row ordered and qY column ordered.

. . . d Let Y:9d

1 2 c

If R(C) is row (column) ordered and we construct the

correponding column ordered qR (row ordered pC) we can

believe a priori that every ordering will be lost in qR (pC)!

However, this is not the case! because of the following:

BOERNER‘s footnote lemma:

The column ordered (row ordered) qR (pC) is standard.

s. Footnote ( * ) in page 155, Kap. V. 45 of [B].

This property plays a crucial rble to construct

immediately our YM (cf.41 and 3) directly (rather that by a

recursive method) when M comes from a canonical flag basis.

(n z d zd 2 . . .- >dc > 0 ) is

a sequence of vector subspaces of V ordered by decreasing

1 2 DEF. 3 A flag F = F d d . . , d

1 2 c

(2.8) F = F d d = v‘”2 v(2)2.. - 2 v(c)

1 2...dc

not - +1+1) vith dirnKV‘” = di (with the possibility V‘” -

excluded) . A flag F contains the flag F 1 (and we write F2F’) iff

VTIV“ for every r=1,2, . . . DEF. 4 . A YOUNG tableau with vertical shape dl,d zl...,dc

and values in V is called a flag tableau if for every column

r ( 7 = 1,2, ..., c) the corresponding vectors xlr,xzrl . . . , x drr

202 F . Gaeta

form a b a s i s B~ o f t h e space v") of a f l a g Fd . * , d 1 2 c

(s. DEF. 3 ) .

The flag conditions can be expressed in terms of V as

follows

qhc- l l . . ' PZPl where h2 I hl means A2=xlh. . AX^ -d hAl if h l , A 2 are exterior

monomials with dl (resp. d2) factors.

2 1

DEF. 5. A f l a g b a s i s of F d d . . . d is c a l l e d canonical 1 2 c

i f f

B(') = (e1,e2'. . .,ed 7 = 1,2, ... ,c 7

where B"'- - ( elle2, ... ,ed ) i s a b a s i s o f t h e maximal

component v"), thus B");? ~"'2 ... 2B 1

(C) . The YOUNG tableau YB associated with a canonical flag

basis B has the simplest form:

(2.11) YB(il) = Y (i2) = ... = Y,(iAi) = e, B A&. . A r

for every row i = 1,2,. . .,r where Fdl ,..., = F

(cf. (1.6)), i.e. iff (AllAz,...,Ar) is the horizontal

partition of m conjugate with dl,d2r...rdc; we shall write

C

(2.12) Y p d . . . d + V-{OI 1 2 c

where


4 el el ..... el

e e ... e A -2

2 2 2 YB(3)= . . . . . . 2

ed

1 ed

If we use the @VL approach (cf. 51) we notice that

@YB(F) has degree equal to A l in el for i = 1,2,...,r, r=dl.

It is clear that for any F we can construct a canonical

basis taking an arbitrary basis Bc for V(') and then if B1 is

defined and Vl-l = Vl, Bl- l = B l . Otherwise if V1-l$ Vl we

can add dl-l-dl linearly independent vectors in Vl-l-Vl to Bl

in such a way that

(STEINITZ theorem) . EXAMPLE. Let n be any integer >a. If the vectors of the

three columns of a Y ( I J 4 , 3 , 1 ) with values in V are linearly

independent, i.e. if they define subspaces of dimension 4,3,1

respectively such that V(')= V'2'> V'3' Y(3) is a flag

tableau. In particular if e1,e2,e3,e4 are linearly

independent the YOUNG tableau

el el el

e2 e2

e3 e3

e4

is associated to a canonical flag basis of type

In the formal statement (cf. 9 3 ) of the algorithm

204 F. Gaeta

M b YM the vectors do not belong to V but to the V ( t ) (with

just one vector for every t ) and the algorithm will associate

with any M an MY = YH for an Fd with the dI uniquely 1 2 " ' . c

determined; for instance we can obtain

e , (U e1(3) eJ5) eJ6)

e2(2) e2(4) e p )

e3(7)

with eI "flag like" - associated to a canonical flag basis and the labels forming a standard YOUNG tableau.

3. FORMAL STATEMENT: CASE OF A CANONICAL FLAG BASIS. ORBITAL

COMPONENT OF @M,

The condition M(t)+O, We E L) is equivalent to the non

vanishing of the tensor product: @M+O. Besides M ( t ) + O is

necessary to insure the existence of ( M ( t ) ) E P ( V ) (cf. 5 2 )

for every t . Then (@M) is the image of the ordered m-tuple

(M(1)x.. . x M ( m ) ) E (P(V))m in the canonical SEGRE embedding

( P ( V ) ) n U P(?@), cf. [H-PI. Again V = K", K of arbitrary

characteristic.

(cf. 52) I Ll

FORMAL STATEMENT. Let M l E (V-{OI)

L1 = {1,2,. . .,i}, L = Lm = with Ml equal to the restriction

of M:L + V-(0) to Ll: MI = M I L I .

There exists a sequence of frames Z1 with i entries (cf.

52) and bijective maps Yi:31 + LI such that Yl(31) is a

standard tableau (cf. DEF. 2) with entries l12,...,i and the

composition YM = MIYl is a f l a g tableau (cf. DEF.2). I


Y

In the particular case that M(L) contains dl linearly

(e h e h . . .hed*O) with el independent elements e , e 2 , . . . repeated h l times such that (3.1) holds YM is a (generally

twisted) canonical basis (cf. g2) of a f l a g

= F i.e. whose shape is given by the

partition dl,d2,.. . ,dc of m conjugate with the

1 2 1

‘ ed 1

A l . . . h

Fdld2.. . dc

hi 2 h2 2 ... 2 An.

Proof. For i = 1 0 ++ +-+ 1-1 all the

conditions are trivially satisfied. If FS, Y s - l , Ms-l are

constructed and YH is associated to a flag F(‘-l) the only

possible place for M ( s ) in Y is at the bottom of the first

s-1

column 7 of Fs-l such that M(s)@ V(s-l’T) I M(s) E Vs-l”-l (*

M ( s ) E Vs-l” for any j = lt2t..f7-l). If M ( s ) is contained

in all the components of F(‘-l) its place should be at the

top of the next new column, in the first row

M(s)eV7 w M ( s ) EV*-~ ’ ’ for e v e r y J = 1 , 2 , .

?- (3.2)

M ( s ) eViY-’

In fact M(s) cannot be in a column j<.a because

206 F. Gaeta

M ( s ) E V('-'") , and the corresponding new column r of

F'would contain linearly dependent elements. M ( s ) cannot be

in a column j>.a because this would imply M ( s ) E V('-'*')

against our assumption. On the other hand M ( s ) llfits" at the

bottom of the rth column because V ' s ' r l is the union of

and M ( s ) ( c V ( s - l ' r ) ) thus the linear independence

in the .ath column of YH is insured. The new Ys is still

standard (it is a Itregular application of nodes") with no

entries either to the right in the same row, other below in

the same column. The flag conditions in F(') is preserved in

the first 7-1 columns and in the columns r+llr+2,...,

's-1, r )

S

unafected by the addition, but V's") c V(s-l'J-') - - V(s"-')

and v's ,7) , v(s-l,r+l)- - v(s,r-l)

In the particular case of dl different and linearly

independent vector entries e (j = 1,2, ...,dl) the

alternative (linearly independent or linearly dependent) is

simplified to (elements all different in a column, or some

repeated e ) . Accordingly in the previous construction the

J

J

insertion of M ( s ) = e in YH to construct Yn is done as s-1

J S

follows: M ( s ) = e fits at the bottom of the first column ir

where e did not occur, i.e. e E V(s- '* l ) for

i = 1,2,...,y-l but e J d V ( s - l ' r ) .

J

J J

It is clear that the vectors in the column r form a

basis B of components V ( s ' r ) of F(') with B1 > Bz 2 . . . 3 Bcl

q.e.d. This canonical basis is generally twisted (cf. 9 2 ) ,

i.e. the el do not occur necessarily in the i-th row.

EXAMPLE. A1 = 4, h = 2, A = 1, m = 7. 2 3

A nutural association of PGL(V)-orbits 207

M ( 1 ) = e2, M ( 2 ) = el M ( 3 ) = el, M ( 4 ) = e3 M ( 5 ) = e2

M ( 6 ) = M ( 7 ) = el

We shall prove the following property:

If M(L) contains dl = r different and linearly

independent elements with el repeated A i times, and

A l 2 A 2 ... 2 A r > 0, the labels of ei in Yn contained in

the first, second, third,.. column form an increasing

sequence. The same as their appearance in M ( l ) , M ( 2 ) , ... For instance in the previous example we have

Labels of el: 2 < 3 < 6 < 7 e2: 1 < 5 e3: 4

The property holds in general in el has label l in the

column r and we drop all the labels >1 we have an standard

tableau with l entries and the label of el in the column 7-1

is surely < l .

This property enables a drastic simplification of our

insertion algorithm:

Let us write the labels of el,e2, ..., e in a YOUNG A l A 2 , , .

tableau of shape F

208 F. Gaeta

where in the row i el appears with the indexes

e l l c e12 < . . . c e r h , in the same order appearing in M(1)

M ( 2 ) .. . M(n) (we can drop all the e f e , ) . Then the label

YOUNG tableau is row ordered. According to BOERNER's footnote

lemma we have:

J

The tableau Yn is obtained from (1.4) by column ordering

of ( 3 . 4 ) ! Namely:

Let q E Q be the vertical permutation transforming the

Y(tl,) in a standard tableau. Then the same s applied to

YM = (e,(tlJ)) gives the final Yn with a standard label

tableau and with a canonical flag basis obtained by

q-twisting of the horizontal one in (3.4).

In the previous example (3.4) becomes

2 3 6 7

1 5

4

and the column ordering produces (3.3).

It is obvious that the previous construction M + Yn

is invariant by the GL(V) action; cfr. 51 for similar

statements regarding sM and (@M) (The (@M) __j Y(@n,) in the

SEGRE variety is invariant by the action of the projective

group PGL (V) ) . Let us check now that every standard tableau

) actually appears (provided dl = n). In fact we

can choose the M(l),M(2), ... step by step in such a way that the conditions hold. The dl 5 n conditions insures that there

are enough linearly independent elements to fit the first

'('xi d . . . d 1 2 c


column.

The embedding of Fd ...d in a suitable projective 1 2 c

space leads naturally to the YOUNG operators

p = C P Q = 1 (sgn q1.q P E P v Q

because the p E P leaves invariant the conditions ( 2 . 1 1 ) .

The antisymmetrization Q is essentially equivalent to the

construction of the tensor product Al@A2@. . .@Ac where

A E A V, (Al@. . .@Ac) is the image of Fdl...dc in the

projective space Ip (Id . . . ) (irreducible when K has zero

characteristic).

d r r

1 2 c

The construction @M Yen associates to any monomial

tensor a well defined component (irreducible for

characteristic zero). We call it the orbital component.

Let us check a few examples with the projective

interpretation.

Let m = 2 ( x ) @ ( y ) . If (x) = (y ) w y = Ax for some k*O

we have the trivial tableau x ( 1 ) y ( 2 ) . It corresponds to a

point (x@x) of the VERONESE variety of I P ( V ) .

If (y)*(x) w XAY+O the attached tableau is x(1 ) and we

have the variety of pair of different points. They define the

Grassmannian (locus of the ( x ~ y ) E IP (A%) .

Y ( 2 )

Let m = 3 , and the case [ f 2'; ( 3 ) ) . The corresponding orbital component defines the projective flag (Fzl ) (line,

point) where (XAY) defines a line and (z) belongs to this

line.

210 F. Gaeta

4 . A TENTATIVE CONNECTION WITH SCHENSTED ALGORITHM.

This famous construction was introduced in [S] to

improve a correspondence discovered by ROBINSON [RJ. It is a

recursive procedure to attach (in a bijective way!) to any

permutation S of the symmetric group Sm = S(L) acting on the

naturally ordered set L an ordered pair Y ' l ) , Y'2' of

standard YOUNG tableaux of the same shape F:

( 4 . 1 ) S:L L w Y:": F 4 S Y:'): F+ S

In this way he could find a combinatorial proof of the

famous formula

where fh= dim IA (the irreducible representation of Sm over

p: correspodding to 3. =Pd (h,d any two conjugate partitions of

m). It is amazing the depth of the result with so few

prerrequisites (only group theory and ordering in L). This

was the source of hundreds of articles trying to extend it. A

natural extension to maps S:L -+ r (in our notations)

between two ordered sets, where n=#r is not necessarily equal

to m and S is not necessarily injective. This was done by

KNUTH in [K] and it is particularly well adapted to the study

of the geometry O E flag manifolds Fd or more generally to the

study of the tensor representation GL(V)+ G L ( v @ ) in order

to replace ( 4 . 2 ) by

h

( 4 . 2 ) ' n'" = 1 f h F~ ( A + m r=d sn)

1

h where F =dim IA (IA irreducible representation space

A natural association of PGL(V)-orbits 21 1

h attached to an allowable pair 3 , Yd (r=d sn), taking into

account that this representation is Ilrepeatedll fA times. FA

equals the number of standard tableaux (in r) with possible repetitions in rows and no repetitions in any column of FA.

Accordingly we should try to replace (4.1) by

1

s: L + r - ( Y ( ' ) ( F ) , Y ' * ' ( F ) ) ( S non necessarl l y InJective)

with Y'" : f---1 L injective (labelling irreducible components

in the same isotypic block) and Y'":F+ r (with repeated

indices, but not in any column).

Since we want to study the canonical embedding of the

r is needed as the flag manifold Y d ( = 2 ) in P ( I d ,,,d 1,

label set for the homogeneous coordinates x1,x2,...,xn in

Pn-l=P (V) and we should Schenstedls abstract

h

1 2 c

constructions in terms of the m-monomials of the form

(where (el,e2,...,er) is a canonical flag basis

(r=dl=dimension of the maximal component of an Fd). In order

to do that we interchange rows and columns in the Schensted

constructions, we avoid repetitions in any column and we

accept KNUTHIS reasonable modifications.

In order to establish the link of this procedure with

our M d Yn it becomes necessary to study the formal laws

armonizing the vertical antisymmetry (use of the signed

(sgn q) .q of YOUNG) and the "twisted symmetry" allowed in any

212 F . Gaeta

pseudorow (qrl) where the e are the basis vectors of our

canonical basis B. J

I must confess that I am disappointed that the

foreseable natural results are not deeper than the already

known (independent of Schensted construction) via "standard

Grassmann coordinate productsll of the p

the V('), V(3) ,...,) (cf [H-PI, the straightening formulas,

etc). For instance we can dualize the M+ YH algorithm

replacing V by the dual of V'": (V"~)*=Horn(V~",K) by taking

the restrictions El=xll V'" of the coordinate functions in V

(i si 5 . . .sin) and studying the monomials E , . . . , E

and its canonical location by the algorithm ( E W Y ) . But a lot of work must be done yet.

I 1 1 1 2"' J of v ( l ) (or

I I 1 I2

1 2

E

A s a compensation for this disappointment I found that:

The explicit linear combinations with integral

coefficients

of a non standard Grassmann monomial p(N) of shape FA in

terms of the standard ones ~(2,) can be done combinatorially

form : in determinantal

( 4 . 4 )

where F = FA and

-1 according to

the step number E takes the values 0,l or

this rule: I J


E = 0 if it is not possible to find p E P I q E Q such 1,

that

C, = pq X I or C = pqN)

E = sgn q otherwise 1,

Proof: Cf. BOERNERIs book [B] and develop the determinant

(4 .4 ) resulting from the expansion of p(N), p(X,) as linear

combination of C monomials with values in e1,e2,. . . ,er. The

determinant E ( i , j = 1 , 2 , . . . ,F) is lltriangulartt with all

diagonal elements equal to 1, where the Cl,C2,.. . ,XF are ordered lexicographically; accordingly we obtain ( 4 . 3 ) from

1,

( 4 . 4 ) .

REFERENCES

[B] BOERNER, Darstellungen von Gruppen, Grund. Math. Wiss.

[Bou] BOURBAKI, Algebre multilindaire Hermann, Paris

[Gl] GREENE An extension of Schensted's Theorem, Advances Math. 14, 254-265.

[G2] Some order-Theoretic properties of the Robinson-Schensted correspondence, LNM 574, page 114. Springer Verlag, 1977

[H-PI HODGE-PEDOE Methods of algebraic Geometry I, 11,111, Cambridge University Press.

[K] KNUTH, Permutations, matrices and generalized Young Tableaux, Pacific J Math. 34 ( 1 9 7 0 ) , 709.

[L] LITTLEWOOD, The theory of group characters, Oxford, Clarendon Press, 1958.

[PI PROCESI, A primer of invariant theory, Brandeis University Lecture Notes

[R] ROBINSON, On the representations of the symmetric group, Amer. J. Math. 6 0 ( 1 9 3 8 ) ; 745-760; 69 (286-298) ; 7 0 (1948) , 277-294.

[ S ] SCHENSTED , Longest increasing and decreasing subsequences, Canad. J. Math. 13 ( 1 9 6 1 ) ,

Band 74, Zre Auflage, Springer, 1967.

179-191.

214 F. Gaeta

[Sch] SCHUTZENBERGER, La correspondance de Robinson, Combinatoire et representation du groupe symm&trique, Strasbourg , 1976, LNM 579, Springer, 1977.

[W] WEYL, The classical groups, Princeton University series.

[Y] YOUNG, Collected papers, Toronto.

Combinatorics '90 A. Barlotti el al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. 215

Decomposing Steiner Triple Systems Into Four-Line Configurations

T.S. Griggsa, M.J. de Resminiband A. Rosac

aDepamnent of Mathematics and Statistics, Lancashire Polytechnic, Preston PR1 2TQ, England

bDipartimento di Matematica, Istituto "Guido Castelnuovo", Universita di Roma "La Sapienza", Piazzale Aldo Moro 2,140185 Roma, Italy

cDepartment of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S4K1

1. INTRODUCTION

A Steiner triple system (STS) is a pair (V,B) where V is a v-set of elements, and B is a collection of 3-subsets of V called triples or lines such that every 2-subset of V is contained in exactly one triple. The number v = I V I is called the order of the STS. It is well known that an STS of order v (STS(v)) exists if and only if v z 1 or 3 (mod 6). If in the definition of an STS "exactly" is replaced with "at most", we have apartial triple svstem. As in [ti], we will use the term configuration to describe a partial triple system with a fixed small number of lines.

In a recent paper by Hor& and Rosa [8], the following problem was addressed: Let C be any configuration of three lines. Can the set of triples of any STS (of order 1 13, possibly with one or two triples deleted) be partitioned into copies of C? It has been shown in [8] for three of the five possible three-line configurations that the answer is in the affiiative but for the remaining two configurations (including the "mangle") the question remains largely open.

This paper deals with decompositions of STSs into four-line configurations. Unlike in [ 8 ] , we consider here only "exact" decompositions, i.e. we restrict ourselves to STSs with a set of triples B such that 4 divides I B 1 , the cardinality of B. First of all, it is certainly no longer true that any STS of sufficiently large order can be decomposed into copies of a configuration C if C is any configuration of four lines. In fact, if C is the Pasch configuration P (quadrilateral [9] or arrow 131; see also Figure 1 below), there exist STSs of every order

216 T.S. Griggs er al.

Fig. 1

Decomposing Steiner triple systems 217

v = 3 (mod 6) without a single copy of P. Such STSs are called anti-Pasch (cf. [3], [6], [9]). Anti-Pasch STSs are also known to exist for infinitely many orders v s 1 (mod 6), and it has been conjectured [3], [6] that they exist for all orders v I 1 (mod 6), v 2 19. Similarly, there exist STSs of arbitrarily high order without a single copy of the configuration C14 (see Figure 1 below): the projective spaces PG(d,2) which is an STS of order 2d+l-l. And although each STS of sufficiently large order must by necessity contain at least one copy of each of the remaining 14 four-line configurations Ci (cf. Figure l), it is by no means clear that every such STS must be decomposable into copies of Ci for all i. On the other hand, this is likely to be the case for several of the four-line configurations. In fact, it follows from the results of [8] that this is so for configurations C1 and C7. and we prove it here for another two configurations C3 and C4.

However, we are concerned mainly with the following problem: given any four-line configuration C, for which orders v does there exist an STS(v) whose set of triples is decomposable into copies of C?

2. FOUR-LINE CONFIGURATIONS

There are 6 disconnected, and 10 connected configurations of four lines, with three points per line. The 16 nonisomorphic four-line configurations are given in Figure 1. Denote N(n) = (1,2, ..., n). Let S(Ci) = {v: there exists an STS(v) decomposable into copies of GI. We call S(Ci) the spectrum for the configuration Ci.

An obvious necessary condition for an STS(v) to be decomposable into copies of any four- line configuration C is that the number of triples be divisible by 4. Thus we get:

Lemma 2.1. S(Ci) c (v: v I 1 or 9 (mod 24)) for i E N(16). Proof. Trivial.

The next two lemmas show that every STS of sufficiently large admissible order can be decomposed into copies of Ci if i=3 or 4.

Lemma 2.2. Every STS(v), v I 1 or 9 (mod 24), v 2 57, can be decomposed into copies of

Proof. Let (V,B) be an arbitrary STS(v), v = 1 or 9 (mod 24). Decompose B into v(v-1)/12 (disjoint) 2-windmills (i.e. pairs of intersecting lines); this is always possible by [8]. Form a graph G whose vertices are the v(v-1)/12 2-windmills, and where two vertices are adjacent if the corresponding 2-windmills have no elements in common. Then the degree of each vertex

c3.

218 T.S. Griggs et al.

of G is greater than or equal to q = [v(v-1)/12 - 11 - [5(v-1)/2 - 101. Clearly, in order to prove the statement of the lemma it suffices to show that G has a 1-factor. But if v 2 57 then q 2 I V(G) 1 /2, and thus G has a l-factor by Dirac's theorem (cf. [2]).

Lemma 2.3. Every STS(v), v = 1 or 9 (mod 24), v 2 169, can be decomposed into copies of

Proof. Let (V,B) be an arbitrary STS(v), v I 1 or 9 (mod 24), v 2 169. Find in B a set of v(v-1)/24 disjoint copies of a 3-windmill (i.e. a configuration of three lines intersecting in the same point). This is always possible since by [8], every STS(v) with v 2 19 (possibly with one or two triples omitted) can be decomposed into copies of a 3-windmill. Form a bipartite graph G where V(G) = VluV2; vertices of V1 are the v(v-1)/24 disjoint 3-windmills, and vertices of V2 are the remaining v(v-1)/24 lines of B. A 3-windmill (in V1) and a line (in V2) are adjacent if they have no elements in common. Obviously, to prove our statement it suffices to show that G has a 1-factor. For this we need to show that for every set of vertices S such that ScV2, I N(S) 12 I S I where N(S) is the neighbourhood of S (cf. [2]). Since each vertex of V2 is not adjacent to at most 3(v-1)/2 - 3 vertices of V1, it follows that if v 2 73, then each vertex of V2 has degree 2 v(v-1)/48 = I V2 I /2. Thus if ScV2 is such that 1 S I I 1 V2 I /2 clearly I N(S) 12 I S I . Assume now S is such that I V2 I /2 < I S 1 s I V2 I but I N(S) I < I S 1 . This means that there exists a vertex in V1\ N(S) (i.e. a 3-windmill) which has a nonempty intersection with every line of S. But any 3-windmill can have a nonempty intersection with at most 7(v-1)/2 - 9 other lines. Thus 7(v-1)/2 - 9 > v(v-1)/48 which implies v < 169. Thus for v 2 169, G has a 1-factor.

c4*

3. SMALL ORDERS

The unique STS(9) admits decompositions into isomorphic copies of only 4 of the 16 four- line configurations.

Lemma 3.1. A decomposition of the STS(9) into copies of Ci exists if and only if i = 11, 13, 14, or 15. Proof. An exercise.

Lemma 3.2. 25 E S(Ci) for any i E N(16). Proof. The following is a cyclic STS(25) decomposable into copies of Ci for any i E N( 16). Let V = 225, and take as the base mples: 0 1 6,O 9 11,O 3 10,O 4 12 (here and in what follows we omit set brackets for brevity). To see that it is decomposable into copies of Ci,


i E N(16), it is sufficient to exhibit a "base configuration" consisting of four triples, one from each of the four orbits of mples. The remaining configurations are then obtained from the cyclic group of order 25 acting as i 4 + l (mod 25) on this base configuration. The base configurations are as follows: C1: 0 1 6 , 2 11 13,4 7 14,5 9 17. C2: 0 1 6,O 9 1 1,2 5 12,3 7 15. C3: 0 1 6 , 0 9 11,2 5 12.2 15 19. C4: 016 ,0911 ,0310 ,4816 . C5: 0 1 6 ,6 15 17,7 10 17,4 8 16.

C7: 0 1 6,O 9 1 1,0 3 10,O 4 12. Cg: 0 1 6 ,6 15 17,7 10 17,5 9 17. C9: 0 1 6 ,6 15 17,7 10 17,lO 14 22. C10: 0 1 6,O 9 1 1,6 21 24.9 13 21. C11:O 1 6 , 0 9 11,621 24,0821. C12:016,0911,6916, 1513. C13: 0 1 6.0 14 23,14 11,6 10 18. C14: 0 1 6.1 15 24,6 21 24,O 8 21. C15:Ol 6 ,6822,62124,0821. C16: 0 1 6 , l 3 17,3 6 13,O 13 17. Note that the line corresponding to C7 is redundant as any STS(v) with v 2 25, v I 1 or 9 (mod 24) is decomposable into copies of C7 (cf. [8] and Theorem 5.1. below).

c6: 0 1 6 ,2 11 13,6 21 24,O 8 21.

Lemma 3.3. 33 E S(Ci) for any i E N(16). Proof. (a) The following is a cyclic STS(33) decomposable into copies of Ci for i=1,2,4,5,9,13. Let V = 233, and take as the base triples: 0 1 4,O 2 14,O 5 15,O 6 13, 0 8 17, 0 11 22. Base configurations: C1: C13: C2: C4: C5: C9: 0214,0515,1714,152332; O14,125,236,01122(step3) . The second of the base configurations represents the short orbit of configurations; in the case of C2, C4, C5, C9, the configurations of the short orbit are obtained from the base configuration using step size 3 (i.e. adding 3i modulo 33 to its representative, for i=1,2 ,..., 10).

0 1 4 , 3 5 17,6 11 21,7 13 20; 0 1 4 , l 3 15,4 9 19,O 6 13; 0 1 4,O 2 14,3 8 18,6 12 19; 0 1 4,O 5 15,O 6 13,2 10 19; 0 1 4,O 2 14,14 19 29,3 9 16;

0 8 17,11223,62230, l l 1928. 0 8 17,O 11 22,622 30.11 19 28. 0 8 17, 1 9 18,2 10 19,6 17 28 (step 3). 0 2 14, 1 3 15,2 4 16.2 13 24 (step 3). 0 8 17, 1 9 18,2 10 19,8 19 30 (step 3).

220 T.S. Griggs et ai.

(b) By [8], any STS(33) is decomposable into copies of C7.

(d) The following is an STS(33) decomposable into copies of C16=P. Let V=Zllx( 1,2,3) and take as both base triples and configurations: O1O2O3,0162103,3102103,316203; 011131,8111102,813163,0110263; 021242,621233,6242101,0233101; 037353,837361,8353102,0361 102.

Lemma 3.4.49 E S(Ci) for any i E N(16). Proof. (a) By [8], any STS(49) is decomposable into copies of C1 or into copies of C7.

(b) The following is a cyclic STS(49) decomposable into copies of Ci for i E N(15)\{1,7).LetV=Z4gandtakeasthebasemplesO 112,0221,039,0426, 0 5 18,0717,0824,01429. Base configurations: C2: 0 4 2 6 , 0 8 2 4 , 1 3 2 2 s 193% 0 112,039,2720.4 11 21. C3: 0 1 12,039,2720,23441. C4: 0 1 12 ,039 ,05 18,41121.

0 4 26,O 8 24, 1 3 22, l 15 30; 0 4 26,O 8 24.0 2 21,l 15 30;

Decomposing Steiner triple systems 22 1

0 4 26,O 8 24,5 7 26,l 15 30; 0 4 26.0 8 24,24 26 45,l 15 30; 0 4 26,O 8 24,O 2 21,7 21 36; 0 4 26,O 8 24,12 2641,23 24 35; 0 2 21,O 8 24,102439,2 3940; 0426,O 8 24,4 24 38,4 5 16; 0 426,O 8 24,4 18 33,24 26 45; 0 4 26,O 8 24,4 18 33.14 15 26; 0 1 12,O 10 42,l 17 42,lO 12 3 1; 0 426 ,05 18,O 25 33,4 18 33;

0 1 12,039,91427,41121. 0 1 12,9 12 18.0 5 18,3 10 20. 0 1 12 ,039 ,0518 , l l 1828. 0 2 21,O 3 9,9 14 27,4 11 21. 0 4 26,O 3 9,8 13 26,3 13 45. 0 2 21,O 3 9,3 8 21,3 10 20. 0 1 12,9 12 18,05 18,5 37 44. 0 2 21,O 3 9,9 16 26,3 34 39. 0 3 9,O 14 29,3 7 29.9 14 27. 0 1 12,01042,1410,101231.

(c) The following is a cyclic STS(49) decomposable into copies of C16=P. Let V = Z49 and take as both base triples and configurations:

0 1 12.0 3 20, 1 20 25,3 12 25; 0 2 10,O 4 18.2 18 25,4 10 25.

Lemma 3.5.57 E S(Ci) for any i E N(14) u (16). Proof. (a) By [8], any STS(57) is decomposable into copies of C1 or into copies of C7, and by Lemma 2.2, into copies of C3.

(b) The following is a cyclic STS(57) decomposable into copies of Ci for i = 2,4,5,8,9,13. Let V = 257 and take as the base triples 0 1 27,O 2 15,O 3 24,O 4 16,O 5 25,O 6 17,O 7 29, 0 8 18,O 9 23,O 19 38. Base configurations: C2:

C4:

0 1 27.0 3 24,2 6 18,4 9 29; 0 2 15, 1 3 16,2 4 17,5 24 43 (step 3). 0 1 27,O 3 24,O 4 16,4 9 29; 0 2 15, 1 3 16,2 4 17.2 21 40 (step 3).

0 6 17,O 7 29,l 9

0 6 17,O 7 29,O 8

9,2 11 25;

8 , l 1024;

C5:

c8:

C9:

C13:

The third base configurations represent in each case the short orbit of configurations. The configurations of this orbit are obtained from the base configuration using step size 3.

0 1 27,O 3 24,l 5 17,2 8 19; 0 2 15, 1 3 16,2 4 17,O 19 38 (step 3). 0 2 15.0 3 24,o 4 16,2 6 18; 0 1 27, 1 2 28,2 3 29, 1 20 39 (step 3). 0 2 15,O 3 24,2 6 18,3 8 28; 0 1 27, 1 2 28,2 3 29,O 19 38 (step 3). 0 2 15.0 3 24,2 6 18,15 20 40; 0 1 27.1 2 28,2 3 29,9 28 47 (step 3).

0525 ,0729 , l 1024,5 1323;

0617,0729,0923,61424;

0617,0729,61424,7 1630;

0 6 17,O 7 29,6 14 24,17 26 40,



Lemma 3.6. 81 E S(Ci) for i E (1,2,3,4,5,7,8,9,13). Proof. (a) By [8], any STS(81) is decomposable into copies of C1 or into copies of C7, and by Lemma 2.2, into copies of C3.

(b) The following is a cyclic STS(81) decomposable into copies of Ci for i = 2,4,5,8,9,13. Let V = Z81 and take as the base mples 0 1 38,O 2 21,O 3 34,O 4 22, 0 5 35,0623,07 36,O 8 24,09 41,O 1025,O 11 39,O 12 26,O 13 33,O 27 54. Base configurations: C2: 0 1 38,O 3 34,2 6 24,4 9 39;

0 1 0 2 5 0 11 39, 1 13 27,2 15 35; C4: 0138,0334,0422,2737;

0 1025,O 11 39,O 1226,l 1434;

~ 5 : 0 2 21,O 3 34,2 6 24,4 9 39; 0 1025,O 11 39, l 1434,102236;

01025,01139,01226,102343; 0 2 21,O 3 34,2 6 24,3 8 38; 0 10 25,O 11 39,lO 22 36,1124 44; 0 2 21,O 3 34,2 6 24,2126 56; 0 10 25,O 11 39, 10 22 36,25 38 58;

c8: 0221,O 3 34,0422,27 37;

Cg:

C13:

0 6 23,O 7 36,19 25,2 11 43; 0 2 21, 1 3 22.2 4 23,3 30 57 (step 3). 0623,0736,0824,11042; 0 2 21, 1 3 22.2 4 23,2 29 56 (step 3). 0623 ,0736 , l 1042,61430; 0 1 38, 1 2 39,2 3 40,4 31 58 (step 3). 0 6 23,07 36,O 8 24,6 15 47; 0 1 38, 1 2 39,2 3 40, 1 28 55 (step 3). 0 6 23,07 36,6 14 30,7 16 48; 0 1 38, 1 2 39,2 3 40,O 27 54 (step 3). 0623,0736,61430,7 1648; 0 1 38,l 2 39,2 3 40, 12 39 66 (step 3).

The fourth base configuration represents the short orbit of configurations; the configurations of this orbit are obtained using step size 3.

4. RECURSIVE CONSTRUCTIONS NEEDED

Our main recursive construction that follows is a variant of Wilson's fundamental construction (cf. [ 11 and also for undefined design-theoretic terms).

Lemma 4.1. Suppose there exists a group divisible design GD(V, G, B) such that d G I + 1 E S(Ci) for all G E G where w is a positive integer. Suppose further that for each BE B, there exists a GD(V', G', B') with I G' I = I B I , I G' I = w for all G' E G', I B' I = 3 for all B' E B' and whose set of blocks B' is decomposable into copies of Ci. Then d v I + 1 E s(c~). Proof. Weight every point of GD(V, G, B) with weight w. Let - be a new point. For every G E G, put on the set wG u I-) a copy of an STS(4 G I + 1) decomposable into copies of Ci. For each B E B, replace B with a copy of a GD(V', G', B') decomposable into copies of Ci. The result is an STS (d V I + 1) decomposable into copies of Ci.


One of the three ingredients needed in Lemma 4.1, the "master group divisible design", is readily available (cf. next section). Another one was given in the preceeding section. The next two lemmas provide the remaining ingredient.

Lemma 4.2. There exists a GDD with 12 elements, 3 groups of size 4 and blocks of size 3 decomposable into copies of Ci if and only if i E N(16)\(4,15). Proof. An exercise.

Remark. Note that the GDD in Lemma 4.2 is equivalent to a Latin square of order 4 and thus that there exist only two such nonisomorphic GDDs.

Lemma 4.3. The (unique) GDD with 8 elements, 4 groups of size 2 and blocks of size 3 can be decomposed into copies of C15. Proof. An exercise.

5. MAIN RESULTS

Theorem 5.1. Let i E (1,3,4,7). Then there exists an integer vo(i) such that every STS(v), v = 1 or 9 (mod 24), v 2 vo(i) is decomposable into copies of Ci. Proof. This follows from [8], Theorems 3.1 and 4.1 1, and our Lemmas 2.2 and 2.3. It is currently known that vo(1) ~ 4 9 , vo(3) 57, vo(4) I 169, and vo(7) = 25.

The next theorem settles almost completely the question of the existence of STSs decomposable into copies of Ci when i E N(16).

Theorem 5.2. (a) Let i E (1,2,3,4,5,7,8,9). Then v E S(Ci) if and only if v I 1 or 9 (mod 24),v 2 25. (b) Let i E ( 1 3 , ~ ) . Then v E S(Ci) if and only if v = 1 or 9 (mod 24), v 2 9. (c) Let i E (6,10,12,16), and let v Z81. Then v E S(Ci) if and only if v I 1 or 9 (mod 24),v 2 25. (d) Let i E (11,14), and let v # 81. Then v E S(Ci) if and only if v 1 or 9 (mod 24), v 2 9. Proof. (a) There exists a GDD with n groups of size 6 and blocks of size 3 for all n 2 3 (cf. (71). Apply Lemma 4.1 with weight w=4 using the fact that by Lemma 3.2,25 E S(Ci) for all i E N(16). This shows that v E S(Ci) for v 1 (mod 24), v 2 25, v # 49, for all i E N(16)\(4,15). By Lemma 3.4,49 E S(Ci)) for all i E N(16).


There exists a GDD with n groups of size 6, one group of size 8, and blocks of size 3 for all n 1 3 (cf. [5]). Apply Lemma 4.1 with weight w=4, using the fact that by Lemmas 3.2 and 3.3,25 E S(Ci) and 33 6 S(Ci) for all i E N(16). This shows v E S(Ci) for v I 9 (mod 24), v k 33, v # 57, 81 for all i E N(16)\(4,15). By Lemma 3.5.57 E S(Ci) for all i E N(14) u (16). and by Lemma 3.6,81€ S(Ci) for i E (1,2,3,4,5,7,8,9,13). Further, we have (25,33,49,57,81) c S(C4) by Lemmas 3.2 to 3.6 and v E S(C4) for v = 1 or 9 (mod 24), v 1 169, by Lemma 2.3. Thus the only thing left to prove is that (73,97,105,121,129,145,153) c S(C4). This presents no difficulties, and is left as an exercise.

(b) There exists a GDD with n groups of size 12 and blocks of size 4 for all n 1 4 (cf. [4]). Apply Lemma 4.1 with weight w=2, using the fact that by Lemma 3.2, 25 E S(C15). and Lemma 4.3. This gives v E S(C15) for all v I 1 (mod 24), v # 49,73. Lemma 3.4 gives 49 E S(C15). The following is a cyclic STS(73) decomposable into copies of C15. Let V = 273 and take as both base triples and configurations: 0 1 14.0 2 19,O 9 36,2 14 36; 0 21 68,O 29 62,O 53 57,21 29 53. There exists a GDD with n groups of size 4 and blocks of size 4 if and only if n I 1 (mod 3) (cf. [4]). Apply Lemma 4.1 with weight w=2, using the fact that by Lemma 3.1, 9 E S(C15), and Lemma 4.3. This gives v E S(C15) for all v I 9 (mod 24).

0 3 18.0 6 3 1.0 23 66,3 31 66;

6. CONCLUSION

For 10 out of the 16 nonisomorphic four-line configurations, the spectrum of v I 1 or 9 (mod 24) for which there exists an STS(v) whose set of triples is decomposable into copies of the four-line configuration has been determined completely. For the other 6 configurations including the Pasch configuration C16 = P, just one value (v=81) has been left in doubt.

ACKNOWLEDGEMENTS

The authors would like to thank Franya Franek for programming help in finding an STS(33) decomposable into Pasch configurations, Curt Lindner for several valuable comments, and especially Doug Stinson for pointing out an omission in an earlier version of this paper. Part of this work was done while the third author was visiting Dipartimento di Matematica, Universita di Roma "La Sapienza". He would like to thank the Department for its hospitality.

226

REFERENCES

T.S. Griggs et al.

T. Beth, D. Jungnickel, H. Lenz, Design Theory, Bibl. Inst. Mannheim, 1985. J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, Macmillan, 1976. A.E. Brouwer, Steiner triple systems without forbidden subconfigurations, Mathscentrum Amsterdam, ZW 104/77. A.E. Brouwer, A. Schrijver, H. Hanani, Group divisible designs with block size four, Discrete Math. 20(1977), 1-10. C.J. Colbourn, D.G. Hoffman, R. Rees, A new class of group divisible designs with block size three, J. Combinat. Theory (A) (to appear). T.S. Griggs. J.P. Murphy, J.S. Phelan, Anti-Pasch Steiner triple systems, J. Combin. Inf. & Syst. Sci. 15(1990), 1-6. H. Hanani, Balanced incomplete block designs and related designs, Discrete Math. 11(1975), 255-369. P. Horslk, A. Rosa, Decomposing Steiner triple systems into small configurations, Ars Combinat. 26(1988), 91-105. D.R. Stinson, Y.J. Wei, Some results on quadrilaterals in Steiner triple systems, Discrete Math. (to appear).


Non-symmetric configurations with deficiencies 1 and 2

H. Gropp

Miihlingstr. 19, D-6900 Heidelberg, Germany

Abs t r ac t This paper continues the discussion of the existence problem of configurations ( v r , b L )

started in [ll] and [13]. It is concentrated on configurations with deficiencies 1 and 2 including also symmetric configurations. The existence of many non-symmetric configurations with k = 4 is proved, especially of those with small deficiencies. The results are obtained by either collecting results about divisible designs, regular packings of complete graphs and resolvable configurations or by applying special techniques to construct new configurations. A classification of configurations (125, 203) and (146, 283) is also included.

1 INTRODUCTION AND NOTATION

Definition 1.1 A configuration (v?, bk) is a f inite incidence structure with the following properties:

1 . There are v points and b lines.

2. There are k points o n each line and T lines through each point.

3. Two different lines intersect each other at mos t once and two different points are connected by a line at most once.

R e m a r k 1.2 Zf v = b ( and hence T = k ) the configuration is called symmetric and denoted by Vk.

Configurations (v,,bk) with k = 2 are r-regular graphs on v vertices. Since these graphs are investigated in graph theory it will be assumed that k 2 3 for the rest of this paper.

Configurations have already been defined in the last century. There are also important results, especially about symmetric configurations, obtained more than 100 years ago. For a general report on the history of configurations see [15]. The mathematical results which have been obtained about symmetric configurations are described in [ll].

228 H . Gropp

The Italian abbreviation cfz. will be used in many cases for the long word configuration since Italian mathematicians were important in the research of configurations in the early days. The reader is referred to a paper about the relations between configurations and graphs [16], some facts mentioned in this paper are described in [16] from a graph-theoretical point of view.

Remark 1.3 Of course, the dual of a configuration is also a configuration. I t is conveni- en t only to investigate configurations with b 2 v and T 2 v and to consider configurations without this properties as duals of configurations.

The following conditions are necessary for the existence of a configuration.

Lemma 1.4 If there exists a Configuration (v , , b k ) , the following conditions hold:

1. v s b a n d k s r

2. v r = bk

3. v 2 r ( k - 1) + 1 In this case the parameter set is called admissible.

Proof: The first condition holds by convention ( compare the remark above ). The second condition is proved by counting the incident point-line pairs in two ways. Concerning the third condition consider all the points on the lines through a fixed point. These are r(k - 1) + 1 different points since two different lines intersect in a t most one point.

Remark 1.5 Since fo r given r, k, and suitable v there is at most one possible parameter b only 3 of the 4 parameters are independent. That is the reason why there is a second suitable set of only 3 parameters fo r a configuration. Both parameter sets will be used in this paper.

Definition 1.6 The order m of a configuration (v,, bk) is defined as m = k - 1 analogous to the order of a f inite projective plane. The deficiency d of a configuration (v,, b,) is defined as d = v - ~ ( k - 1) - 1 and measures the "distance " of the configuration f r o m a Steiner

The index t of a configuration (v , , b k ) is defined as t = T f k ( or t = b f v ). If k divides r the configuration is said to have natural index.

system. Each point is not connected to ezactly d other points of the configuration.

If t = 1 the configumtion is symmetric.

In order to change from the parameters m,d,t to the parameters v , ~ , b, k the following formulas are needed.

Lemma 1.7 A configuration with order m, deficiency d and index 1 is a cfz. (v , , b k ) where k = m + 1, T = t(m + l), v = d + tm(m + 1) + 1, and b = l(d + tm(m + 1) + 1).

Non-symmetric configurations 229

Closely related to a configuration is its configuration graph.

Definition 1.8 The configuration graph of a configuration ( v , , b,) has the v points as vertices, and two vertices are connected by an edge if they are not collinear in the configuration.

The following lemma is obvious.

Lemma 1.9 The configuration graph of a cf i . (v,, b k ) is a d-regular graph on v vertices where d is the deficiency of the configuration.

In this paper the research on symmetric configurations ( compare [ll] ) is continued and also extended to non-symmetric configurations. The main aspect of this paper is to investigate configurations with deficiencies 1 and 2. A further paper [13] is devoted mainly to results about configurations with natural index. Its results are summarized in section 5.

However, this partition is not too strict. While an important result of [13] is the existence of all configurations with k = 3 the main aim of this paper is a big step towards the existence of the configurations with R = 4. At least the existence of those with small deficiencies is proved here.

Section 2 contains additional results about symmetric configurations which are not contained in [ll] and an updated existence table. In section 3 some construction methods for configurations with small deficiencies are explained.

The classification of all configurations (125,203) and (146, 283) is discussed in section 4. While section 5 repeats the main results of [13] many configurations with k = 4 are constructed in section 6.

The most results of this paper are obtained by gathering existence results from many older papers and looking for configurations in different parts of combinatorial theory. I hope that the reader can find all the definitions which are not mentioned in this paper in standard books about graph theory or design theory.

2 SYMMETRIC CONFIGURATIONS nk

2.1 General gemarks

In this section results about symmetric configurations are collected which are not contained in [ I l l . The additional results are obtained by applying theorems which were not known to me while writing [ll] although they are up to 40 years old. I would like to apologize for this gap in [ I l l . On the other hand, there are a lot of papers containing much weaker results than [I13 published during the last decades.

2.2

In [ll] the relation between biplanes and configurations with deficiency 1 is explained. In general each symmetric %design ( v , k, A) can be defusable to a configuration ((.A),, b,) with deficiency X - 1.

Defusion of a block design

230 H . Gropp

On the other hand, if the configuration graph of a configuration (v,, b k ) with deficiency d is a union of complete graphs Kd+l (i.e. the configurationis a so-called divisible design ), a symmetric 2-design ( v / ( d + l), k, d + 1) is obtained by identifying all points in the same clique of the configuration graph. For this paper only cerbain divisible designs are of interest, i.e. those with A1 = 0 and A2 = 1. They are defined as G D ( v , b , r , 12, m,n) as follows.

Definition 2.1 A divisible design GD(v , b, T , k, m, n) is a configuration (v,, bk) such that the point set is partitioned into m subsets with n points each where

2 points are collinear i# they belong t o diflerent subsets. Such a configuration is also called divisible.

2.3

The following old, but very important theorem plays an essential role for the existence question of symnietric configurations, especially of those with deficiency 1. It was proved by R.C.Bose and W.S.Connor [4] in 1952 and contains the faiiious theorems of R.H.Bruck, H.J.Ryser of 1949 and of S.Chowla, H.J.Ryser of 1950 as a special case.

Theorem 2.2 ( Bose, Connor I 9 5 2 [4, Theorem 91 )

The theorem of Bose and Connor

If a symmetric divisible design G D ( v , v , I z , k, m, n) with P = k a - v > O , Q = k , n = d + l , m = l + h ( k - l ) / ( d + l ) ezists then ( a ) if m is even then P mus t be a square and if further m = 4t + 2 and n is even

( b ) if m is odd and n is even then Q is a square and the equation P x 2 + ( - I ) ( m - ' ) / 2 n y 2 = z2 has a non-trivial solution in integers z,y,z. ( c ) if both m and n are odd, t hen the 2 equations (-l)(m-l)hi? + P y 2 = z2 and ( -1 ) (n - ' ) / 2nx2 + Q y 2 = z2 both have OT both have

then k is a s u m of two squares;

n o t a non-trivial solution in integers x,y,z.

2.4 Configurations with deficiency 1

As a consequence of this theorem the following corollary can be obtained for configurations n k with deficiency 1.

Corollary 2.3 Each symmetric configuration n k with deficiency I has a I-regular configuration graph. Hence it i s divisible and k OT k - 2 is a square.

Theorem 2.4 There are unique configurations 83 and 144. There i s n o configuration nk with n = k(k - 1) + 2 f o r all 5 5 k 5 10 and if neither k nor k - 2 is a square.

Proof: The result for k = 3 and k = 4 has been known for a long time (see [ I l l ) . For k = 6 the result was proved by Schellenberg [22] in 1975. This was the first non- existence proof for a symmetric divisible design whose existence is not ruled out by Bose, Connor. For k = 9 the result is proved in [ll]. In all other cases k and k - 2 is not a square.


Remark 2.5 The smallest unknown case is k = 11. I t has, however, been proved in [ I l l that the 5 known biplanes (56,11,2) cannot be defused.

There are severalpapers written after 1952 which obtain weaker results than [4]. FOT example, in 1980 N.L.Biggs and T.Ito [3] proved that f o r all k 5 or 7 mod(8) there is n o symmetric configuration v k with deficiency 1 in graph-theoretic language. of course, such a b can never fulfill the condition in Corollary 2.3.

2.5 Configurations with deficiency 2

A symmetric configuration n k with deficiency d 2 2 is not necessarily divisible. If d + 1 divides n it can be divisible. For example, there are 2 non-divisible configurations Q 3

and 1 divisible configuration 93. If d + 1 does not divide n no divisible configuration is possible. As the examples of the configurations 103 and 235 show this does not a t all indicate that the existence itself is in doubt.

For d = 2 the following corollary can be obtained from the Bose-Connor-Theorem.

Corollary 2.6 There are n o divisible configurations 336 and 9310.

2.6 Elliptic semiplanes

Definition 2.7 A n elliptic semiplane S(v,k) is a finite incidence structure with the following properties:

1. S(v,k) is a symmetric configurntion v k .

2. Given a nonincident point-line pair P, L there exists at most one line M incident with P and parallel t o L . Dually there exists at most one point Q incident with L and not connected with P.

The following results on the existence of symmetric configurations are proved by the existence of certain elliptic semiplanes. There is an infinite series of classical elliptic semiplanes (compare [ti]) and until now 2 exceptional elliptic semiplanes constructed by R.D.Baker [l] in 1977 and R.Mathon [17] in 1987.

Theorem 2.8 There is an elliptic semiplane S((p' - ~ ) ~ a ) if p is a prime power. There exist elliptic semiplanes S(45,7) and S(135,12).

Proof: A Baer subplane of a projective plane of order n2 is a subplane of order n. Remove the points and lines of a Baer subplane from a projective plane of order p2 and obtain a configuration with p' + p2 + 1 - (p2 + p + 1) = p' - p points and lines and

The elliptic senliplanes S(45,7) [l] and S(135,12) [17] are divisible symmetric con-

A recent book as reference for the results of this section is [6].

k = p2.

figurations 457 and 13512 resp.

232 H. Gropp

2.7 The existence table for configurations n k

The following table shows the updated knowledge concerning the existence of symmetric configurations with order 5 12 and deficiency 5 9 ( compare Table 4 in [ll] ).

An entry in usual scripture means tha t the corresponding configuration exists. Configurations i n italics exist but the existence of a cyclic configuration is in doubt. The bold configurations d o not exist. A blank spa

Deficiency Order

2 3 4 5 6 7 8 9

10 11 12

-

e shows that the existence problem is open. 0 1 2 3 4 5 6 7 8 9

3 CONSTRUCTIONS OF CONFIGURATIONS

3.1 General remarks

The a i m of this section is to construct configurations from other configurations by omitting some points and lines.

3.2

Lemma 3.1 If there exists a Steiner sys tem S(2,k,v) there is a cfz.

Point deletion of a Steiner system

((. - 1)(,-1), ( b - T)k) .

Proof: Remove a point and the T lines through this point. The obtained incidence structure is a cfz. with deficiency d = 12 - 2.

Example 3.2 A cfz, (247,424) is obtained by deleting a point from a Steiner sy s t em S(2 ,4 ,25) .

3.3 Line deletion of a Steiner system

Lemma 3.3 If there ezists a Steiner sys tem S(2,k,v) there is a cfz.

((v - h ) ( , - k ) i ( b - k ( T - 1) - Ilk)*

Non-symmetric conjigurations 233

Proof: Reniove k collinear points and all lines containing a t least one of these points. The obtained structure is a configuration with the above parameters.

Example 3.4 A cfz. ( 2 l 4 , 2 I 4 ) is obtained by deleting a line f r o m a Steiner system 5 ( 2 , 4 , 2 5 ) .

3.4 Deletion of pairwise non-connected points

Lemma 3.5 If there is a configuration (v,, bk) with deficiency d such that i ts configura- t ion graph contains a complete graph Kd+l as a component there exists also a configura- t ion ( ( ~ - d - - l ) , - ~ - ~ , ( b - r ( d + l ) ) ~ ) whosepoints can bef i l ledinto a (d+l ) -d imens iona l array of length T which i s a generalized Howell design or a similar structure.

Proof: Delete d + 1 non-connected points and all the r ( d + 1 ) lines through them. The fact that the obtained structure is a configuration with the parameters given above is obvious. The r ( d + 1 ) deleted lines are labels for the rows, columns, ... of the ( d + 1)- dimensional array and the position of a point of the obtained cfz. in the array is given by those lines incident with it in the original cfz.

The following 2 partial non-existence results for symmetric configurations are just examples to show the relation to Howell designs.

Definition 3.6 A Howell design H ( s , 2 n ) is a square array of side s with N = { l12 , . . . ,Zn} such that

1 . every cell is empty or contains a %subset of N ,

2. every symbol of N is exactly once in each TOW and column,

3. every 2-subset OCCUTS a t most once.

A Howell design H(2n- l12n) is a Room square. There are natural generalizations: Take k-subsets instead of 2-subsets and/or D-dimensional arrays H D ( s , 2n) instead

of 2-dimensional arrays ( see e.g. [9] ).

Lemma 3.7 There is n o configuration 235 with a triangle in its configuration graph.

Proof: Delete 3 non-connected points as described above and obtain a Howell cube H 3 ( 5 , 8 ) . But even a 2-dimensional Howell design H ( 5 , 8 ) does not exist ( see [21] ).

Lemma 3.8 There is n o cfz. 346 with a K , in its configuration graph.

Proof: By deletion of 4 non-connected points a Howell 4-cube H4(6, 10) would be obtained. But there is no Howell cube of size 6 with 10 symbols ( see [21] ).

234 H . Gropp

4 CLASSIFICATION OF CONFIGURATIONS

4.1 Previously knnwn results

While the existence of all configurations with L = 3 and adniissible parameters has been proved in [13] not too much is known about the classification of all configurations with certain parameters. The numbers of non-isomorphic configurations v3 are known for 7 5 v 5 14 as well as the number of Steiner triple systems S(2,3,v) for v = 7,9,13,15. There are unique configurations 73 and 83, 3 configurations 93, 10 configurations lo3, 31 configurations 113, 229 configurations &, 2036 configurations and 21399 configurations 143 ( see e.g. [ll] ). There are unique S(2 ,3 ,7) and S(2,3, 9), 2 S(2,3,13) , and 80 S(2,3,15) ( see e.g. [18] ).

In this paper the configurations (125,203) and ( 1 4 ~ ~ 2 8 ~ ) are discussed. The configurations (12s,203) have already been classified by J.Novik in [19]. However, [19] is published in Czech language and seems to be relatively unknown. For this reason NovAk’s result is proved here again.

Their total number is 574.

A further classification is done in [14] for configurations (124,163).

4.2 Configurations ( 125, 203)

There are exactly 2 Steiner systems S(2,3,13). This has been proved in 1899 by V. de Pasquale using a result of S.Kantor of 1881 who had classified all configurations lo3. A lot of details about the history of Steiner systems S(2,3,13) from T.P.Kirkman until to-day can be found in [12]. The 2 systems S(2,3,13) are tabled together with a lot of properties in [18].

The abovementioned connection of Steiner systems S(2,3,13) and configurations lo3 is described in detail in [lo].

In the following the Steiner systems 5(2,3,13) No.1 and No.2 are used as printed in [18] on pages 14 and 15. System No.1 has an automorphism of order 13 which acts transitively on its 13 points, system No.2 has 4 point orbits of lengths 6, 3, 3, and 1 under the action of its autoniorphism group.

If two points of S(2,3,13) are deleted ( see above ) which belong to the same point orbit, of course the obtained configurations (lz5, 203) are isomorphic. Thus there are a t most 5 such configurations. In fact, all these 5 configurations are non-isomorphic.

Theorem 4.1 There are exactly 5 configurations (125, 203). They can be constructed b y deletion of a point from S(2,3,13) in the following way.

Cfz. (125,203) No.1: Delete point 13from S(2,3,13) No.1 Cfz. (l&, 203) No.2: Delete point 1 fiom S(2,3,13) No.2 Cfz. (125r203) No.3: Delete point 3 b o m S(2,3,13) No.% Cfz. 203) No.4: Delete point 4 from S(2,3,13) No.2 Cfz. (125,203) No.5: Delete point 7 f r o m S(2,3,13) No.2


4.3 Configurations (146,283)

There are exactly 80 Steiner systems S(2,3,13). This has been proved in 1917 by Cole, Cumniings and White. Again the 80 systems S(2,3,13) are tabled together with a lot of properties in [18] and their notation is used below.

The reader is referred to [18] for information about the autoinorphisni groups and point orbits of the 80 systems. There are altogether 787 point orbits in the 80 systems. Again it turns out that this is also the exact number of configurations (146, 283). In no case it happens that the deletion of 2 points of different orbits in a Steiner system leads to isomorphic configurations.

Theorem 4.2 There are exactly 787 configurations ( 146, 283). They can be constructed b y deletion of a point from S(2,3,15) and numbered according to the point orbits in [18]. I n the following the mmbers of configurations are given for those 80 which are obtained b y deletion of point 1 in the Steiner system. N0.i.

CJz. No. 1,2,5,8,15,19,22,24,33,44, CJz. No. 55,64,71,77,82,89,91,95,102,106, Cfz.No.l13,120,127,142,157,172,187,202,217,224, Cfz. No.233,238,253,268,283,290,295,298,313,328, Cfz.No.343,358,367,371,380,395,410,425,440,455, Cfz.No.470,485,500,515,530,545,560,575,590,597, Cfz. No. 61 2,615,622,629,636,651,666,681,696,711, Cfz. No. 726,74 1,756,761,767,772,775,780,785,787

5 ALREADY KNOWN RESULTS ABOUT NON- SYMMETRIC CONFIGURATIONS

The existence result for h = 3 and the results for h = 4 which are needed in this paper are repeated here again. Also the asymptotic result for cfz. with natural index is given again. The reader is referred for proofs and further results to [13].

Theorem 5.1 There is a configuration ( v r , b3) i f l v 2 2r + 1 and vr = 3b.

Lemma 5.2 There i s a configuration ( v r , b , ) for every v 4b if v # E,

Omod(l2), v 2 3r + 1, v r =

E = {84,120,132,180,21G,264,312,324.372,456,552,648,GG0,804,852,888}.

Lemma 5.3 There is a configuration (v, ,b,) with r = 4t ,v 2 3r + 1, vr = 4b for all 1 5 t 5 15 and t = 18.

Remark 5.4 The statement in the previous lemma is slightly stronger than the statement in Theorem 3.5 in [13].

Theorem 5.5 For given Ic and r with r = t h there i s a v o ( h , t ) such that there is a cJz. ( v r , b k ) for all v 2 vo for which the necessary conditions hold.

236 H . Gropp

The reader should not be confused about some results in [13]. Since [13] was written 3 months earlier than this paper some configurations which are constructed in this paper are said to be in doubt in [13].

6 CONFIGURATIONS WITH k = 4

6.1 General results

In the following the existence problem for k = 4 is attacked. The analogous problem for k = 3 has been solved in [13]. Although not all configurations (w,, b 4 ) are constructed here, it seems quite reasonable to conjecture that they all exist since the most difficult ones ( with small deficiency ) exist.

Lemma 6.1 The following parameters d and r are possible for a configuration (v,, b4): If r E 0 mod(4) then d can have any value. If r = 1 mod(4) then d E 0 mod(4). If r 2 mod(4) then d E 1 mod(2). If r c 3 mod(4) then d E 2 mod(4). Hence, for each value o f d exactly 2 of the 4 congruences for r mod(4) are possible.

Proof: The statement follows from v = 3r + d+ 1 and the fact that v r is a niultiple

In terms of regular packings the following theorem is proved in [2]. A lot of other of 4.

papers are used, especially [5] and [7].

Theorem 6.2 ( Bermond, Bond, Sotteau [2] ) There is a configuration with k = 4 for all d 5 2, for d = 3 , r for r

A very important source for configurations with k = 4 are divisible designs ( see Definition 2.1 ). In [7] it is proved that all divisible designs with k = 4 which fulfill the necessary conditions also exist with two exceptions. Together with the result of Lemma 3.5 the following theorem is obtained.

Theorem 6.3 There is a configuration with k = 4 whenever v i s a multiple of d + 1.

2mod(4) and Omod(4), if d = 4,5 ,6 ,8 ,10 .

Proof: Theorem 6.3 in [7] asserts the existence except of 2 cases. A G D ( 8 , 4 , 2 , 4 , 4 , 2 ) is no configuration in the sense of this paper since v > b. There is no divisible configuration (246,364) but such a configuration can be obtained via Lemma 3.5 from a configuration (268, 52,) constructed in Theorem 6.2.

The existence of resolvable Steiner systems yields the following result ( Theorem 3.2 in (131 ).

Theorem 6.4 All admissible Configurations with k = 4 and v = 4 mod( i2 ) exist.


6.2 Special constructions

At first, some special constructions are given which are very similar to those in [13]. For example, the special solution of a configuration (226,334) in [2] enables the following construction of all configuration with k = 4 a n d T = 6 (compare also Lemma 3.9 in 1131).

Lemma 6.5 There i s a configuration (v6, b4) for v 2 20, v even, b = iv.

Proof: Take the following base lines (00, l o , 30, I]}, {90,01,11,51}, (00, 50, 6], 91) and a n automorphism of order for v 2 22. A solution for v = 20 is

given in Theorem 6.2.

Lemma 6.6 There is a configuration ( q o , b4) for v 2 32, v even, b = 4.. Proof: Take the following base lines {oo,4o,ll0,15~}, {Oo, l o , 30,61}, {80, 01, 11, 51}1

v # 34,36,38,42,44,46. Taking {00,4,-,,9~, 131} instead of {Oo,40, 110,151) yields solutions for all v 2 44.

A further substitution ({lOo,O1, 11,51} instead of { 8 0 , 0 1 , 1 ~ , 5 ~ } ) yields solutions for v = 42.

A configuration (3410,854) is constructed by taking {OO, l o , 50,131}, {Oo, 20, 8 0 , 01}, {OO, 11,31, GI}, {Oo,41,51, 111}, {00,30,21,101}. Since the deficiency of this configuration is 3 its existence follows also from Theorem 6.2.

{20,01,31,91}, { O O , 60,01 ,2~} and an autoniorphism of order for v 2 32,

A configuration (3810, 954) is constructed by taking {OO, 40,50,71},

A divisible configuration (3610, 904) exists ( see Theorem 6.3 ). {Oo,30,90,81}, { 9 0 , 0 ~ , 1 ~ , 4 1 } , {60,01,21,71), { O O , ~ O , ~ I , ~ I } .

Lemma 6.7 There is a configuration ( q 4 , b d ) for v 2 44, v even, b = iw.

Proof: Take the following base lines {OO, 80,90,61}, {OO, 20,60,91},

{ 00,50,01,151} and a n automorphism of order f for v 2 44, v # 46,48,50,52. Taking {Oo, 50,221, 151) instead of {Oo, 50, 01, 151) yields a solution for v = 52. A configuration (4614, 161,) has deficiency 3 a n d exists because of Theorem 6.2 A configuration (5014,1754) is constructed by taking {OO, 10,70,141}, {OO, 30,110,61}~

{oo, 70,100,111}, {00,21,81, {OO, 51,131,141), {00,161,181,211},

{00,40,90,51}, {130,01,51, G I ) , {100,01,71,91), {170,01,81,111), {Do, 20,01,41}. A divisible configuration (4gI4, 1684) exists (see Theorem 6.3). In the following 2 configurations are constructed which cannot be obtained by the

methods above.

Lemma 6.8 There are configurations (329,724) and (4415,1434).

Proof: Take as base lines 0,1,7,11 and 0,2,5,14 mod(32) as well as the 8 lines

Take as base lines 0,8,9,13 and 0,2,12,19 and 0,3,18,24 niod(44) as well as the 11 O+i,8+i,16+i,24+i, i=O, ... 7 to obtain a configuration (329,724).

lines O+i,ll+i,22+i,33+i, i=0,10 t o obtain a configuration (4413,1434).

238 H . Gropp

6.3 Conclusion

In the following some small examples of configurations with k = 4 are given whose existence is not proved in this paper. This remark can be proved by applying all the previous results in this paper.

Remark 6.9 Not ye t constructed are the following configurations wi th k = 4 . These are the sm.allest configurations with respect to d:

(56171 2384)1(G821,3574),*** (d = 4 ) ~ (G218, 2794),(7422r 40T4), ... (d = 7 and T 3 2m0d(4)), (21268, 3G044),(23G76,44844),... (d = 7 and r = Omod(4)).

Acknowledgement: I want to thank Alan Hartman for fruitful discussions and hints which helped me a lo t to find papers containing configurations hidden as divisible designs or graphs.

Added in proof: I want to mention two papers [23], [24] which were written about 20 years ago and which I found quite recently. They contain results about symmetric configurations nk. Especially interesting are non-existence results on cyclic syniiiietric configurations. These two papers together with the references which they contain show another trial to establish a theory of configurations ( compare [15] ). Unfortunately, after a few years this research was stopped again.

References

[l] R.D.Baker, Elliptic semiplanes I: existence and classification, Proc. 8th S.E. Conf. Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, 1977), GI-73

[2] J.-C. Bermond, J Bond, D.Sotteau, On regular packings and coverings, Annals of Discrete Math. 34 (1987), 81-100

[3] N.L.Biggs, T.Ito, Graphs with even girth and siiiall excess, Math. Proc. Cambridge Phil. SOC. 88 (1980), 1-10

[4] R.C.Bose, WSConnor , Combinatorial properties of group-divisible inconiplete block designs, Ann. Math. Statist. 23 (1952), 367-383

[5] A.E.Brouwer, Optimal packings of K4’s into a K,,, Journal of Comb. Th. A26 (1979), 278-297

[GI A.E.Brouwer, A.M.Cohen, A.Neuniaier, Distance-regular graphs, Springer, Berlin, Heidelberg (1989)

[7] A.E.Brouwer, A.Schrijver, H.Hanani, Group divisible designs with block-size four, Discrete Math. 20 (1977), 1-10

[8] P.Dembowski, Finite Geometries, Springer, Berlin (1968)


[9] J.Dinitz, D.R.Stinson, A survey of Room squares and related designs, Research Report 89-10, Univ. of Vermont (1989)

[ lo] J.W.DiPaola, H.Gropp, Hyperbolic graphs from hyperbolic planes, Congr. Nume- rantiuni 68 (1989) 23-44

[ll] H.Gropp, On the existence and non-existence of configurations nk, Journal of Com- binatorics, Inforniation and System Sciences 15 (1990), 40-54

[12] H.Gropp, The history of Steiner systems S(2,3,13), Mitteilungen Math. Ges. Ham- burg 12 (1991), 849-861

[13] H.Gropp, Non-symmetric configurations with natural index, Discrete Math. ( to appear )

[14] H.Gropp, The construction of all configurations (124, 163), Conference Prachatice (CSFR), June 1990

[15] H.Gropp, On the history of configurations, Internat. Symp. on Structures in Math. Theories, San Sebastian, September 1990 ( ed. A.Diez, J.Echeverria, A.Ibarra ), Universidad del Pais Vasco - EusM Herriko Unibertsitatea, Bilbao (1990), 263-268

[16] H.Gropp, Configurations and graphs, Discrete Math. ( to appear )

[17] R.Mathon, A new elliptic semiplane S(135,12,3) ( talk a t the Brit. Comb. Confe- rence, London 1987 )

[18] R.A.Mathon, K.T.Phelps, A-Rosa, Small Steiner triple systems and their properties, Ars Combinatoria 15 (1983) 3-110

[19] J.Novbk, Maxidlnisystkmy trojic z 12 prvku, Mathematics ( Geometry and Graph Theory ), Univ. Karlova, Praha (1970), 105-110

[20] M.O’Keefe, P.K.Wong, The smallest graph of girth 6 and valency 7, Journal of Graph theory 5 (1981) 79-85

[21] A.Rosa, D.R.Stinson, One-factorizations of regular graphs and Howell designs of small order, Utilitas Math. 29 (1986), 99-124

[22] P.J.Schellenberg, A computer construction for balanced orthogonal matrices, Proc. 6th S.E. Conf. Combinatorics, Graph Theory and Computing (Boca Raton, 1975), 513-522

[23] M.J.Lipman, The existence of small tactical configurations, Graphs and Combi- natorics, ed. R.A.Bari, F.Harary, Lecture Notes in Math. 406, Springer-Verlag, Berlin-Heidelberg- New York (1974), 319-324

[24] J.Q.Longyear, Tactical configurations: an introduction, Graphs and Combinato- rics, ed. R.A.Bari, F.Harary, Lecture Notes in Math. 406, Springer-Verlag, Berlin- Heidelberg- New York (1974), 325-329

Combinatorics '90 A. Barlotti el al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. 241

CHARACTERIZING LINEAR SPACES BY BLOCKING SETS

Erika HAHN

Fachbereich Mathematik, Universitaet Giessen Arndtstr. 2, D-6300 Giessen, Germany

ABSTRACT. Let R = { I i ; ,Kz , ...} be a set of configurations. A linear space L is called R-blocking, if any substructure of L, which is isomorphic to one of the configurations of 12, contains a blocking set. Our first result says that an R-blocking linear space is [0,2]-semiaffine, if R is the set of any three confluent lines. We shall also prove that a linear space, where any triangle contains a blocking set, is [0,2]- semiaffine (apart from a few exceptions). Considering the set R of all three lines L,G, H , where L n G = 0 and L n H # 0 # G f~ H , it turns out that such an R-blocking linear space is either a punctured projective plane, an affine plane or an affine plane with an additional point.

1. INTRODUCTION

A finite linear space is an incidence structure of points and lines, such that (i) any two distinct points p and q are contained in a unique line pq, (ii) every line contains at least two points, there exist at least two lines, (iii) the number v of points and the number b of lines is finite. If three lines of L mutually intersect in a common point, then they are called confluent. If they mutually intersect, but not in a common point, then their union forms a triangle. Two lines of L are called parallel, if they coincide or have no point in common. A parallel class Il of L is a set of lines such that each point is contained in a unique line of n. The degree rp of a point p is the number of lines through p , dually the degree k~ of a line L is the number of points on L. We also call p an r,-point and L a kL-line. The number of parallels through p to L is denoted by a ( p , 15) := rp - kL 2 0 , where ( p , L ) is any non-incident point-line pair. Let j be a non-negative integer. A linear space is called [O,j ] -semiafine, if it is a(p ,L) E {0,1, ...,j} for all non-incident point-line pairs ( p , L ) of L. A blocking set 13 of L is a set of points that meets every line in a t least one but not in all points. In order to characterize global structures like linear spaces by local properties like the existence of certain blocking sets we shall introduce the following condition. A configuration is an incidence structure of points and lines such that any two distinct points are contained in a t most one line. Let R = { K l , Kz, ...} be a set of configurations. Then a linear space L is called R-blocking, if any substructure of L, which is isomorphic to one of the configurations of R, contains a blocking set.

242 E. Hahn

If R is the set of configurations of any i 2 3 lines, then the corresponding R-blocking linear space is called i-blocking. i-blocking linear spaces already have been classified in 1989 ([5]).

2. CLASSIFICATION OF R-BLOCKING LINEAR SPACES

In this paper we shall characterize R-blocking linear spaces L, where R is one of the following sets: (1) the set of all configurations of three confluent lines, (2) the set of all triangles of L, (3) R = { L , G , H I L n G = 0 and L n H # 0 # G n H } .

Definition 1 Let L be a linear space and let i 2 3 be an integer. L is called i-star- blocking, if any configuration of i confluent lines of L contains a blocking set.

Theorem 1 Let L be an R-blocking linear space, whe*re R is the set of any i confluent lines o f t . Then L is [ O , i - 11-semia8ne.

Proof Assume by way of contradiction that there exists a non-incident point-line pair ( p , I.), such that p coincides with i lines parallel to L . Then these i lines through p contain a blocking set, which is not met b y L , a contradiction.

Corollary 1 A 3-star-blocking linear space is one of the following structures (of order k 2 3): (a) a projective or a punctured projective plane, (b) an afine plane or an aBne plane with an additional point, (c) an afine plane, where one point or one line is deleted, (d) a projective plane, where one of the following sets is removed: (d.1) s points (2 5 s 5 k - l), no three of which are collinear, (d.2) a line L but one point p on L together with one or two points not on L but collinear with p, (d.3) two lines L , G exclusively either their point of intersection x, or a point p # x on L , or two points p E L, q E G, p,q # x. (See Hauptmann [6], Kuiper-Dembowski [3], Lo Re, Olanda [7], and Oehler [9].)

Definition 2 Let R be the set of all triangles of a linear space C. If L is R-blocking, then L is called A-blocking.

Example 1 A projective plane of order k 2 4, where three collinear points x, y , z are removed, is not A-blocking: Suppose that it is. Let X , Y , Z be three lines of a triangle such that x is removed from X , y from Y and z from Z . Hence X u Y U Z contains a blocking set. If it is k 2 4, we can choose three non-blocking points u E X, v E Y and w E Z fulfilling the following conditions:

Characterizing linear spaces by blocking sets 243

(1) u, v, w are non-cdinear, (2) u @ YU Z , v @ XU Z and w @ X U Y (if it was { u } = X n Y , then the line u z would not have a blocking point), (3) uv, vw and uw are lines of degree k + 1. Then any line but L contains a blocking point, a contradiction.

First we shall deal with A-blocking linear spaces with constant line degree. Note that a linear space where any line has degree 2 is either K3 or the affine plane of order 2. Since both of these structures do not admit blocking sets, we shall furthermore suppose that the minimum line degree is 3.

Lemma 1 Let C be a 2 - ( v , k , 1) design. If any triangle of C contains a blocking set, then C is a projective plane or an afine plane.

Proof Step 1. C is [O,2]-semiafine: Assume t o the contrary that there exists a non-incident point-line pair (q , G ) such that q lies on three lines H1, H 2 , H3 parallel to G. Since we have constant point degrees, it is n(p, L ) 2 3 for all pairs ( p , L ) with p @ L . Let 2 1 , ..., z k - 1 , ~ be the points on H I . Then each of these points z; (a = 1 , .., k - 1) coincides with two lines G l ( x , ) , Gz(x; ) which are parallel to G . Moreover, G l (x ; ) , G ~ ( x ; ) are also disjoint from H2 (assume that G l ( x ; ) meets H2 in a point s. The triangle consisting of the lines Gl (x i ) , HI , Hz contains a blocking set that is missed b y G , a contradiction). Arguing in the same way we obtain

We proceed in two steps.

G , ( z , )nGl ( s j )= l f o r a l l s , l E { l , 2 } , i , j E { I , . . , k - 1 } , w h e r e i # j .

Now choose two points y E Gl(z1) and y‘ E Gz(x1). Then the lines Gl (z l ) , G2(x1), yy’ form a triangle that contains a blocking set. Since G l ( x l ) and Gz(x1) are disjoint from all other lines G,(x;) (s = I, 2 and i = 2, ..k - I), it follows that yy’ meets and blocks all lines G, (z i ) and G. Consequently, yy’ has at least 2k - 1 points, a contradiction. Step 2. There is no 2-semiafine 2 - (w, k , 1 ) design with k 2 3, which is A-blocking: W e assume the contrary. Since the point degree r is k +- 2 we have v = r ( k - 1 ) + 1 = k2 +- k - 1. From v r = bL follows that k divides ( k +- 2)( k2 +- k - l ) , thus k divides 2 and k = 2. All in all we have proved that L is [0 , 11-semiafine, hence it is a projective or an afine plane.

Next we consider A-blocking linear spaces with non-constant line degree.

Theorem 2 Let C be an R-blocking linear space, where R is the set of all triangles of C . If any line contains at least k 2 7 points, then C is either (a) one of the [0,2]-semiafine structures described in corollary 1, or (b) a [0,3]-semia$ne plane, where k: 5 kr, 5 k + 2 and k +- 1 5 rp 5 k +- 3 f o ~ all lines L and all points p and k2 +- k - 1 5 v 1. k2 + 3k +- 2, or (c) a [0,4]-semiafine plane with k 5 kL 5 k + 2 and k 5 rp 5 k -+ 4 for all points p and all lines L , and k2 + k - 1 5 v 5 k2 + 3k + 2. If k = 6 , then L is a [0,5]-semiafine plane, where any line has 6,7 or 8 points, any point lies on at least 6 and at most 11 lines, and 41 5 v 5 55.

244 E. Hahn

Remark in 2 b) and c).

There exists only afinite number of [0,3]- or [0,4]-semiafine planes described

Corollary 2 If k 1 616, then in the cases (b) and (c) C is embeddable in a projective plane of order k -+ 3 (cf. theorem 2.6 in [,I).

Proof In several steps we examine the parameters of C. We suppose that it is k 2 6. Step 1. It is IkL - kG1 5 1, i f L , G are two parallel lines of l, and lkL - kGl 5 2, i f L and G intersect: Let first L and G be disjoint, and let p be a point outside of L U G. Considering the lines connecting p with the points of L it is obvious that at most one of these connecting lines is parallel to G (if two such lines L1, Lz were parallel to G , then G would miss the blocking set contained in L U L1 U L2). Therefore, L meets at least kG - 1 lines through p and we have k L 2 kG - 1. Exchanging the roles of L and G we can suppose that kG 2 k L - 1, and together we have IkL - kGl 5 1 . Now let L intersect G . Intending to show that lkL - kGl 5 2 we can suppose that G is the shortest line of degree k and L is the longest line of degree k t s, s 2 0. Assume that s 1 3. Ifp1 is a point outside of both L and G , then pl lies on a line H parallel to G . Since we have lkG - k H l 5 1 and kG 2 k , it is k H 2 k and H meets L . Denote by p l , ...,pk-l some points on H - L. Then their degree is at least k + s and each of them lies on s - 1 lines Pl(i), ..., Ps-l(i) (i = 1 , .., k - 1) disjoint from G and diflerent from H . Moreover, it is

Ps(i) n q ( j ) = 0 for all s, 1 E ( 1 , ..,s - l}, i , j E (1, .., k - 1 ) with i # j .

(Assume that S ( l ) meets Pl(2) in a point x. Then Pl(1) U P I ( 2 ) u H contains a blocking set which is disjoint from G, a contradiction.) Now consider two points y1 E Pl(1) - ( L U H ) and y z E PZ(1) - ( L U H ) . Again the triangle consisting of the lines P1(1), Pz(1) and ylyz contains a blocking set. Since both Pl(1) andPz(1) m i s s t h e l i n e s P 3 ( i ) f o r a l l j = 1 , . . , s - 1 a n d i = 2 , . . , k - 1 , itfollows that y1y2 meets and blocks all lines Pj(i) and G. Since y l y z might intersect H - L it follows that y l y z has at least (s - l ) ( k - 3) t 4 points. In view of k~ = k t s we obtain (S - 1)(k - 3) t 4 5 k t s, thus sk - 2k 5 4s - 7. Together with s 2 3 this yields k 5 5, a contradiction to k 2 6 . From now on let any line of C have degree k , k t 1 or k t 2. Step 2. Let C satisfy the following condition (*): (*) If G is the shortest line of degree k 2 6 , then any point outside of G lies on at least k + 3 lines. We shall show that C is not A-blocking: Let H be a line parallel to G which contains k points X I , ..., x k of degree a t least k t 3. Then each point xi, i = 1, ..., k , coincides with two lines H1(i) , Hz(i) # H disjoint from G. Choose two points x E Hl(1) - H and y E HZ(1) - H . Then xyU HI(1) U Hz(l) contains a blocking set. This implies that xy meets and blocks all the lines HI ( i ) , H2(i) and G for i = 2, ..., k. Considering that xy might intersect H , we conclude that kz, 2 2(k - 1) i- 2. In view of kz, 5 k t 2 it follows that k 5 2, a contradiction to k 2 6 .


Step 3. If L does not fulf l l condition (*), then L is one of the structures described in

Not satisfying (*) implies that there exists a point s outside of G whose degree is at most k + 2. Counting the number of points of L in view of s we obtain ( k + 2 ) ( k - 1 ) + 1 5 v 5 k + ( k + l ) ( k + l ) + l = k 2 + 3 k + 2 . L e t p be anypoint, then

(41 (b) and ( 4 :

v - 1 k 2 + 3 k + 1 5 r <-- - = k + 4 + - ' - k - l k - 1 k - 1'

If k = 6, we get rp 5 k + 5 and C is [0,5]-semiafine with 41 5 v 5 55. Now let k 2 7 , then we obtain r p < k+4. If k+2 is the maximum point degree, then L is [0,2]-semiafine. Assume that there exists a point of degree k + i and a point of degree k + i + 3. Since rp 5 k + 4 for all p , necessarily i 5 1. On the other hand we have ( k + i + 3)(k - 1) + 1 5 v 5 ( k + i ) ( k + 1 ) + 1. This yields k - 3 5 2i, and with k 2 7 it follows that i 2 2, a contradiction. Together we showed that it is Irp-rqI 5 2 for all points p , q of L. If the point degrees are between k + 1 and k + 3, then L is [0,3]-semiafine, whereas it is [0,4]-semiafine, if the maximum point degree is k + 4. Finally, applying Ostrom's lemma ([lo]) it follows that L is a plane, and the whole proof is complete.

We shall examine another set R of configurations of three lines.

Definition 3 Let C be an R-blocking linear space, where R := { L , G , H E LIL n G = 0 , L n H # 0 # G n H } . Then L is called u-blocking.

Obviously this definition does not lead to a characterization of projective planes, in which such a configuration of three lines L , G , H does not occur.

Now we study u-blocking linear spaces with constant line degree.

Lemma 2 Let L be u 2 - (v, k , 1) design. If L is u-blocking, then L is a projective plane or an afine plane.

Proof Step 1. Assume to the contrary that L is not [0, 11-semiufine. Since the point degree r is constant, this yields T 2 k + 2 and I I (p , L ) 2 2 for any non-incident point- line pair ( p , L ) . Moreover, it is II(p,L) < k - 3. (Denote b y G,G',H three lines with G n G' = 0 and G n H # 0 # G' n H . Let B C G U G' U H be a blocking set and let x E G , y E G', z E H be three points outside of B. Then it is q := G' n H E t?, because otherwise the point q of degree at least k + 2 lies on a further line GI' # G' parallel to G with G " n B = 0, a contradiction. This implies that u := y x n H E B and u' := yznG E B . Any line through y parallel to G contains a point of B n H . Hence these lines miss u,z or H n G. Therefore it is I I ( y , G ) 5 k - 3 and r - k 5 k - 3. ) We conclude that k + 2 5 r 5 2k - 3. Let r = k + j , where j E (2 , .., k - 3 ) . Then we have v = ( k + j ) ( k - 1) + 1 and b = Step 2. The maximum number of disjoint lines is two: Suppose there exist three mutually parallel lines X , Y, 2. Any other line G intersects exactly one or all of these lines (if G only met X , Y but not Z, then Z would miss the

= i ( k $ j ) ( k z + j k - k - j + 1 ) .

246 E. Hahn

blocking set in X U Y U G). Some point x on X has at least degree k + 2, thus it lies on a line Y' parallel to Y . Then Y' is also parallel to Z . Through a point y E Y' - X passes a line X' disjoint from X . Since X U X' U Y' contains a blocking set, X' intersects and blocks the lines Y, 2, a contradiction. Step 3. Since it is r > k there exist two parallel lines G and H . The number of lines that intersect G is k ( r - 1 ) + 1 = k2 + k ( j - 1 ) + 1 . These are exactly the k2 lines connecting the points of G with the points of H and the k ( j - 1 ) lines parallel to H . Together with the k ( j - 1 ) lines parallel to G we obtain b = k2 -t 2 + 2k( j - l ) , since due to step 2 there are no further lines. Comparing this with the parameter b from step 1 we ge t ( k - l ) j z - (2k - 1 ) j + k2 - k = 0. The fact that this equation has no solution for k 2 3 shows that OUT linear space is [0, 11-semiafine.

Finally we classify u-blocking linear spaces with non-constant line degree.

Theorem 3 Any u-blocking linear space C is either an afine plane, an afine plane with an additional point, or a punctured projective plane.

Proof First we study the case that there exist at least three mutually disjoint lines. From now on denote b y t the maximum number such that there are t mutually disjoint lines GI, ..., Gt. We proceed in several steps. Step 1. GI, .., Gt have the same degree k: The lines connecting a point p E GI with the points of Gz intersect each of the lines G3, ..,Gt (suppose that p g , q E Gz, is parallel to Gs. Then GI U G2 U pq contains a blocking set missed b y Gs, a contradiction). Consequently we have kGz 5 kG3,..,kG,. Exchanging the lines we obtain kG, = ... = kG, =: k . These arguments imply that Step 2. Any line other than GI,..,Gt meets exactly one or each of the lines GI,..,Gt. (If a line G # GI, ..., Gt met say GI and G2 and was disjoint from GS, ..., Gt , then GJ, ..., Gt would not coincide with a point of the blocking set contained in G U GI U Gz, a contradiction.) Step 3. Let C satisfy the following condition: (**) For i , j E { l , . . , t } , i # j there exists a point x E Gi, such that x lies on a line G # Gi parallel to G,. Then C is not u-blocking: Let y E G - x be a point. Then it is ry 2 k + 1 and y is incident with a line G' parallel to Gi. Obviously G U G' U G; contains a blocking set and therefore G' meets and blocks all lines G,, s = 1 , ..., t , s # i. This is a contradiction to step 2. Step 4 If C does not satisfy condition (**) of step 3, then C is [0, 11-semiafine: The fact that any line other than GI, .., Gt meets each of the lines GI, .., Gt implies that - rp = k t 1 if p E GI, .., Gt, rp = k i f p 4 GI, .., Gt, - any line has at most k + 1 points, and

Case 1: If any line other than GI, .., Gt has degree k t 1, then L is a punctured projective plane of order k: This is obvious, if there exists no point outside of GI U ... U Gt. But i f there are points

- b = t + k2 .


outside of G I U ... U Gi,then they all have degree k and therefore lie on a common line H # G1, ..,Gt of degvee k + 1 =: t + s , s 2 1. Since any line ti' other than H , GI , .., Gt contains at most one of the s points of H - (G1 U .. U G t ) it follows that kH = t + s = k + 1 = kHt 5 t + 1, hence s = 1. Let G be a line through H n GI and H ' n G2 . Thus t = kG = k + 1 = kH = t + 1, a contradiction. Case 2: If there exists a line X # GI, .., Gi of degree h 5 k , then C is an afine plane or an afine plane with an additional point: First we shall show that all lines disjoint from X lie in a parallel class of k lines: Any point of degree k + 1 coincides with a line parallel to X . Suppose that X and Y are parallel and that them exists no line disjoint from both X and Y . Hence b is the number of lines that intersect X or Y :

b = hky + t ( k - k y ) + t ( k - h ) -t ( h - t ) ( k - 1 - k y ) + ( ky - t ) ( k - 1 - h ) + 2.

Inview o f b = t + k 2 and t 5 h < k weobtain k 2 - k ( k y + h ) + h k y + k y + h - 2 = t 5 h. It follows that k < 1/2(ky + h + ky - h ) and k < ky . This implies ky = k + 1, and there is no k-point outside of Y. Furthermore we get h = t . We conclude that

h 2 - k(k + 1 + t ) + t ( k + 1) + k + 1 + t - 2 = t

and t = 1. This contradiction to t > 2 shows that there exists a line Z disjoint from both X and Y . By step 1 the lines X, Y and Z have the same degree h. Now consider a line X ' parallel to X which passes through a point x E G1 - ( X U Y U 2). By step 2, X' meets at most one of the lines Y , Z . Let p = X' n 2. x also lies on a line Z' parallel to 2, and X'U Z u 2' contains a blocking set. Consequently 2' meets the lines X , Y , a contradiction to step 2. Hence there exists a parallel class of k lines XI, ..,Xk, all of degree h (refer to step 1). Finally let h = t + s , where s 2 0 is the number of points of degree k on X I , . . ,Xk outside ofG1,. . ,Gt. Thenwehavet+s= k . (Supposek>t+s. L e t x = G l n X l a n d y = G 2 n X 2 . Since X1 U X2 U xy contains a blocking set, xy blocks all other lines X3, .., xk. But x y misses exactly k - ( 1 + s ) 2 1 of these lines, a contradiction.) In case that there is no point outside of G I u .. U Gt , it follows that h = t = k , and b y definition C is an afine plane of order k . In case that there exists exactly one point x of degree k at is x 6 X I , . . , X k , and we have h = t = k . Obviously, C is an afine plane with an additional point x . Suppose there exist at least two points x , y of degree k . If s = 0 , then the line xy (meeting G1, .., Gt) has at least t + 2 = k + 2 points. This contradicts kL 5 k + 1 for all lines L of C . If s 2 1, then a point of degree k lies on the line Xi parallel to X j for i # j and i, j E { 1, ..., I c } . This is a contradiction.

This finishes our examination of (at least) three mutually disjoint lines. Linear spaces withoul three mutually disjoint lines have been classified by Metsch [8]. Among these only the projective planes are u-blocking. Summarizing, any u-blocking linear space is [0, 11-semiafine, and the proof is complete.

Acknowledgement We are grateful to the referee for improving the quality of the paper.

248 E. Hahn

References

[l] A. Beutelspacher, Einjuehrung in die endliche Geometrie Z,U, Bibliographisches In-

[2] A. Beutelspacher and K. Metsch, Embedding finite linear spaces in projective planes,

[3] P. Dembowski, Semiaffine Ebenen, Arch.Math. 13 (1962), 120-131. [4] F. Eugeni and E. Mayer, Blocking sets of index two, Ann.Discr.Math. 37 (1988),

(51 E. Hahn, Blocking sets in linear spaces, Math.Sem.Giessen 201 (1991), 73-81. [6] W. Hauptmann, Endliche [0,2]-Ebenen, Georn.Dedicata 9 (1980), 77-86. [7] M. Lo Re and D. Olanda, On [0,2]-serniaffine planes, Simon Stevin 60 (1986), 157-182. [8] K. Metsch, Classification of linear spaces without three mutually parallel lines, Journal

[9] M. Oehler, Endliche biafhe Inzidenzebenen, Geom.Dedicata 4 (1975), 419-436. [lo] P. De Witte, Cornbinatorial properties of finite linear spaces 11, Bull. SOC. Math.

stitut, Mannheim/Wien/Zuerich, 1982.

11, Discrete Math. 66 (1987), 219-230.

169- 176.

of Geometry 37 (1990), 128-141.

Belg. 27 (1975), 115-155.


Partial Geometries of Rank n

Daniel R.. Hughes

School of Mathematical Sciences, Queen Mary and Westfield College, Loiidon E l 4NS, U.K.

1. Introduction and background

In the theory of Buelcenhout diagrams and their geometries, some partial geometries arise very naturally: the circle geometries, generalised quadrangles, linear spaces and their duals were quickly recognised as being important. (In fact, most of the “important” rank 2 geomet,ries a,re partial geometries.) Extensions (which froin a diagram point of view involve adjoining a circle geometry to some other geometry) of projective geometries and of polar spaces were studied because of their connections with the sporadic simple groups, and for polar spaces in particular were perhaps first investigated as such by Buekenhout in (51 and by Buekenhout and Huhaut in [7]. But extensions in a more general set.ting have been around for a long time, for example inversive planes, or the geometries associated with the Mathieu groups; see also [28]. A natural generalisa- tion of this was to consider extensions of partial geometries, where an interesting theory has developed. Another step takes us to “partial geometries of rank la”, which includes most of the geometrirs already studied and belonging to some Buekenhout, diagram.

In this paper we describe some of these ideas and indicate some open questions that might be of interest, Init we do not attempt to list everything. The reader will find inore in one of the many recent, papers. Perhaps one of the most important questions still to be settled for many of these geometries concerns either finding their a.utoinorphism groups, or characterising those wit,h, say, a,n automorphisni group transitive on maximal flags. We begin by describing partial geometries of rank 2 , and then in $3 we survey the results about extended partial geometries. In 54 we consider other classes of rank 3 partial geometries. The basic concepts about diagrams and diagram geometries can be found in, c.g., [4].

2. The pGs

A rank 2 structurf,, or geometry, is a set of points and blocks, with some incidencc relation between them, for which we use any convenient intuitive terminology (“point 011

block”, “block on point”, “block contains point”, etc.). If the geometry is semi-linear, meaning that any two points are on at most one common block, then we often call thc blocks lines. In any rase, if P and Q are distinct points lying on some common block, then we say that P arid Q are collinear. The point-graph r = r(S) of siich a structurc is the graph with the points of S as vertices, with two adjacent if they are collinear. We use a lot of graph terminology: thus S is connnected if r(S) is, the distance d(P, (2)

250 D.R. Hughes

between two point,s is the distance between them in r(S), the clianieter of S is the diamet>er of r(S), etx. In this section only, a “geometry” or “structure” has rank ’2 unless otherwise specified (e.g., at the very end of the section), and is finite. Usually our geometries will be coniiected too.

A a-partialgeometry oforder (s, t ) , where a, s, t are positive integers, is a coiinected semi-linear geometry wliose blocks are called lines, such that (i) every point is on t + 1 lines, (ii) every line contains s + 1 points, and (iii) if P is a point not on the line y, then there are exa.ctly cy points on y col1inea.r with P . If S is such a. geomet,ry, then we call 5 a pG,(s, t ) , or merely a pG. Clearly tlie dual of a pGm( s, t ) is a I&,( t , s). 0l)viously we must have

and not, so obvious, but not difficult, is: Theorem (2.1). A p G m ( s , t ) contains ‘o = (s + l)(st/Cy + 1) points and its point graph is strongly regular with parameters ( u , s ( t + l ) , s - 1 + f ( a - 1 ) , ( t + 1)a).

We use the rank two diagram o--o for tlie family of pa.rtia1 geometries ant1 then And of o-..---l;;c=o for the part,icular family of tr-pa.rt,ial geometries. course

7 S

will mean the fa.mily of cy-prtial geoiiietries of order (s, t ) . Certa.in sub-families a1rea.dy have, or will have, special diagrams of their own.

There are some important existence and noii-existence theorems about, pGs, for which we refer the reader to e.g., [2,15,lG] especially; see also [13,34]. The partial geometries can be divided into several types, which we give below, togcther with thcir diagrams.

(1) Linear spares, or 2-designs wit,li X = 1; these a.re exa.ctly tlie pGs wit,li cy = s + 1. The 2-design has paramet,ers 2-(s( t + 1) + 1, s + 1, I) , with diagram:

L - - 0 Y=-s+l-? S t

Perhaps the most, interesting examples are projective and f i n e geometries (including in pa.rticular projective and affine phnes), and tlie circle geometries, which are the trivial 2-designs for ( u , 2 , l ) . The circle geometry has its own diagram:

Furthermore, projcctivr planes, which are the intersection of type (1) and (2) , have a standard diagram:

= - r s + 1- S S S

and &ne planes, the intersection of (1) and (3), are often represented by:

- - A f S S - S S s- 1 S

Partial geometries of rank n 25 1

(2) Dual linear spaces, or dual %designs with X = 1; these arc exactly the pGs with N = t + 1. A general diagram is:

There are some especially important examples (see [21,34]): (a) The dual of the point-line structure of a projective geometry PG(n,q) , with

(b) The dual of the point-line structure of a n f i n e geometry AG(n,q) , with .'i =

(c) The dual of a, unital. Here J = q2 - 1, t = q. (d) Thc dual of the circle geometry. The dual, whose order is (s, l), is represented

by the natural dual of the circle geometry diagram:

s = q(qn-1 - l ) / ( q - l ) , t = y.

q ( f - 1 - l ) / ( q - l), t = q - 1.

13 - - 2-

0 1 S I S

(3) Netas: tliese art' exactly the pGs with N = t . (See [3,21,34].) A p G t ( s , t ) is a net, and has type (.Y + 1, t + 1). The diagram for a net of type (.s + l , t + 1) is:

Of particular interest is a doubly affine plane of order q , which is an a,ffiiie plane (of orderq) withoneparallelclassremoved, andis a l soadualne t . It i s a p G q - l ( q - l , q - l ) .

Dual nets have a simple structure, and are sometimes called 2- transversa.1 designs. The dual nets have diagrams:

(4) D i d nets; these are exactly the pGs wit,h N = .s.

- I.1 - 0 Y s ? S t

? for a dual net.) Of special interest is a. ciual

( 5 ) Generalised qua.clrangles, or GQs; these are exactly the pGs with N = 1. (See

N" (But some people prefer affine plane, which is a pGq(q, q - 1).

[38].) The standard diagram for a G Q ( s , t ) is:

= - T I - ? S t

The lcnown GQs all come in certain sub-types. (a) The first set of t,hese have cla.ssical pa.rameters, where q is a prime-power: ( i ) (r l lq); Gv) ( q 2 , q 3 ) ; (ii) (Q, q 2 ) ; ( v ) (9"q2) .

( q 2 , q ) ; (13) The second set have semi-classical parameters, where again q is a prime-power: (i) ( 4 - l , q + 1); (c) Finally, where s and t are arbitra.ry positive integers, we have: ( i ) (s, 1) (the grids); (ii) (1 , t ) (the dualgrids). (6) Proper pGs: these are the pGs with 1 < N < min{s, t } . We sha!! not say anytliing

Now we can ma.ke a very general definition:

(ii) ( q + l , q - 1).

about these.

252 D.R. Hughes

Definition (2.2). A (residually connected) geometry S of rank 11 is a partial geometry of rank n if S is a geometry for tlie rank n diagram

- Q ... -d - with n nodes. (Note that we cannot speak of a constant, “a” or of an “order” ( s , f ) if 71. > 2, except perhaps in some special ways which must be defined.)

The nodes will represent, from left to right, points, lines, planes, . . . as usual, and wc say that two points are collinear if they lie 011 a coininon line. The graph r = r(S) is the collinearity graph of the points. The diameter of S is the diameter of r.

In the remaining sections, we discuss some special cases.

3. The EpGs

The EpGs, extended partial geometries, are partial geometries of rank 3, with diagram -

d * ’ Q

In the literature EpGs are often discussed as if they are of rank 2: this is because the lines, on the middle node, are in fact nothing but pairs of points which lie in a common plane; the planes are often called blocks, a terminology we use in this section. The EpGs have been extensively studied, and in this section we give an outline of what is known. An EpG,(s, t ) is a geometry with diagram

Chamber transitivity in a rank 3 structure (which is what an EpG “really” is) means t,ra.xisitivity on “2-flags” when we consider the structure a.s r7 rank 2 geometry: a 2-flag is a set consisting of a lilock and two points (ordered) on that block. Let us call that strong Aag transitivity, as opposed to Aag transitivity, which ineans transitivity on (simplc) flags. The determina.tion of EpGs with strongly flag transitive automorphisni groups is a long way from finished, although a lot is known in some special cases; see for instance Theorem (3.14) below. In what follows we sometimes comment on strong flag tmnsitivity.

In an EpG a very important role is played by the antiflags, and also by the residue intersect ions: Definition (3.1). Suppose S is an EpG,(s, t ) .

(a) For any point P aiid any block y in S, with P not on y, we define p(P, y) to be tlie number of points on y which a,re collinear with P; p,(P) a id p*(P) are, respcctivcly, the iniiiiinum a.nd tlie maximum of all non-zero cp(P, y), as y varies; 9 0 = po(S) and p* = y * ( S ) a.re, respect,ively, tlie minimum and the inasiiiiuin of all p o ( P ) and y * ( P ) .

(b) S is 1ocall.y y-uniform a.t P if po(P) = p*(P) = 9, a.nd is p-uniform if it is locally 9-uniform at all its points.

(c) S is 1ocalJ.y triangular at P if whenever two points A a.ncl B in Sp are collinear in S, then they are collinear in Sp. S is tria.ngular if it is locally triaiigular a t all its points.

Partial geometries of rank n 253

(d) If P and Q are points a t distance two in S, then we define S p Q to be tlie set of

Then we have a number of important results:

(a) For all P and y, Q . p(P, y) = 0 (mod3). (b) For all P and y, either cp(P,y) = 0, or cp(P,y) 2 a + 1. (c) S is locally triangular a t P if and only if vo(P) = v * ( P ) = a + 1, and so S i -

triangular if and only if ( P O = p* = cy + 1. Theorem (3.3). The diameter d of the EpG,(s,t) satisfies d 5 niax{3,s + 5 - 2 ~ ” ) . where 90 = vo(S) (2 a+l). So there are only finitely many EpGs with fixed para.metcrs a , s, t .

The large diamet,ers have been extensively studied, and the bound in Theorem (3 .3) greatly improved in special ca.ses; see [16,17] for more deta.ils.

Triangular EpGs are the EpGs with minimal p; a t the opposite extreme, we have: Definition (3.4). The EpG,(s,t) S is called a one-point extension if any two points are collinear.

And immediately (the last sentence of the next theorem conies from computing tlic number of blocks in S): Theorem (3.5). The EpG,(s, t ) is a one-point extension i f , for some point P , we liavc~ po(P) = s +2, and only if this is true for every point P. Furthermore, the EpG,(s, t ) is a. one-point extensioii if and only if it is a 2-design for (1 + (s + l)(st/cr + l), s + 2, t + 1). So if S is a one-point, extension, then s + 2 divides 2t(t + 1)(2t - a).

This last result has an analogue for triangular extensions, also from counting tlic number of blocks: Theorem (3.6). Suppose S is an EpG,(s,t), locally triangular at one point. Then Q’(Q + 1) divides s t ( s + l ) ( t + 1 ) ( s t + a ) .

A lot is known about the two extreme cases: triangu1a.r and one-point extensions. We shall say a little about this below. Furthermore, bounds on the residue intersections are known (Theorem (3 .7)) and have been investigated (see [7,19,29,24]); these minimal cases are perhaps close to classifica,tion. Theorem (3.7). If P and Q are two points at distance two in tlie EpG,(s,t) S , if cpo = po( P, Q) is the minimum of

points in S p n SQ, and p(P, Q) to be the number of points in S ~ Q .

Theorem (3.2). Let Sp be an EpG,(s, t ) .

P ) a.nd PO( Q ) and

= P o P + t ( P o - l ) / ~ ) , P 2 = ( s + l ) ( p o ( t + l ) - s - t + a - l ) / a ,

then p(P,Q) 2 max(i31,P~). We review the situation for EpGs, according to the type of the pG. Type (1). The geometries for 0 C ” n L 0 are merely 3-designs

with X = 1, and it seems hopeless to aim for a classification of such geometries. They will include the Ma.thieu design for 3-(22,6,1), a.11 affine geometries AG(n, 2) , all inversive planes, all Steiner ciua.druple systems and ma.ny more. We have of course some non- existence theorems, mostly derived from count.ing the number of blocks in the extension and noting that it must be integral.

Type (2). The geometries for 0 c J 0 are also difficult to classify in the general case. Here we will find all semibiplanes: the geometries for thc diagram 0 C 3 0 . I f o J represents a “good” dual linear space such as the dual to the point-line structure of a projective or affine geometry,

n “

254 D.R. Hughes

then the situation is ~nucli iinprovetl. Tlicw aicl exarnples a n d noii-exihtrnce tlicwcms kiiown ([31]): Theorem (3.8). (a) If S is i~ geoiiietiy foi

locally triangular at P , then Sp is the dual of a projective gcoinctry PG(n, 2).

and S p is the dual of a PG(n, q ) , then q = 2 or 4. For q = 2 and each 17, there cxists at least one example of such a. geometry; for y = 4 an example is kiiown only for 71 = 2. Tlie exaniples for q = 2 will be strongly flag transitive if the autoinorphisin group of the appropriate Grassinaiiri variety is transitive on the flags (consisting of an iiicidciit point and maxiinal sulxpace pair) on that, variety. The exa.mple for q = 4 lias automorpliisin group M 2 . 2 and is strongly flag transitive.

v 0 , locally triaxi- gular at P and S p is the chal of a.n affine geoniet,ry AG(n , q ) , t h i q is even. For 71 = 2 there is a n example for every q which is a power of 2, a,nd t,liere is ail exairiple for (I = 2 tuid for every 77 > 2. (See also Exa.mple (3.12) for the particu1a.r ca.se n = 2, q = 4 . )

is much more complica.ted, and ever not well understood. There a.re lots of theorems “limiting” the possibilities, and some exa.inples. The following result is from [28]; iri t ha.t paper sollie iriore iion-existence tlieorems a.nd examples arc given. Theorem (3.10). Let S 1 x 1 n geoiiietry for

(b) If S is a geometry for c n A 0 , locally triangular at I‘

n Theorem (3.9). If S is a geometry for 0 C J

Q p e (3). The situation for geometries lielonging t,o C v r\ N 0 0

0 C N U 0 1 S t

(a) If S is a one-p in t extension, then S is an inversive plane and t = s + I , or S is a ’-design for ( $ 2 + 3s + 2, s + 2, (s + 2)/2) with t = s / 2 - 1. Examples of each cxist for every s + 1 equal t o a prime-power, and both with and wit,liout, strongly flag transitive groups.

(13) Triangii1a.r examples exist which a.rc 3-transversal dcsigns, for every s + I equal t o a prime-power and with ( i ) t = s and (ii) t = s - 1.

Type (4) . Here t,liiiigs are much clewer (see [28]). Theorem (3.11). Let S he a geoinctry for

(a) If S is a one-point extension, tlieii S is cithcr the 2-design for (21,6,4) which is the external restriction of the Mat.liieu design M22, or the liplane (7,4,2). In lmtli cases t,he geoinet,ries are strongly flag tra.nsitive.

(1.)) If S is triangular, tlieii S is a 3-transversal design, and examples exist (for instance) for every t = q - 1,

(c) If S is not one-point or tria.ngular, then it is one of a rest,rict,etl faiiiily of divisible designs defined in [2S] a.nd for which exaiiiples are given t,here.

There is even a. muitiple extension of one of the geometries in Theorem (3.11)(ii): Example (3.12). (See [28].) Let S2 be the 4-(23,8,4) which is the external restrictioii of the Mathieu 5-design for (24,6,1). If S1 is a residue of S2 then S1 is an exteiisioii of

q a prime-power.


S, tlie 2-(21,G,4) of Tlicoreni (3.11)(a). Hence tlie Matliieu group M23 is a maximal-flag transitivr automorpliisni group of a geometry for

c - c - c - M - c . . C A c - J o 1 1 1 4 3 I 1 1 4 3

” - v

Type (5). The extmded generalised quadrangles (EGQs) c ” 0

1 S t

have a very rich theoiy, and there are many papers dealing existence theoleins are known, but the exaniples mostly have t = 1 (and a le extensions of grids). We give h(,re only an indication of some of the interesting examples, but the non-existence tlieoicnis are mostly for tlie triaiigular case, the one-point case, and

for

with them. A lot of lion-

sometimes for the case t1ia.t t,lie constant p(P. Q ) above takes on one of the iniiiiiiiun values /31 or 132. See [12] for a detailtd description of all tlie examples lciiown at this time, and see [7,11,12,1G,17,19,29,23,24,25,43,4G] for most of the noii-existence theorems, as well as more about t,lie exa.mples. All t,he lcnown exaniples with t > 1 are included in the next t,lieorem (wc also give the EGQs of order (2 , l ) ) . Theorem (3.13). ( a ) For s = 2, all EGQs are known, aiitl are either “ f i n e polar spaces” or quotients of these. They are all either one-point extensions or are triangular. and including t,lie grid (2,1) , they are 10 in number (t,hree for the grid (2,1), four of order (2,2) antl t,hrec of order (2,4)).

(b) There is a,lso at, 1ea.st one EGQ for ( s , t ) = (3 ,3) antl (3 ,9) , a i d at, lea.st. two each for ( s , t ) = (4 ,2 ) aiid (9 ,3) ; all of these are triangu1a.r except one EGG) of order (9,3).

( c ) For each odd 1)rinie-power y 2 5, tliere is a one-point EGQ of order ( q - 1, (I + 1); there a.re also uniforiii examples (not, triangubr in general) of dia.niet,er three of the sa.nie order.

The EGQs with “good” groups have been studied, and with some succes. We have for instance (see espc*cia.lly [12,20,4G]): Theorem (3.14). If S is an EGQ whose residues a.re all “classical” GQs (see [43]), and whose aut,oiiiorphisiii group is strongly flag transitive, then S is one of tlie seven EGQs of Theorem (3.13)(a) with t > 1, or one of the six EGQs of Theorein (3.13)(11).

T l i~ is if we knew t,liat, a G Q whose a.utomorpliisni group is fla.g transitive must be chssical, or a grid or a dual grid, t h i Theorem (3.14) woiild provide us with it coiiipletc classificat,ioii of tlie strongly flag transitive EGQs with t > 1 (the strongly flag tra.iisitive extended grids would still remain as a separate probleiii perliaps). It seems likely that if the hypotheses of Theorem (3.14) were weakened to dema.nd merely flag-transit,ivity, then the conclusioii would be the same. The groups which arise in Theorein (3.14) arc interesting: t,liey include some classica.1 groups in “unexpected” roles, as well as the sporadic groups M c L , Szcz and H S .

Dual grids always liavc unique extensions, which is easy to see, and indeed, they can be ext,entlcd iiidefiriitely (as can the GQs of order (2,l) and (2.2)). Grids have a. very rich structure of extcLiisioiis, about which we say no more.

The proper pGs (when 1 < a < niin{s, t } ) have not, Iieeii stutlied in any great detail, aiid it is lioped t1ia.t soon that sit,iiat,ion will lie remedied. The liest obvious step 1113 from EpGs would seem to lie what might he called “L.pGs”, tliat is, geometries belonging to

c 0 L+ 0 . These have been studied in some iiitercsting special u 0 , and cases like L L and 0 L

L n n

v

256 D.R. Hughes

a.re presumably a lot more difficult to analyse than EpGs.

4. The pG.L geometries

Chmging the diagmin for an EpG slightly, we get

0 - C 0

and we might call such a geometry a pG.c geometry. These geometries do not seem to be as complex as the EpGs, but they certainly contain some interesting examples ant1 difficult problems. In [32] there is the following:

0

The geonietxies for (a ) are called Cz. c geometries (see the nest tlieorem). The geometries belonging to ( b ) are truncations of a thin projective geometry of appropriate rank, or equivalently, their points, lines and planes are the i-sets, ( z + l)-sets and ( i + 2)-sets in an N-set (see [42] in particular). Theoreill (4.2). If S is a C2.c geometry for the diagram

Theorem (4.1). All rank 3 geometries for the diagra.m 0 . . c: C either belong to (a)o C 0 , or to (I ) ) 3 - C I-- v

0 r\ 0 S I U

then the diameter of S is lmunded by '11 + 2, and the number of point,s in S is bouiided by (s + 1)"+'. S attains one of these bouiids if and only if it attains the other, and then if and only if it is a. truiicatiori of the dual of a thin polar space of rank 11 + 2:

C

0 4- A .... v v

S I 1 1 1

But there are ma.ny other C2.c geometries. A semibiplaiie with E; points on a I h A c and 11 points lea.& t,o a C2.c geometry with % J points, s = 1 a.iid 11. = X- - 2. In addition, tliere are inany quotients of the maxiinal geometry of Theorem (4.2). See [32] for I T ~ O ~ C .

well understood than pG.c geometries. Spmgue's result in [42] classifies some of those

or when the two linear spaces are truncated projective geometries; and

The geometries for 0 n L , the pG.L geometries, are less

belonging to 0 _I W n L 0 , such as 0 3 e u C 1

0 L n L 0

for instance, is interesting and includes

0 c f-. L 0

and geometries such as

A L A f 3

0 " 0 and L


which give projective and affine geometries, respectively. But in any case we have: Theorem (4.3). ([32]) Let S be a geometry for the diagram

h L 0 r a t U

and let Q > 1. Then the diameter d of S satisfies d 5 m a x ( 2 , ~ - 20 + 4).

such characterisation for the general case of a pG.L geometry. Unlike the situation for C2.c geometries of maximal diameter, there seems to be no

References

1 A. Blokhuis and A.E. Brouwer, Locally 4-bay-4 grid graphs, J . of Graph Theory, 13,

2 R.C. Bose, Strongly regular graphs, partial geometries and partial1.y balanced designs, Pa.c. J. Math., 13 (1963), 389-419.

3 R.H. Bruck, Finite nets. I. Uniquei~ess and imbedding, Pac. J. Math., 13 (1963):

4 F. Buekenhout, Diagram geometries for sporadic groups, in “Finite Groups Coining of Age”, AMS Series Contemporary Mathematics, 45 (1985), 1-32.

5 F. Buekenhout, Extensions of polar spaces and the doublj7 transitive symplectic groups, Geom. Ded., 6 (1977), 13-21.

6 F. Buekenhout, The basic diagram of a geometry, in “Geometries and Groups”, Lecture Notes 893, Springer (1981), 1-29.

7 F. Buekenhout, arid X. Hubaut, Loca.11.y polar spaces a n d related rank 3 groups, J . Algebra, 45 (1977), 391-434.

8 P.J. Ca,meron, Dual polaa spaces, Geom. Ded., 12 (1982), 75-85. 9 P.J. Cameron, Quasi-symmetric designs possessing a spread, Proceedings of the 1988

Combinatorics Conference in R.avello, (to a.ppear). 10 P.J. Ca,meron, Covers of graphs and EGQs, (to appear). 11 P.J. Cameron and P.H. Fisher, Small extended generalized quaclrangles, Europ. J .

Comb., ( to appear). 12 P.J. Cameron, D.R. Hughes and A. Pssini, Extended generalised quadrangles, Geom.

Ded., 35 (1990), 193-228. 13 P.J. Cameron and J.H. van Lint, “Graphs, codes and designs”, LMS Lecture Note

Series, 43 (1980). 14 F. De Clerck, R.H. Dye and J.A. Thas, An infinite class of‘ partial geometries as-

sociated with the hyperbolic quadric in PG(4n - 1,2), Europ. J. Comb., 1 (19SO),

15 F. De Clerck and J.A. Thas, Partiad geometries in finite projective spaces, Arch.

16 A. Del Fra and D. Ghinelli, A classification of extended generalized quadrangles with

17 A. Del Fra and D. Ghinelli, Large extended generalized quadrangles, Ars Comb.,

18 A. Del Fra and D. Ghinelli, Point sets in partial geometries, in “Advances in Fi- nite Geometries and Designs”, Ed. J.W.P. Hirschfeld, D.R. Hughes and J.A. Thas, Oxford University Press, (1991), 93-110.

NO. 2 (1989), 229-244.

421-457.

323-326.

Math., 30 (1978), 537-540.

maximum diameter, Disc. Maths, ( to appear).

29A (1990), 75-89.

258 D.R. Hughes

19 A. Dcl Fra, D. Gliinelli a.11~1 D.R. Hughes, Extended partial geometries with iiiiriinial

20 .4. Del Fra, D. Ghinelli, T. Meixner aiid A. Pasiiii, Flag transitive extensions of C,,

21 P. Denhowski, “Finik Geometries”, Springer-Verlag, Berlin, 196s. 22 J.C. Fisher, J.W.P. Hirsclifeld and J.A. Thas, Complete arcs iii planes of .scpai.c

23 P.H. Fisher, Extending generalised quadrangles, J. Comb. (A), 50 (1989),

24 P.H. Fisher a.ntl S.A. Hobart, Triang~~lar extended gencralised quadrarigles, Georn.

25 P.H. Fisher and T. Pcnttila, One-point extensions of polar spaces, Europ. J . Comb.,

26 J .W.P. Hirshfeld, “Projective Gcoiiiet,ries over Finite Fields”, Oxford University

27 J.W.P. Hirsclifeld a.nd J.A. Thas, “General Galois Geometries”, ( t o appear). 28 S.A. Hobart, ancl D.R. Hughes, Extended pa,rtia,l geometries: nets and dual nots,

29 S.A. Hobart, and D.R. Huglies, EpGs with minimal 11: 11, Geoi~i. Decl., ( to appear). 30 D.R. Hughes, Extcnsio~is of designs and groups: projective, symplectic and certain

31 D.R. Hughes, Exteiidcd part,iaJ geometries: dual 2-desigiis, Europ. J. Coinli., 11

32 D.R.. Hughcs, On ,some ~ m i k 3 part ia l geometries, in “Advances in Finite Geornetrics and Designs”, Ed. J.W.P. Hirschfeld, D.R. Hughes ancl J.A. Thas, Oxford Uiiivcrsit,y Press, (1991), 195-225.

33 D.R.. Huglies and F.C. Piper, “Projective Planes”, GTM 6, Springer-Verlag, Bwlin, (1972).

34 D.R.. Hughes aiicl F.C. Piper, “Design Theory”, Cambridge University Press, (1985). 35 C.W.H. Lain, L. Thiel and S. Swiercz, The non-existence of finite projective planes

36 V.C. Mavron and M.S. Shrililiande, On designs with intersection nL1Inber.s 0 and 2,

37 -4. Pa.sini, Some remarks on covers and ap r tmen t s , in “Finite Geometries”, Dekker,

3s S.E. Paync and J.A. Tlias, “Finite generalised qua.drangles”, Pit,man, Boston, (1984). 39 S.S. Sane and M.S. Shrikhande, Finite~iess questiom in qna.si-,s.y;vnmietriic desig~is, J.

40 B. Segre, "Lectures on Modern Geomet,ry”, Cremonese, Rome (1961). 41 E.E. Shult,, G~OUJIS, jiola,r spaces and related structures, Math. Centare Tracts, Am-

42 A. Sprague, h’a.~ik 3 incidence structures admitting diial-linr?ar diagram, J . Coiiih.

43 J.A. Tlias, Extensions of finite generalized quadraagles, Synip. Math., 28 (IgSG),

44 0. Velden and J.W. Youiig, “Projective Geometry”, vol. 1, Giiiii aiid Co., Boston

45 P. Wild, 011 semibiplnnes, Pli. D. Tlirsis, University of Lolidon (1980). 46 S. Yoshiara, A c.Cz-geometry associated with the ~ I O L I J I H S , (to appea.’.). 47 F. Zara, Graphs liis aux espaces polaires, Europ. J . Comb., 5 (1984), 255-230.

/ L , Geoni. Dcd., ( t o appear).

geometries, Geom. Dcd., 37 (1991), 253-273.

order, Anii. Discr. Math., 30 (198G), 243-250. Th.

165-171.

Ded., 37 (1991), 339-344.

(t,o appcar).

Press, 1979.

E I I ~ o ~ . J . C O I ~ I . , 11 (1990), 357-372.

nffiiie grotps, Ma.th. z., S9 (1965), 199-205.

(1990), 459-471.

of order 10, Can. J . Math., 41 (19S9), 1117-1123.

Arch. Math., 52 (19S9), 407-412.

NCW York, (1985), 223-250.

Comb. Th. (-4), 42 (1986), 252-258.

stertlain, 57 (1974), 130-161.

Th. (A), 3s (19S5), 254-259.

127-143, Acatleinic Press, London.

(1938).

Cornbinatorics '90 A. Barlotti el al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. 259

The minimal parabolic geometry of the Conway group Col is simply connected

A. A .Ivanov

Institute for System Studies, Academy of Sciences of the USSR, 9 Prospect 60 Let Oktyabrya, 117312, MOSCOW, USSR

Abstract We present an approach for proving simple connectedness of diagram geometries re-

lying on consideration of their simply connected subgeometries. This approach is applied to the minimal 2-local parabolic geometry Q(Col) of the Conway group Col belonging to the diagram notes a triple cover of the classical generalized quadrangle of order (2,2). A crucial role in our proof is playad by a subgeometry in Q(Col) stabilized by the group Coz. This subgeometry is provcd to be simply connected by S.V.Shpectorov.

I . Here the rightmost edge de- N

1. INTRODUCTION

The paper concers an investiga.tion of diagram geometries mainly related to sporadic simple groups. In particular we are interested in geometries belonging to diagrams

P p and h 5-' ' ' -2 h

In the first of the above diagrams the rightmost edge stands for the geometry of the edges and vertices of the Petersen graph; the corresponding geometries are called P-geometries. In the second diagram the rightmost edge denotes a triple cover of the classical generalized quadrangle of order (2,2). In this case w e have T-geometries (T for tilde). A systematic treatement of P- and T-geometries and results on their classification will be presented in a forthcoming monograph [IS3]. One motivation for these geometries is that a number of sporadic simple groups can be realized as their flag-transitive automorphism groups. These groups are M Z L r M23, (302, ,J4, F2 for P-geometries and M24, H e , C O ~ , FI for T- geometries. Recently S.V.Shpectorov and the author [IS51 constructed an infinite family of simply connected flag-transitive T-geometries. The automorphism group of the rank

260 A.A. lvanov

n member of the family has autoinorphism group isomorphic to a nonsplit extension 3"(") . Sp2,(2) where u ( n ) = (2" - 1)(2" - 2 ) / 6 .

The framework for the classification of P- and T-geometries carried out by S.V.Shpectorov and the author is formed by the amalgam method. This method is realized in the following two stages. On the first stage we prove that the amalgam of maximal parabolics arising from a flag-transitive action on a geometry under consideration is isomorphic to the amalgam corresponding to a known example. On the second stage we calculate the universal covers of the known examples.

Nowadays the amalgam method and its modification known as the method of generators and relations are the most powerful tools for classification of flag-transitive geometries in terms of their diagrams. As a very impressive achievement of this method, the classification of flag-transitive extensions of classical generalized quadrangles should be mentioned (cf. [PasB]).

On the second stage of the amalgam method one should usually prove that a concrete geometry has no proper coverings, i.e. that it is simply connected. This problem admits a pure group-theoretical interpretation (cf. Theorem 2.1 in the next section). In particular, if I; is proved to be simply connected, one can obtain a presentation for its flag-transitive automorphism group in terms of elements of parabolic subgroups and relations which hold in parabolics i.e. a geometries presentation (cf. [IvnZ], [Ivn5]). In turn geometric presentations can be used for the uniqueness proof of the corresponding sporadic simple groups. So the second stage of the amalgam method is of an independent interest.

C

'23

c0 2

J4

F2

C0 1

Fl

9 F

[241. A,

[2'+81. Sp,(2 U,(2)

[2 1 1 1,M2, 2

c02

[21.F2

Table 1

s

exceptional C3

C,-building truncated build.

T-geome t ry

C4-building truncated build.

P-geometry

P-geometry

[ IS21

[ Shpl

Ivn21

[ Ivn3 1

Sct. 5

[ Ivn41

In the present paper we develop a method for proving simple connectedness of geometries. This method relies on consideration of simply connected subgeometries and on

The minimal parabolic geometry 26 1

study of an intersection graph of subgeometries. The latter is a graph on a class of simply connected subgeometries with two subgeometries being adjacent if they have a nonempty intersection and if certain additional conditions are satisfied. This method enabled us to obtain uniform simple connectedness proofs for geometries presented in Table 1. In this table G denotes the flag-transitive automorphism group of Q; in the second column we indicate the type of Q (P-geometry or T-geometry); F denotes the stabilizer of a simply connected subgeometry the last column contains the reference for the first simple connectedness proof for Q. (Notice that the simple connectedness proof for the P-geometry of h423 is actually due to S.V.Shpectorov.)

The central result of the paper is the following theorem whose proof is located in Section 5.

Theorem 1.1. The T-geometry Q(Col) of the Conway group Gol is simply

The geometry Q(Co1) is the minimal parabolic geometry of Go1 described in [RSt]. It is also closely related to the maximal parabolic geometry of Go1 from [RSm]. In [Seg] a characterization of the latter one was given. In particular, it was proved thatl the maximal parabolic geometry is simply connected. An independent proof of this result follows from Theorem 1.1. Some results restricting the structure of the amalgam of maximal parabolics corresponding to a flag-transitive action on a rank 4 T-geometry are contained in [Row], [Par].

In Section 2 we recall some basic definitions and results concerning diagram geometries. Section 3 is devoted to a general procedure for proving simple connectedness via consideration of intersection graphs of subgeometries. In Section 4 we give a description of Q(C.1) and its Goz-subgeometry in terms of the Leech lattice. In Section 5 we prove the simple connectedness of B(Col).

connected. 0

2. DEFINITIONS AND PRELIMINARY RESULTS

In this section we recall some basic definitions concerning diagram geometries and also formulate the fundamental principle of the amalgam method (Theorem 2.1).

Let Q = (6, I , A, t ) be a geometry, that is a set Q of elements together with a symmetric reflexive incidence relation I and a type function t : Q + A. A geometry is supposed to satisfy the following axiom: the restriction o f t to any maximal set ofpairwise incident elements is a bijection onto A.

Let @ be a Jag of Q, i.e. a set of pairwise incident elements. Let Qa = (00, I*, A,, t o ) where 40 is the set of elements which are not contained in CP and are incident to all elements of CP, A, = t(CP); I@ and to are the restrictions to 48 of I and t , respectively. Then 00 is a geometry in the above sense and is called the residual geometry of B with respect to CP.

Let Q and Q' be geometries over the same set of types. A mapping 4 : Q' + B is a morphism of geometries if 4 preserves the incidence relation and the type function. A morphism is called a covering if its restriction to any proper residual geometry is

262 A.A. lvanov

an isomorphism. A morphism from a geometry onto itself is an automorphism. Let G 5 Aut(Q) be an automorphism group of Q. G is said to be Jag-transitive if it acts transitively on the set of maximal flags of 4.

We usually assume that A = {0,1, ..., r - 1) where r is the rank of the geometry. The set of elements of type i in Q will be denoted by @, 0 5 i 5 r . A geometry Q is connected if the graph with 4 as vertices and I as edges is connected. All geometries we will consider are supposed to be connected. A considerable amount of information about a geometry is carried by its diagram. The latter is a graph on A where the edge joining i and j symbolizes the rank 2 residual geometries of type { i , j } . The empty edge stands for the generalized digons (any two elements of different types are incident), an ordinary edge stands for projective planes, etc.

Let Q be a geometry, G be a flag-transitive group of automorphisms of Q and @ = { a o , c q , ..., ar-1} be a maximal flag of Q. Let G, = G(a; ) be the stabilizer of a, in G and A be the amalgam of these stabilizers. The members Gi of A are the maximal parabolics. In the flag-transitive case Q ca.n be reconstructed from G and A. Namely the elements of type i in Q are the (right) cosets of Gi in G, 0 5 i 5 r - 1; two cosets are incident if they have a nonernpty intersection. In this situation we write Q E S(G, A).

A geometry Q possesses a universal covering 4,, : d -+ Q such that for any other covering 4 : Q’ -+ 6 there is a covering II, : c --+ Q’ such that 4,, = 44. Let G act flag-transitively on Q. Then the automorphisms from G can be lifted to automorphisms of c and all those liftings form a group G which acts flag-transitively on c. Let d be an amalgam. By definition an A-group is a group which contains A and is generated by the elements of A. An A-homomorphism is a homomorphism of A-groups whose restriction to A is the identity mapping. If the class of A-groups is nonempty then there exists a universal A-group U ( A ) such that any A-group is a n image of U ( d ) under an d- homomorphism. One can define U ( d ) as a group having all elements of A as generators and all equalities valid in the members of A as relations.

The following fundamental principle was proved almost simultaneously in [Pasl], [Tit] and in an unpublished manuscript by S.V.Shpectorov.

Theorem 2.1. Let Q be a geometry, G be a jlag-transiti.ue automorphism group of 6 and A be the amalgam of m.axima1 parabolics. Let q5u : 0 -+ Q be the universal covering, G be the group of all liftiiigs of the automorphisms froin G , and U ( d ) be the universal A-gro*up. Then

A geometry whose universal covering is an isomorphism is said to be simply connected.

Q(U(A),A) and G E U(A) . 0

3. PROVING SIMPLE CONNECTEDNESS

In this section we consider some aspects of proving simple connectedness of diagram geometries.

In view of Theorem 2.1 to find the universal cover of a flag-transitive geometry it is sufficient to calculate the universal closure of the corrcsponding maximal parabolics amalgam. So a flag-transitive geometry is siniply connected if and only if this closure

The minimal parabolic geometry 263

coincides with the original group. Having an amalgam one can present its universal group in terms of generators and relations. In this way the simple connectedness can be checked by coset enumeration on a computer by use of the Todd - Coxeter algorithm. But we are interested in more combinatorial approaches to the simply connectedness problem.

A combinatorial proof of simple connectedness basically consists of the following threee steps.

Step 1. One defines a rule which assigns to each geometry I; in question a graph [(I;). This rule should possess the property that a covering 4 : @ --f 4 induces a covering 4 : r(6) -+ r(I;) of the corresponding graphs. This property will be refered to as coverabilit y.

Step 2. Starting with the condition that 4 is a covering of geometries one distinguish in r(4) a class K of cycles such that a cycle C E K is contractible with respect to 4 in the sense that C can be lifted to an isomorphic cycle in r(G).

Step 3. One proves that the fundamental group of r(I;) is generated by the normal closure of the cycles from K. Such a proof can be performed by splitting all cycles of I'(I;) into cycles from A'.

There is a universal way to associate a graph to a geometry, namely the graph of maximal flags. This is a graph on the set of maximal flags where two flags are adjacent if their intersection is a premaximal flag. In this case the coverability always holds and the class of Contractible cycles contains all cycles whose vertices are flags with a common element. In spite of the universality of the construction, in practice it is not convenient to work with the graph of maximal flags since it is too large.

we distinguish two types: the type of points and the type of lines. In accordance to this choice we define the point graph of I;. This is a graph on the set of points where two points are adjacent if they are incident to a common line. In this situation the coverability condition can fail in general but it always holds if two points lie on at most one common line. The class of contractible cycles contains the cycles whose points and lines are all incident to a fixed element of the geometry.

If I; contains a simply connected subgeometry F then the class of the contractible cycles can be extended. For instance, if we work with the point graph then each cycle contained in the point graph of 3 is contractible. This strategy was applied in [IS21 for the P-geometry of M2:, containing the exceptional C3-geometry of AT as a subgeometry. Nevertheless here we choose another strategy for Step 1. We will define an intersection graph of subgeometries.

Another approach is the following. In the considered geometry

First let us formulate our hypothesis.

Hypothesis 3.1. Let I; be a geometry over the set A = (0, I , ... , r - 1) of types, r 2 3; G be a flag-transitive automorphism group of I; and F be a samply connected subgeometvy o j G over a set A3 C A of types. W e assume that the stabilazer F of 3 in G acts flag-transitively o n 3, that 0 E AX and that distinct zmages of 3 under G have distinct sets of elements of type 0.

Throughout the paper a subgeometry is a quadraple ( 3 , 1 3 , A ~ , t ~ ) where F C I;, I F C I , A+ C A and t 3 is the restriction of t to F, which satisfies the axiom of the geometry. In fact in all situations we meet, I , is the restriction of 1 onto 3, but we

264 A.A. Ivanov

prefer the above definition. Let 4 = {a;li E A,} be a maximal flag in the subgeometry F. Let G, and Fi be the

stabilizers of a(, i E A, in G and F , respectively. Let 4 : t 8 be the universal covering of and G be the group of all liftings of automorphisms from G to automorphisms of 6. Let 6 = {&ti E A,} be a flag of 6 such that d($) = 4 and let Gi be the stabilizer of &; in G. Then 4 induces an isomorphism x; of Gi onto G;. Let p, be the preimage of Fi under xi and let $‘ be the subgroup of G generated by these preimages for all i E AT. Let $ be a subgeometry of consisting of the images under $’ of all elements of the flag 4 and of the incidences between elements of 4. Since 4 is a covering, the amalgam {Piti E A,} is isomorphic to the amalgam iF;li 6 AF}. Since F is simply connected by our assumption, Theorem 2.1 implies that F E F and 9 2 .F. Thus a simply connected subgeometry of E can be lifted to a simply connected subgeometry of c.

Now let S be the set of all images of 3 under the action of G and let S” be the subset of S consisting of all subgeometries passing through the element a = a0 (recall that 0 E A,). Then Go acts on S” and this action is similar to the action of Go on the cosets of Fo. Let R;, 0 5 j 5 s be the set of all orbits of Go acting on the ordered pairs of subgeometries from S”. Let us pick an index j such that R,” is symmetric and irreflexive (we assume that such an index exists). Let v j be the valency of the relation R,” and let Cj be a graph on S where two subgeometries are adjacent if they have an element ,B of type 0 in common and are in the relation R;. Then C j is called the (j-th) intersection graph of subgeometries. Notice that C, and Cj can coincide for distinct indices z and j . Let .F1 and F2 be subgeometries from S which are adjacent in C,. Let mj be the number of elements ,B of type 0 in .Fl n F2 such that (F1, F2) E R;. It is clear that the following lemma holds.

Lemma 3.2. The valency of C j is equal to vj . IF’l . mi’. 0

We can apply the same procedure to 6 and construct a graph g j that is the j - th intersection graph of subgeometries in 6. There is a natural bijection between the sets S“ and s”. In view of this bijection and the isomorphism .F 9, it is easy to see that C$ induces a covering of gj onto C j if and only if f i j = mj where f i j is the parameter analogous to mj and defined for Cj. Notice that in any case mj 5 mj. Let us give a sufficient condition for the coverability, which will be applied below

Lemma 3.3. Let vj = 1 and suppose that Fo is a maximal subgroup of F . Then f i . - m . - 1

Proof. The condition wj = 1 implies that R,” is an equivalence relation with classes of size 2. Let F E S. Then for each element ,f3 E F’ there is a unique subgeometry F ( p ) such that ( F , F ( p ) ) E Rf. If mj > 1 then F ( p ) = F ( y ) for some /3 # y. Now since Fo is maximal in F we conclude that F(p) is the same for all p E 3”. So we have two subgeometries with the same set of elements of type 0, a contradiction. 0

Now let us turn to the second step in our simple connectedness proof. Thus, suppose q5 induces a covering dj : ej -+ Cj. Which cycles from Cj are contractible with respect to dj? Let E,” be a subgraph of C j having S” as vertices, where two subgeometries are adjacent if they are in the relation R,”. Notice that in general E,” is not an induced

3 - 3 - .


subgraph. If 2; is the analogous subgraph of 9, then it is clear that dj induces an isomorphism of Zq onto Z?, so we have the following.

Lemma 3.4 Zf a cycle C is contained in Ej” then it is contractible with respect

So we can define a class K: of contractible cycles and proceed to the third step in our scheme.

In the cases we meet in this paper X: contains a subclass 7 from the class of all triangles in Cj. So the considered step can be reduced to splitting of an arbitrary cycle from Cj into triangles from 7. It is clear that if d is the diameter of Cj then it is sufficient to consider cycles of length up to 2d + 1 only. In all examples we come across in this paper we have d = 2.

Let us give two sufficient conditions for triangulability (compare Lemma 5 from [Ron]). In lemmas below A is a generic graph and 7 is a class of triangles in A. For a vertex z of A the set of vertices of A adjacent to x is denoted by A ( x ) .

to q5j. 0

Lemma 3.5. Let x , y be vertices o f A and d ( x , y ) = 2. Let @(I, y ) be a graph on the set A ( z )nA(y ) where u,v are adjacent ifboth { z , u , v } and { y , u , v ) are in 7. ZfO(x,y) is connected then any cycle of length 4 passing through x and y splits into triangles from 7. 0

Suppose that each cycle of length 4 in A splits into triangles from 7. Suppose also that Jor each triple z, y , z of vertices such that d ( z , y ) = d(x , z ) = 2, d ( y , z ) = 1 the following condition holds: either A(x) n A ( y ) and A(x) n A(.) have a vertez in common or there are vertices u E A ( z ) n A(y) , v E A($) n A(z ) such that {I, u , v} E 7. Then each cycle of length 5 in A splits into triangles from 7. 0

Lemma 3.6.

4. THE CONWAY GROUP GEOMETRY

Here we recall some properties of G(Col). Basically our description of this geometry goes back to [RSt] and its particular form was inspired by a description for the (702-

geometry given in [Shp]. The Steiner system S(5,8,24) consists of a 24-element set P of points and a collection

0 of 8-element subsets of P which are called octads with the property that each 5-element subset of P lies in a unique octad. This system is unique up to isomorphism. A trio is a partition of P into three disjoint octads and a seztet is a partition of P into six 4-element subsets (tetrads) such that the union of any two tetrads is an octad. An octad 0 will be identified with the partition (0, P - 0) of P. With an octad 0 E (3 a structure n ( 0 ) of projective space PG(3,2) is associated. The points of a(0) are the trios and the lines of x ( 0 ) are the sextets refining 0. A point of x ( 0 ) lies on a line of n ( 0 ) if the corresponding sextet refines the corresponding trio.

Let A be the Leech lattice (cf. [Con]). Recall that the set of components of vectors from A is identified with the point-set P of the S(5,8,24)-system. As usual the support of a vector is the set of its nonzero components. Let A, = { X l X E A,(X,X) = 16n) and

266 A.A. lvanov

A = A/2A. Then A carries the structure of a 24-dimensional vector space over GF(2) and = A, U A 2 U A3 U A4 (recall that in the Leech lattice A1 is empty). If x E A,, n = 2,3 then x is the image of exactly two vectors from A, which differ just by the sign. A vector from A4 is the image of 48 vectors from A4 forming a coordinate frame. The Conway group Col acts on A with orbits An, n = 0,1,2,4 and with the stabilizers of vectors from these orbits isomorphic to Col, Coz, C03 and 211.M24, respectively.

Let A: denote the set of vectors from A4 having only one nonzero component equal to f8 and A: denote the vectors from A4 with supports of size 4 and with nonzero components equal to f 4 . Then IA:I = 48, 1A:I = 24 . (:').

L e m m a 4.1. Let X E A4 and suppose that all components of X are divisible by 4. Then X E A: U A:. 0

The set A: consists of a unique vector. If X E A: then all the preimages of x in A4 are contained in A:. The corresponding set of 48 preimages can be characterized as follows: the supports are the tetrads of some sextet and the numbers of minuses are of a fixed parity. In particular we have an equivalence relation on A: whose classes of size 2 are indexed by sextets.

As was mentioned, G 2 Col acts transitively on A 4 and the stabilizer G(A:) is isomorphic to 211.M24. It is easy to show that this stabilizer acts transitively on A: so the latter set is a suborbit of G acting on A,. Let A be a graph on A 4 which is invariant under the action of G and in which A: is adjacent to all vectors from A:. In view of Lemma 4.2 one can give an adjacency criterium for A in terms of inner products of preimages in A4.

The above defined equivalence relation on A! induces an analogous relation on the neighbourhood A(p) of an arbitrary vertex ,!i of A.

Let x, p be adjacent vertices of A and let x * p be the vector in A which is the sum of x and p.

L e m m a 4.2 1 * ji E A(x), moreover p and x * ji are equivalent. Proof. It is sufficient to check the claim for

Thus we have a special class of triangles in A of the shape { x, p, x * p } whose vertices are the nonzero vectors of a 2-dimensional subspace of A. It is clear that each edge is in a unique triangle from this class.

'

= A: and ji E A!. 0

L e m m a 4.3. Let P, fi be nonequivalent vertices from A:. Then these vertices are adjacent in A if and only i f the corresponding sextets are refinements of a common trio.

Proof. By Lemma 4.1 ji and fi are adjacent if and only i f the tetrads of the corresponding sextets have intersections of even size. This can happen only if the sextets are refinements of a common trio. 0

L e m m a 4.4. Let F , fi be nonequivalent adjacent vertices from A:. Then ji * V E A:. Proof. By Lemma 4.3 the tetrads from the sextets corresponding to p and fi have

even intersections. Let p and Y be vectors from A: whose supports have intersection of size 2. Then the image of p -t v in A is contained in A!. So the claim follows. 0

R e m a r k 4.5 Let ji, i. be as in Lemma 4.4 and let S1, S2 and S3 be the sextets


corresponding t o p , P and p*b, respectively. Then there is a unique trio T = {01, 02, 03} such that S, are refinements of T for i = 1,2,3. Moreover, S1, Sz and S3 are lines contained in a hyperplane h and passing through a point p E h in the space ~ ( 0 , ) for j = 1, 2 and 3.

Let It' be a cliquc in A. This clique is said to be *-closed if x , ,G E I( implies * f i E It'. Since * corresponds to addition in A, a *-closed clique is the set of nonzero vectors of a subspace in A. In particular the size of such a clique is 2" - 1 for some m. By Lemma 4.4. any maximal clique is *-closed.

Let I( be a *-closed clique containing A:. Let k be the set of equivalence classes on A: contained in I(. So k is a set of sextets. Clearly, if I(3 is a *-closed clique of size 3 then k3 consists of a unique sextet. The shape of k~ for a *-closed clique of size 7 follows from Remark 4.5. It is easy to see that G acts transitively on the set of *-closed cliques of sizes 1, 3 and 7. The following lemma whose proof is omitted gives a description of the maximal cliques in A.

Lemma 4.6. A maximal clique in A has site 15 and G has two orbits on the set of maximal cliques. Let It'15 and It':, be representatives of these orbits containing A:. Then k15 consists of all sextets which are refinements of an octad 0 and are contained in a fixed hyperplane o f ~ ( 0 ) and k:, consisits of all sextets refining a fixed trio. 0

Now we are in a position to define the geometry Q = Q(Col). The set 9' consists of *-closed cliques in A of size 2'+' - 1. For i =0, 1 and 2 we take all cliques while for i = 3 only those from the orbit of G containing K 1 5 . The incidence relation between elements of the geometry corresponds to the inclusion relation between cliques.

Since the vertices of a *-closed clique of size 2" - 1 are the nonzero vectors of an rn-dimensional subspace of A, Q is alredy defined in its natural GF(2)representation (see definition in [IS4]). If fact the representation of Q in A is the universal one. This claim can be easily proved using the fact (cf. [IS4]) that the universal natural representation of a subgeometry .F = S(Co2) in Q is that in a 23-dimensional subspace of A. The author was informrd by %.Smith that he obtained an independent proof of the universality of the representation of Q in A.

Let us proceed to consideration of subgeometries in Q. Let X be an arbitrary vector from A2, say X = (4,4, 022). Let Il be the set of vectors p E A4 such that ( A , p ) = 32 and let l=I be the image of Il in A. Thus n is a subset of vertices of A. Let F be a subgeometry of Q consisting of all *-closed cliques containing vertices from l=I only. Then comparing the definition of .F with the definition of the P-geometry Q(C02) (cf. [Ivnl], [ISl], and especially [Shp]), we come to the following

Lemma 4.7. The subgeometry .F of Q(Co1) is isomorphic to the P-geometry Q(C02). 0

For convenience of the reader we present below diagrams of maximal parabolics of E(Co1) and G(Co2). In these diagrams under the node of type i the structure of the stabilizer of an elemelit of type i in the corresponding flag-transitive group is given. On the top we indicate the action induced by the stabilizer on the residual geometry and at the bottom the kernel of the action. In our notation we follow [ATLAS].

268 A.A. Ivanov

i

0

1

2

3

N .

I ri (XI I (A, p) C ( X , j i )

4 , 6 0 0 316 U,(2) c0 2 1 232

46 ,575 0 2 l o . M2, . 2

47 ,104 +,8 McL

"2 4 s 3 x 3 . Sp4(2 1 L, ( 2 1 x s 3 L 4 ( 2 )

24+12 22+12+3 2 1 + 8 + 6 2'l

P . s3xs5

L 3 ( 2 ) X 2 L*(2 ) 24+10 21+7+6 2 1+4+6 2 l o

Let S be the set of subgeometries of 0 which are images of .F under the action of G. It is clear that S is in a natural bijection with A 2 . At the same time Qo is in a bijection with A 4 . In these terms a subgeometry from S contains a given element of type 0 if in the corresponding preimages there is a pair of vectors with inner product 32. Suppose that (Y = (YO corresponds to the vector A:. Then the vectors corresponding to S" are the images in A of the vectors from A:. The latter set consists of the vectors whose supports are of size 2 and the nonzero components are equal to f4. Then IAiI = 2 . (i4) and Go % 21'.M24 acts on this set as it acts on the cosets of the subgroup Fo 2lo.M22.2. So OZ(G0) has orbits of length 2 on S" and M24 Z Go/02(Go) acts on the set on these orbits as it acts on the set of 2-element subsets of P. A simple calculation shows that Go has 4 orbits on the ordered pairs of subgeometries from S". Let R;, 0 5 j 5 3 be these orbits. Then we can assume that Rg is the diagonal orbit; RY corresponds to equal subsets from (r); R; to disjoint subsets and Rg to intersecting subsets. All orbits are symmetric and if v J , 0 5 j 5 3 are their valencies then vo = v1 = 1, v2 = 462, 713 = 88.

Now let us turn to the action of G on the set J? = A2. An information about this action is given in Table 2. Here r;(j) is an orbit of the stabilizer G(X), p is a representative of this orbit and A, p are preimages in 12: of and p , respectively.

Table 2

Lemma 4.8. The Jirst and the second intersection graphs of subgeometries in Q The third intersection graph of subgeometries has coincide and have valency 46,575.

vakency 4,600.


1

Proof. Let A, p be vectors from A: = S" which are in the relation Rj" and A, p be their preimages in A:. Then ( A , p ) = 0 for j = 1,2 and ( A , p ) = f 1 6 for j = 3. So the claim follows from Table 2 . 0

On Fig. 1 and 2 we present diagrams describing the structure of the intersections graphs of subgeometries in 6.

46,575 1 46,575

1

Fig. 1

4 , 6 0 0 1 4,600

c, : 275+2,025

2 , 4 6 4

Fig. 2

5. THE SIMPLE CONNECTEDNESS OF G(C.1)

Let q5 : -+ 6 bc the universal covering of G = G(Col). Let %j and Cj be the j - th intersection graphs of subgeometries in 6 and G, respectively, 0 5 j 5 3. Our first lemma

210 A.A. Ivanov

1

here is a direct consequence of Lemma 3.3.

Lemma 5.1. The graph C1 is coverable. 0

Lemma 5.2. g1 = g2. Proof. We saw in the previous section that C1 = Cz. Let F,.? be subgeometries

from S which are adjacent in gl. Then there is a unique element ,fl of type 0 in Fn.? such that (p,.?) E Rf. To prove the lemma it is sufficient to show that there are more than one element of type 0 in n fl. Since (F,.?) E Rf, there is an element g E OZ(G(P)) which maps .f onto fl. Since g acts trivially on the residue of p, we conclude that the residues of p in F and are the same. So these subgeometries cantain a common element y of type 1. Now in each of the geometries 6, F and fl the element y is incident to 3 elements of type 0. So the claim follows. 0

Now our goal is to show that C3 is coverable. But first of all we consider in more details the graph C1 (which is equal to C2) . This is a graph on the set r = A; where x is adjacent to all elemcnts of rz(x) (cf. notation in Table 2) the elements of the latter set are in a bijection with the elements of type 0 in .F 2 G(Co2). Coz acting on F’ has rank 5 and subdegrees 1 , 462, 2,464, 21,120 and 22,528 (cf. [Con], [El]). Comparing t,hese subdegrees and Fig. 1 we conclude that G has two orbits and ‘T; on the set of triangles in El. Each edge is contained in 462 triangles from 71 and in 21,120 triangles from 7 2 . Let us decide which triangles are contractible with respect to the covering of C1 induced by C#J? Let 11s consider t,he subgraph EF of C1 induced by the set SO, CY E So. The structure of this graph is given on Fig. 3 and by Lemmas 5.1, 5 .2 this subgraph is isomorphic to analogous subgraph in 91. So we have the following

$1 : 21 + C1 induced by the universal covering of 9. 0 Lemma 5.3. The triangles in 71 are contractible with respect to the covering

46 2 1 8 0 420 ~ 462 ’ 88

1

46 2

Fig. 3

Lemma 5.4. Lel x a n d ji be vertices of& such that ji E I‘l(x). Then any cycle in

Proof. Let 0 = 0 ( x , p ) be a graph on the set of vertices of El adjacent both to x C1 passing through and ji is contractible wzth respect to $l.

The minimal parabolic geometry 27 I

and ji and where two vertices V l , V2 are adjacent if the triangles {A, PI, F 2 ) and { j i , PI, fi2) are from 71. In view of Lemmas 3.5 and 5.3 it is sufficient to show that 0 is connected. Without loss of generality we can assume that x, ji E Zy. Now it follows from Fig.3 that (0 n ZyJ = 420. One can easily conclude from the description of a;l that the subgraph of 0 induced by 0 n Zy is connected. So each connected component of 0 has at leat 420 vertices. G(X,ji) g US(2) acts naturally on 0 and this action is transitive. If 0 is disconnected we can consider the action of G(X, a ) on the set of connected components. In this way we would obtain a faithful permutation representation of US(2) of degree less than 60 > 24,948/420, a contradiction. 0

Corollary 5.5. The graph C3 is coverable.

Let us consider the covering $3 : $3 -+ C3 induced by the universal covering of g. The subgroup G(X) acting on the set rl(1) of vertices adjacent to x in C3 preserves an equivalence relation with classes of size 2. Two vertices ji, V E rl(X) are equivalent if ,E + fi = as vectors of A. One can check that equivalent vertices are adjacent. G(1) induces a rank 3 group with subdegrees 1, 891 and 1,408 on the set of equivalence classes. This information together with Fig. 2 implies that G has exactly two orbits on the set of triangles in C3. At the same time it is straighforward to see that 2; contains representatives of the both orbits. So we have the following.

Lemma 5.6. All triangles in C3 are contractible with respect to $3. 0

Let j i ; E ri(x) and Hi = G(X,p) for i = 2,3. It follows from Table 2 that H2

2lO.Af22.2 and H3 g M c L . The following two lemmas describe the action of H;, i = 2 and 3 on the set rl (x) of vertices adjacent to in C3. Recall that two vertices j i , fi E r,(X) are said to be equivalent if j i -t V = x in A.

Lemma 5.7. H2 has exactly three orbits on rl(x) with lengths 88, 2,048 and 2,464.

Lemma 5.8. H3 has exactly four orbits on rl(x) with lengths 275, 275, 2,025, 2,0025.

Now we are in a position to prove the following

Lemma 5.9. All cycles of length 4 in C3 are contractible with respect to 43. Proof. By Lemma 3.5 it is sufficient to show that the subgraph 0; = @(),Pi) of C3

induced by the set rI(x) n!?l(jiz) is connected for i = 2 and 3. Without loss of generality we can assume that ji2 together with 0 2 are contained in E;. It follows from the structure of the latter subgraph that any two distinct equivalence classes from 0 2 are adjacent. So 0 2 is connected. H3 induces a primitive rank three action on the set of vertices of 03. On the other hand it is easy to check that 0 3 is nonempty. So the claim follows. 0

I f two vertices are equivalent then they lie in the same orbit. 0

I f two vertices are equivalent then they lie i n different orbits of the same length. 0

Let us proceed to consideration of length 5 cycles in C3.

Lemma 5.10. A length 5 cycle in C3 passing through Proof. Direct calculations in the Leech lattice show that each vertex from rl(x) is

adjacent to a vertex from 0 2 . This implies that a cycle of length 5 passing through x and ji2 is contractible.

212 A.A. Ivanov

and j i 2 splits into triangles and a cycle of length 4 (compare Lemma 3.6). 0

Lemma 5.11. All cycles of length 5 in C3 are contractible. Proof. In view of Lemma 5.10 it is sufficient to prove contractability of a cycle

C = {& = 1, i j l , &, f i 3 , 64) where &, fi3 E I'3(1). Let Ir;, be the vertex from I'l(fi2) which is equivalent to 9. Then T = {&, i&w} is a triangle and one can see from Fig. 2 that w E I'i(1) for i = 1 or 2. In the former case C splits into T and a cycle of length 4. In the latter case C splits into T and two length 5 cycles passing through 1 and some vertices from I ' Z ( 1 ) . By Lemmas 5.6, 5.9 and 5.10 in any case C is contractible. 0

So all cycles in C3 of length less than or equal to 5 are contractible. Since the diameter of C3 is 2 we have the following

Proposition 5.12. The covering 43 is an isomorphism. 0

Thus, the universal covering of 6 is an isomorphism as well and Theorem 1.1 is proved.

References

[Con1

[ATLAS]

[Ivnl]

[IvnZ]

[Ivn3]

[Ivn4]

[IvnS]

[IS11

[IS21

~ 3 1

J.H.Conway, Three lectures on exceptional groups. In: M.B.Powel1 and G.Higman (eds.), Finite Simple Groups, pp. 215-247, Acad. Press, New York, 1971.

J.H.Conway et al., Atlas of Finite Groups, Oxford Univ. Press, 1985.

A.A.Ivanov, On 2-transitive graphs of girth 5, Europ. J. Comb. 8 (1987) 393- 420.

A.A.Ivanov, A presentation for J4, Proc. London Math. SOC. (to appear).

A.A.Ivanov, A geometric characterization of Fischer's Baby Monster, J. Al- gebraic Comb. (to appear).

A.A.Ivanov, A geometric characterization of the Monster, Proc. Symp. "Groups and Comb.", Durham, 1990 (to appear).

A.A.Ivanov, Geometric presentation of groups with an application to the Mon- ster, Proc. ICM-90, Kyoto, Japan, August 1990, pp. 385-395, Springer Verlag, 1991.

A.A.Ivanov and S.V.Shpectorov, Geometries for sporadic groups related to the Petersen graph. 11, Europ. J. Comb. 10 (1989) 347-361.

A.A.Ivanov and S.V.Shpectorov, The P-geometry for M 2 3 has no nontrivial 2-coverings, Europ. J. Comb. 11 (1990) 373-379.

A.A.Ivanov and S.V.Shpectorov, Geometries of Sporadic Groups (in preparation).


A .A. Ivanov and S.V.Shpectorov, Universal representat ions of P-geometries from Fz-series. Preprint, Institute for Systems Studies, Moscow, 1989.

A.A.Ivanov and S.V.Shpectorov, An infinite family of simply connected flag- transitive tilde geometries, Geom. Dedic. (to appear).

.- C.Parker, Groups containing a subdiagram *-*-' 7

Preprint, 1989.

A.Pasini, Some remarks on covers and apartments, In: C.A.Baker and L.M.Batten (eds.) Finite Geometries, pp. 233-250, Marcel Dekker, New York, 1985.

A.Pasini, A classification of a class of Buekenhout geometries exploiting amalgams and simple connectedness, Preprint, University of Napoli, 1989.

M.Ronan, Coverings of certain finite geometries. In: P.Cameron, J.Hirschfeld and D.Hughes (eds.), Finite Geometries and Designs, pp.316-331, Cambridge Univ. Press, 1981.

M.A.Ronan and S.Smith, 2-local geometries for some sporadic groups, In: B.Cooperstein and G.Mason (eds.) Proc. Symp. Pure Math. No. 37, pp. 283- 289, AMS, Providence, 1980.

M.A.Ronan and G.Stroth, Minimal parabolic geometries for the sporadic groups, Europ. J. Comb. 5 (1984) 59-91.

P.Rowley, Minimal parabolic system with diagram ~ N , J. Algebra, 141 (1991) 204-251.

Y.Segev, On the uniqueness of the Col 2-local geometry, Geom. Dedic. 25 (1988) 159-219.

S.V.Shpectorov, The universal 2-cover of the P-geometry ~ ( C O Z ) , Europ. J. Comb. (to appear).

J.Tits, Ensembles ordonnks, immeubles et sommes amalgameks, Bull. SOC. Math. Belg. A38 (1986) 367-387.



Coset Enumeration in Groups and Constructions of Symmetric Designs

Zvonimir Janko

Dept. of Mathematics, Heidelberg University, Germany

Let D be a symmetric block design with parameters (v, Ic, A) on which operates an automorphism group G of order g . Let P be the set of v points and let B be the set of v blocks of D. Then P partitions in the G-orbits of points PI, P2, . . ., P, and B partitions in the G-orbits of blocks B1, Bz, . . ., B,. These two partitions define a tactical decomposition of D (see BEUTELSPACHER 111, p. 210). Set mi = (Pi1 and ni = lBil for i = 1,2,. . . , c and let bi be a block (line) in the block orbit Bi. Then we have:

ml + mz + ... + m, = v,

n1 + n2 + . . . + n, = v.

Let a,; 2 0 be the number of points on the line b, from the point orbit P;. Then we obviously have:

ail + ai2 + . . . + a; , = k for all i = 1,2, . . . , c .

On the other hand the numbers mi and ni are indices of the point or block stabilizers in G and so mi and ni are divisors of g . Therefore mi = g/mi and n: = g/n; are also integers for all i = 1,2 , . . . , c. The c x c matrix (ai;) is called the matrix of the G-orbit structure of the design D. For the rows of this matrix we have the following relations:

(3)

m: . a x + m: . a h + . . . + m: . a:, = ( ~ n i + n ) . n:

m: . ail . ajl + mk . ai2 . a j z + . + ' + m: . a;, . a;, = X . g ,

(4)

for all i = 1 ,2 , . . . , c, nnd for each i # j , i , j = 1,2, . . . , c we have also the relations:

(5)

where n = k - X is the order of the design D. We give the group G in terms of generators X and relations so that each point

or block stabilizer H , can be expressed with the same generators X . We have [G : H,] = m, or n,. The Coxeter-Todd coset enumeration method with respect to the subgroup H , gives not only the number [G : H , ] of cosets but gives also explicitely the permutation representation of G with respect to the (right) cosets of the subgroup H , . The corresponding programs have been made by HRABE DE ANGELIS. We do it for each stabilizer subgroup. To construct the design D will mean only to put together all these permutation representations according to the orbit structure matrix (a , ] ) .

216 2. Janko

In JANKO-TRAN [2] a symmetric design for (70,24,8) was constructed whose full automorphism group is Fzl x 22 of order 42.

As an example for the above method we shall construct all symmetric designs D for (70,24,8) on which operates the group G = Eg. Fzl of order 168 (a faithful extension of an elementary abelian group E8 of order 8 with the non-abelian group Fzl of order 21) so that an automorphism of order 7 operates fixed-point-free on the design. The group G is given in terms of generators and relations: G = ( a , b, c, d , elc2 = dz = e2 = 1, (cd)' = (ce)' = (de)2 = 1, a7 = 1, b3 = 1, b-'ab = a', a-'ca = d, a-'da = e, a-'ea = cd, b-'cb = c, b-'db = e, b-leb = de). We consider the following subgroups of G whose indices in G are divisible by 7:

a) G7 = (b, c, d, e) has index 7 in G, b) G14 = (b, d, e) = Ad has index 14 in G, c) G28 = (b, c ) 2 2 6 has index 28 in G, d) G56 = (b) 21 2 3 has index 56 in G. If G operates in more than three orbits on D, then G operates in c = 4 orbits and

there are exactly five orbit matrices which satisfy the necessary relations (1) to (5). But no one gives rise to a design.

If G operates in three orbits on D , then there are exactly three orbit matrices of which only the following matrix gives rise to designs:

m 1 = 7 77x2 = 7 m3 = 56 4 4 16 n1 = 14 1 3 20 n2 = 28 3 1 20 n3 = 28.

We apply the coset enumeration method for G with respect to subgroups G7 and G56

to obtain permutation representations of G of degrees 7 and 56 respectively. We "put" these representations in the matrix and obtain up to isomorphism exactly four designs which are obviously non-self-dual.

If G operates in two orbits on D, then there are exactly two orbit matrices of which only the following matrix gives rise to designs:

8 16 n1 = 14 4 20 722 = 56.

A. GOLEMAC has shown as a big surprise that this matrix gives rise to only one design for (70,24,8) which turns out to be non-self-dual. Here we use the representations of G of degrees 14 and 56.

In this way the following theorem was proved. Theorem. If a group G = E8 . Fzl operates o n a symmetr i c design for (70,24,8),

t h e n there are exactly (up t o ~somorph~sm and duality) f ive such designs which are all non-self-dual and G is the fu l l automorphism group of any one of t h e m .

Groups and constructions of symmetric designs 211

REFERENCES

1

2

A. Beutelspacher, Einfuhrung in die endliche Geometrie I, Verlag Mannheim, Wien, Zurich, 1982. Z. Janko and Tran v. Trung, The existence of a symmetric block design for (70,24,8), Mitt. Math. Sem. GieBen, vol. 165, 1984, 17-18.


Cornbinatorics '90 A. Barlotti el al. (Editors) 0 1992 Elsevier Sciencc Publishers B.V. All rights reserved. 219

ON THE UBIQUITY OF DENNISTON-TYPE TRANSLATION OVALS

IN GENERALIZED ANDRE PLANES

V. Jha* and N. L. Johnson

ABSTRACT

We show that to each non-prime integer N > 1 there corresponds a non-Desarguesian Andrk plane of order 2N that admits translation ovals which are natural analogues of the Desar- guesian translation ovals, and whose existence in certain Andrk planes was first detected by R. H. F. Denniston in 1979. We refer to these ovals as A-conics, and in a sequel we

shall characterize them combinatorially. In the present article, we shall show that if a non-

Desarguesian generalized Andrk plane T of order 2N exists (i.e., N is non-prime), then n can

be chosen so as to admit a A-conic, and that if N is a prime power then every generalized

Andrk plane of order 2 N admits A-conics. However, we shall also establish that quite often

a second generalized Andrk plane A' of order 2N exists (e.g., whenever N is even but not a power of 2), such that T' does not admit A-conics.

* Dept. of Mathematics, Glasgow College, Cowcaddens Road, Glasgow G4 OBA, Scotland

280 V . .Iha. N.L. Johnson

1. INTRODUCTION

Let R be an oval in a projective plane A of order n, and let e be a tangent to R. If A' admits a translation group 7 such that the n f i n e points of R form a 7-orbit then R is called a

translation oval (or a 7-oval), relative to e. It is well known that A admits a translation oval only if 7 E Z r and, in particular, n = 2N. Although it is conceivable that all planes of

order n admit ovals, it is obviously not the case that all planes of order 2N admit 7-ovals, if only because some f i n e planes of order 2N just do not have enough translations.* On the other hand the Desarguesian plane of order 2N certainly possess T-ovals (and these have been classified by Payne, c.f. Hirschfeld [4, 8.5.41). Thus a basic question is the following

PROBLEM (P). admits a 7-oval!'

Does there always ezist a non-Desarguesian plane of order 2N (> 8 ) that

One of the main goals of this article is to contribute to this question by showing (c.f., theorem 4.6):

COROLLARY. Desarguesian) Andre' plane of order 2N that admits a T-ova l .

If N as a non-prime positive integer > 1 then there exists a (non-

REMARK. If N above is assumed even then the above result is a consequence of the ex-

istence of the Denniston 7 - o v a l s [2], which live in certain Andre' planes, and also follows

from the ezistence of the Korchmhros ovab of even order in the Hall planes 171.

*

The result above is a consequence of our investigations concerning a certain type of T-oval, which we call a A-conic, that may exist in generalized Andrk planes. Roughly speaking, a A-

conic is an analogue of the generic 7-ovals that exist in Desarguesian planes (and have been shown by S. Payne to be of type y = fz2 , [4]). These A-conics may also be viewed as being generalizations of the Denniston ovals indicated above. Although the natural definition of a

A-conic is algebraic, it is easily seen to be a geometric concept, and in particular both, the existence, as well as the non-existence of A-conics have geometric significance. In a sequel to

As in the case of the Figueora planes of order 2N.

M

On the ubiquity of Denniston-type translation 28 1

this article we shall show that planes obtained from Desarguesian planes by replacing certain

lines by translation ovals (using a procedure of Assmus and Key [l, section 51) are precisely

the generalized Andrk planes that admit A-conics. As far as this article is concerned our

main results about A-conics may be partially summarized in the following terms (yielding

in particular the result above). For the sake of convenience, we shall nearly always focus on

A-conics y = z Z M , the normalized versions of A-conics y = fz2 , with f = 1. M

THEOREM 4.6.

Desarguesian) afine Andre' plane x , of order n, such that x admits a A-conic.

If n = 2N > 8, and N is not a prime integer, then there is a (non-

REMARK.

planes (viz., the Hall planes) possess 7-ovals that are not A-conics.

The ovals of Korchmbros 1'71 in the Hall planes show that at least some Andre'

In spite of the above theorem, many generalized Andrk planes do not possess any A-conics,

although they may have other types of 7-ovals, as indicated in the remark above. Thus, for

instance, our study of A-conics implies that:

THEOREM 5.6. If n = 2N > 4 is a square integer (but N is not a power of 2 ) then there

are Andrk planes, x l and x2, of order n such that x l admits A-conics (as stated in the

previous theorem) but x2 does not.

Although such x2 without A-conics exist for many non-square n, e.g., when N has a large

number of distinct prime factors, they cannot exist when N is a prime power, i.e., all generalized Andrk planes of order 2 P b admit A-conics (c.f., theorem 4.7).

Our approach to proving the above theorems, and to the study of A-conics in general, is based

on exploiting a fundamental equivalence between translation planes admitting 7-ovals, and

what we have chosen to call transversal spreads. Although this equivalence seems part of the

folklore (tacitly used in Assmus and Key [l, section 5 ] ) , and is just a natural analogue of the

Andr6 connection between translation planes and spreads, we feel it has not been sufficiently

exploited as a computational tool in the study of 7-0vals in translation planes. One of the

goals of this article is to stress the usefulness of this method by using it in the context of

generalized Andrk planes. In particular, section 2 is devoted to a detailed development of

282 V. Jha, N.L. Johnson

the And& connection for translation planes admitting 7-ovals, and the related concept of a

transversal spread-set, a natural analogue of the usual concept of a spread-set of matrices.

2. THE ANDRE CONNECTION FOR 7-OVALS

The well known necessary conditions for the existence of T-ods are readily seen to be sufficient aa well. The And& type connections for spreads admitting T-ovds that we develop below (c.f., theorem 5 and 7 below) are easy consequences of this observation.

RESULT 1.

to some translation group 7 whose axis is a line e . Then

( 1 ) n = 2N for some N ; and

( 2 ) 7 is an elementary abelian 2-group of order n and every involution in 7 has a different

center.

( a ) Let a be any projective plane of order n(> 2 ) admitting a ?-oval 0, relative

(b ) Conversely, a projective plane A of order n = 2N admits a 7 - o v a l if an a f ine version

A' contains a translation group 7 of order n such that every involution in 7 has a distinct

center on e. The 7 - o v a b of a, invariant under the chosen 7, are the sets of the f o r m

O r b T ( 0 ) U { X } where 0 is any a f ine point of a' and X = en R.

PROOF. Any elation subgroup E of 7, with a common center, partitions the affine points of a' into E-orbits consisting of IEl collinear points. But now R cannot be left invariant by E unless IE( = 2, and so the n - 1 nontrivial elements of 7 are all involutions with distinct centers on e. Thus (a) follows, and (b) is equally straightforward.

It is worth stressing that the problem of finding 7-ovals is essentially that of examining the action of translation groups on parallel classes. Thus the above result implies

COROLLARY 2.

admits a translation group 7 of order n such that

( 1 )

A plane ?r of order 2N = n admits a 7 - o v a l iff an a f i n e version A'

7 is regular (and hence transitive) on precisely two parallel classes of lines, [XI and

[U] , where X , Y E e; and

every other parallel class of e is the union of two 7-orbi ts , each of size n / 2 . ( 2 )

On the ubiquity of Denniston-type translation 283

We now specialize to translation planes. Let T be the full translation group of a', an affie

translation plane of order n = 2N. Let Ell,] be the group of (L, C)-elations in T , for all L E e. Hence S, := ( T , E [ L ~ ) is the spread associated with T . Now a' admits a translation oval

with axis e iff T contains a subgroup 7 of order n such that

(1) 7 n E~ = 1 = 7 n E ~ ~ X , Y E C, x # Y; and

( 2 ) 17 n Ezl = 2Vz E a\{x, Y } .

But since (1) is an easy consequence of ( 2 ) , we have the following result, letting F2 = GF(2)

throughout the article.

THEOREM 3. Let S = ( V , r ) by any GF(2)-spread, with V = E N and r a collection of

2N + 1 pairwise disjoint subspaces, each of rank N over GF(2) . Then the T-ovals of the

a f ine translation plane T S , arising from the spread S , are precisely the translates of all the

rank N subspaces of W < V satisfying:

3 X , Y E r such that IW n XI = 2VX E r \ { X , Y } (*)

The condition (*) above motivates the following

DEFINITION. N subspace W of V is called a transversal to r, with carrier set { X , Y } c r, if

Let S = (V,r) be any &-spread an V = E N ; thus = 2N + 1. A rank

rank [W n A] = I V X E r \ { X , Y } .

Now = U { W} is a transversal spread with azis W, carriers X , Y , and r = soc le ( r ) or

3OC(r).

REMARKS 4. A n y transversal W , to a spread r, determines its carrier set C =

{ X , Y } uniquely, and C consists of two distinct components of r, both disjoint from W . ( 2 )

A n y transversal spread r has a unique socle I', and a unique axes W .

(1)

PROOF. (1) is immediate, and (2) is a consequence of (1).

Theorem 3 above asserts, in effect, that a translation plane admits T-ovds precisely when its spread l? admits a transversal W. This And&-type connection can also be stated in terms of transversal spreads as follows.


THEOREM 5.

iff its coordinatising spread r is the socle of a transversal spread fi. A n a f i n e translation plane admits a 7 - o v a l relative to the iranslation azis

Isomorphisms between transversal spreads f i l and fiz, in the vector space V, may be defined in the obvious way:

FI s' fiz @ 3g E GL(U,+) such that g ( r 1 ) = f 2

However, this definition is probably too fussy unless one wants to distinguish between different types of 7-ovals that live in the same plane. As this is not our present objective, we focus on a weaker definition, regarding transversal spreads as being equivalent if they arise from the same translation plane:

DEFINITION. 71 equivalent to % if their 3 0 C k 3 , viz., sot(%) and sot(%) are isomorphic qua spreads.

Zf % and 72 are transversal spreads on the same vector space V , we consider

Thus we may restate the previous theorem in slightly different terms.

REMAFtK 6. planes are coordinatized b y equivalent transversal spreads.

Zf x1 and x2 are translation planes admitting 7 -ova l s then xl s' x2 iff the

The basic open question as to whether every even order translation plane admits a 7-oval

(c.f., Problem (P)) may be expressed as follows in our terminology:

(*) 13 every spread r, of order 2N, a Jock of a transversal spread fi?

A more provocative version of this question asks whether every pair of distinct points on the translation axis of any translation plane is the carrier of at least one T-oval.

(**) Given a spread r, of even order, and two distinct components { x , Y } C r is there

always a transversal spread r such that: s o c ( f ) = r and { X , Y } is the carrier set o f f ?

We now turn to the coordinatization of transversal spreads f by sets of matrices; this permits a computational study of P, in much the same way as spread-sets of matrices are used to study ordinary spreads I".


DEFINITION.

set i f ? U { A } is such that

(a) r is a GF(2)-spread-set; i.e., 0,2 E r and the difference between any two distinct

A collection 7 of ZN + 1 distinct N x N matrices over G F ( 2 ) is a 7-spread-

elemenb of r i s non-singular; and

(b) A is a non-singular matriz such that rank [ A - M ] = 1VM E r+.

Using f one obtains a transversal spread on V = Fp in the same way as spreads are obtained

from spread-sets. Thus xf is the transversal spread on V = F; @ 3; specified by:

where

r, = {y = 7 x : 7 E 7)

is the spread associated T . We regard ri as being coordinatized by ?; i t is clear that any

transversal spread can be coordinatized by a 7-spreadset.

Thus we have established the following computationally useful version of the And& con-

nection for 7-ovals. This connection, when specialized to A-conics (c.f., theorem 3.2 ahead)

underpins all the results of this article. Note also that there is no reason to believe that the

usefulness of the Andre connection is restricted to the study of generalized Andre planes.

THEOREM 7. Given a spread-set r , of N x N matrices over GF(2) , the corresponding

spread xr, o n V = Fy @j F;, admits y = Ax as a transversal iff T U { A } is a transversal

spread-set. Conversely, any spread r on U has a transversal with carrier set { X , Y } iff r is of type rI where r is a spread-set that is contained in some transversal spread-set T U { A } .

For the remainder of this article we specialize to generalized And& planes. In the next

section, after introducing our notation and some basic results concerning generalized Andre

systems, we apply the above result to the study of A-conics, the translation ovals of interest

in this article.

286 V . Jha, N.L. Johnson

3. A-CONICS IN GENERALIZED AND& PLANES

For the remainder of this article N = sr is a composite prime, where integers s, r , are both

> 1, and 7 = G F ( 2 N ) > K = G F ( 2 ” ) = GF(q). We assume the function A : 3* -+

( A u t 3 ) ~ defines a generic generalized Andre system &A = (3,+,-) with kern K = GF(q);

now d imKQA = r > 1, QA is not a field, and so 131 2 16. Thus A is specified by a unique

integer valued map X:3* -+ I ~ : = {o ,I , ... , r -1)

such that: ,,“m)

m o x = m x A ( m ) = mz V m E F , X E 3

Now the corresponding spread on U = 3: @ 3: has component set

where X = { y = 0) is the X-axis, and Y = {z = 0) is the Y-axis. We consider potential

ovals that are subspaces of the following type.

DEFINITION. A subspace of U , 8 , , , = { ( x , ~ x ~ ~ ) : z E U}, or “y = f x Z M , ” for f E F*

and M E I N , i s called a A-conic of K X if it is a transversal to the spread K A ; in particular

the X and Y - a x i s of K A must be the carriers for the A-conic.

M M It is easily seen that “y = fz2 ” is a A-conic, if and only if “ y = g x Z ’’ is a A-conic, whenever g E 3*. This allows us to restrict attention to normalized A-conics in which the

scalar f = 1.

M DEFINITION. A A-conic of type y = zz is called a A-conic.

For the remainder of this article, we shall almost exclusively focus on A-conics rather than on A-conics. But we note that although the notion of a A-conic is a geometric invariant

(c.f., proposition 3.3 ahead), the concept of a A-conic is not. It will also become evident

that the degree M of a A-conic is not a geometric invariant.


We first recall the well known fact that all collineations in the translation complement of a

generalized Andrt5 plane, other than a Hall or a Desarguesian plane, are either autotopisms

or anti-autotopisms.

RESULTS 1.

set { X , Y } , and XU Y is left invariant by (Autrx), , .

If xx is not a Hall-spread then all the A-conics of xx have the same carrier

PROOF. Apply Foulser's results [8, corollary 14.10 and 14.11, p. 661.

Our principle tool for investigating A-conics is the following necessary and sufficient condition

for their existence; it is a specialization of the existence condition for transversal spread-sets

(c.f., theorem 2.7) in the context of A-conics.

THEOREM 2 . Let Qx = (F = G F ( 2 N ) , + , o ) be a generalized Andrk system, with kern

X: = GF(2 ' ) , and specified b y x : F* -+ I,.. Then @M := y = x 2 , for M E (1,. . . , N - l},

i s a A-conic of the spread 1 ~ x i f i

M

(*)(sX(m) - M , N ) = 1Vm E F*

PROOF. By theorem 2.7 OM is a A-conic precisely when:

m E .F +- 311 E F* such that m o x = x 2 M

Thus OM is a A-conic iff to each m E F* the equation

has a unique solution for I E F*.

homomorphism of the group F', OM is a conic iff:

But since the LHS corresponds to a multiplicative

H (sx(rn) - M , N ) = 1Vm E 7'

as required.


If the spread ?TA is recoordinatized by another generalized AndrC system &Ix, isotopic or

anti-isotopic to QA, then by Foulser [3, propositions 5.3 and 5.6)

X'(3') = fX( .F*) + B (mod r)3B E I , (3.1)

and this means that the chosen "M" satisfying (*) of theorem 2 may, and often does, fail

for some other legitimate choice of A', and hence M is therefore not in general a spread invariant.

We now verify that A-conics, as opposed to A-conics, are permuted among themselves by Autnx, the translation complement of A X .

PROPOSITION 3. A-conic8.

If 7 r ~ is not a Hall spread and g E AutrA then g maps A-conics to

PROOF. By result 1 above g(X UY) = XU Y, and so g is an autotopism of type (2, y ) -+

(R(b o x), R(y)) or an anti-autotopism of type (x,y) + (R(y), R(b o x)), where in both cases a, b E .F* and R is of the following form (see Foulser [3, proposition 5.3, 5.61):

R : 3 + 3

x -+ (.(a 0 u)-1)7

for some field automorphism y of 3 = GF(2N) . Now it is easily verified that in both cases g

maps sets of type y = axp onto sets of the same type, y = u'xa'. In particular, if y = UP is additionally a 7-oval (and hence a A-conic) then so is y = a'z"', since collineations certainly

map 7 - o d s to 7-ovals. The result follows.

Note that the proposition continues to be valid for Hall spreads as well, but we shall refrain from attempting to prove this in the present article, as the proof seems rather awkward at this stage. Instead we shall deduce this in a sequel (61 by giving a combinatorial characterization of A-conics that applied to Hall spreads aa well.

We now illustrate theorem 2 above by applying it to the two-dimensional case. To put our conclusions in context we remind the reader that all two dimensional generalized Andre planes are actually Andre planes [8, corollary 12.51.


COROLLARY 4.

A a d m i h a A-conic iff q is a square.

Let A be a (generalized) AndrC plane of order q2 with kern GF(q). Then

PROOF. We consider (*) of theorem 2 with T = 2. Since A is non-Desarguesian A(F*) =

(0,l) . Now for (*) to hold we must have M E # ( N ) , and also (s - M , N ) = 1 where q = 2'

and N = 2s. Thus we require M E $ ( N ) such that (s - M,2s ) = 1, s 2 2, and this holds

precisely when s is even, and M E $ ( N ) .

The following consequences of the corollary are worth observing.

COROLLARY 5.

N

Hall plane of even order q4 possesses a translation oval.

( a ) I n any two-dimensional generalized Andrk plane of order 2 N , where

0 (mod 4) , every subspace of type y = x2 , where M E qh(N), is a A-conic. ( b ) Every M

Note however, that the work of Korchmkos [7] implies the stronger result that all Hall planes of even order admit translation ovals. When combining his result with corollary 4

we observe that certain Hall planes admit T-ovds that cannot be viewed as being A-conics. The full geometric significance of this fact will be explained in a sequel [6].

4. CLASSES OF A-CONICS

The principal aim of this section is to show that least one AndrC plane of order 2 N admits a

A-conic if N > 3 is non-prime. In particular, this means that at least one non-Desarguesian plane of order 2N > 2 admits an oval if N is non-prime. To demonstrate all this it is

sufficient to focus on (generalized) AndrC systems &A with A ( 3 ' ) = {0, 1).

DEFINITION. The generalized Andre' system QA = (F, f, 0) is called monic if A ( 7 ' ) =

{0,11.

An immediate consequence of the definition of an And& system is the following

*LEMMA 1. To each S C IT, such that 1.91 > 1, there corresponds QA such that i ( F * ) =

s U (0) . I n particular, a monac Andre' system & A of order 2 N exists whenever N = rs


and both r , s > 1; this Q x can be chosen so that kern Q x = GF(q) = GF(2'), and hence

d imQx = r .

PROOF. map such that p ( K * ) = S , with p(1) = 0. Then x = pv will do (c.f., [5, section 9.31).

Let v : F* + K* = GF(q)* be the canonical epimorphism and p : K* -t S any

REMARK 2.

[8, corollary 18.51.

If r i s pr ime, then all monic genemlized Andrd planes will be Andrd planes

We begin by showing

LEMMA 3. A monic Andre' sys tem of square order admits a A-conic.

PROOF. We can write N = 2r, i.e., s = 2. Now choose any monk AndrC system &A

such that kern Qx = GF(2') = GF(4). Now y = x2 is a A-conic, since the condition (*)

of theorem 3.2 with M = 1 becomes in our context: (s - M , N ) = (2 - 1,N) = 1 and

(0 - M , N ) = 1.

The above lemma implies that A-conics can be chosen when 4"; we now generalize this

fact by showing that A-conics exist whenever N is not square free.

LEMMA 4.

generalized AndrC plane AX (= A n d r t plane as t=pr ime) of order 2 N admits A-conics:

fact , if M E q5(N) t hen y = x2

Suppose N = st and t is a pr ime divisor of s. T h e n every t -d imens ional

in M .

18 a A-conic of A X .

PROOF. 7'. This cannot fail unless:

By theorem 3.2 we must show that if M E q5(N), then (sx(f) - M , N ) = 1Vf E

3 a prime divisor u of N such that ul(sx(f) - M ) 3 f E F'

But now u 4 sx(f), since otherwise uIM, contradicting M E q5(N). Hence u 1 s + ult +. u = t . But by hypothesis t l s , and we have a contradiction. The proposition follows.

We now consider the case when N is odd and square-free.

On the ubiquity of Denniston-type translation 29 1

LEMMA 5.

Andr t system of order 2N and prime dimension admits a A-conic.

Suppose N = p1 . . . pk is a product of k > 1 distinct primes. Then any monic

PROOF.

2N (c.f., lemma 1). Thus x : 7' + It with X ( 7 ' ) = {O, l} . Now we show that y = x z

the required A-conic when

Suppose t E { P I , . . . , p a } is the dimension of a monk Andre system & A of order

is M

where wlog we assume t = PI.. First observe that none of p 1 , . . . ,pk-l divides Md, since all

pi's are odd and d 5 2. Thus (Md, N ) = 1 or t . In the latter case, since t cannot divide two

consecutive numbers, we have:

Now using M = k f d , as defined in (i), and noting that is monic, and writing s = pl . . . pk-1,

we have (sx(f) - M , N ) = (s - M , N ) or ( - M , N ) = 1 and so by theorem 3.2 we have the

required result unless

( s - M , N ) > 1 (ii)

Since none of the P I , . . . , p k - ] divide s - M , (ii) implies t = pk divides s - M = -d. But

this is a contradiction as d 5 2 and t is an odd prime.

The collection of A-conics constructed thus far (lemmas 3-5) show

THEOREM 6. If N > 1 is any non-prime integer then there is a non-Desarguesian Andre'

plane of order 2N that admits a A-conic. I n particular, if a non-Desarguesian (generalized)

Andrk plane of order 2N exists then the plane can be chosen so that it admits a A-conic.

I f N is a prime-power we can go much further:

will admit A-conics.

all generalized AndrC planes of order 2N

PROPOSITION 7.

in every generalized Andre' plane A X , for every choice of M E $ ( N ) .

If N = p k > p is any power of the prime p , then y = zZM i s a A-conic

292 V. Jhlha, N.L. Johnson

PROOF. pJM, since M E q5(N). Thus t = 1, and now theorem 1* applies.

If (sx(f) - M, N) = t > 1 =+ p ( t and P I S , and hence we have the contradiction

We have already seen that certain two-dimensional Andrk planes do not admit A-conics. In the next section we show many other classes of T A do not admit A-conics. In particular, it will follow that the above proposition certainly fails for arbitrary composite integers N.

5. THE INDEX OF Q A

The main purpose of this section is to show how to generate families of T A that cannot contain A-conics. In a sequel [6], we shall demonstrate that the non-existence of A-conics in T A means that the plane cannot possibly be obtained by replacing a partial spread of

translation ovals in a Desarguesian spread, i.e., not all generalized Andre planes are “ovally

derived” from Desarguesian spreads, using the term derivation in the sense of Assmus and Key [l]. Our approach here is to define the “index” 1x1 of a generalized Andre plane and to show that if 1x1 is sufficiently large T X cannot admit a A-conic.

DEFINITION. If the generalized And& plane AX is coordinated b y Q l , the indez of T X is

1x1 = li(F*)l.

We now note that index 1x1 is a geometric invariant of the plane.

REMARK 1.

I , coordinatizes T A , then lA’(.F*)l = IA(.’F*)I. If, in addition to QA, the generalized Andre‘ system QA, defined b y A’ : F* +

PROOF. anti-isotopic (c.f., result 3,1), we can deduce that lA‘(F*)l = I A ( F ) l from eqn. (3.1).

Since all generalized Andre systems coordinatizing a given plane are isotopic or

Thus the index 1x1 is bounded above by T , the dimension of the corresponding quasifield QA, and we can attain the upper bound (XI = r for many (generalized) And& systems as the following remark shows.


PROPOSITION.

(1)

(2)

If &A is a nearfield then xx has indez 1x1 = r .

For any N = .w, r > 1 a proper divisor of N, there ezists Andrk systems &A such that

1x1 = r .

PROOF.

4.1.

(1) is implied by Foulser [3, Example 3.2, p. 3831, while (2) is implied by lemma

The following result implies that many, but by no means all, of the generalized Andre

systems meeting the above bound 1x1 = r cannot admit A-conics.

THEOREM 3. kern K = GF(q) = GF(2'). If the index 1x1 > d ( N ) then K A contains no A-conics.

Suppose & A = (F = GF(2N) , +, 0 ) is a generalized Andrk system, with

PROOF. Since for all f E F we have X(f) # X(g) + sX(f) f sA(g) (mod N) and so

But by hypothesis 1x1 > d ( N ) , and so the RHS of (1) contains a zero divisor of ZN,

regardless of the choice of M. Hence to each M E 1. . . N there corresponds f~ E 7 such

that ( s A ( f ~ ) - M, N ) > 1. Hence, by theorem 3.2, AX admits no A-conics.

It is important to note that planes with 1x1 = r very often do not satisfy the hypothesis

of the above theorem, essentially because the constraint 1x1 > d ( N ) implies a surprisingly

severe number-theoretic constraint, viz.,

which means, using proposition 2 that

LEMMA 4.

for some r dividing N iff d ( N ) < 6 ( N ) ; = greatest proper divisor of N. N is non-prime, a generalized Andrk system & A with 1x1 > d ( N ) can be chosen

294 V . Jha. N.L. Johnson

To get an idea of when the above inequality holds, recall that the Euler function of N =

p:' . . . p k n b (where pl < . . . < pk are primes and d ai 2 1) is given by

k

and obviously 6 ( N ) = N / p l . Hence for k > 1

1

6 ( N ) i=2 Pi

k !w < 1 *(pl -1)l-I (1- -) < 1

But as the LHS of (2) fails whenever N is a prime-power, we have shown

LEMMA 5.

T h e n 4 ( N ) < 6 ( N ) i f l

Suppose N = nfZl p;' i s the pr ime decomposition of N , wi th pl < . . . < Pk.

k 1

i=z Pi (P1 - 1)l-J (1 - - < 1

Since the lemma always holds if p = 2, provided N is not a power of 2, we obtain the

following corollary of theorem 3 above, plus the lemmas 4 and 5.

THEOREM 6. Andre' plane of order n that admits n o A - conics, unless s is a power of 2.

FOT every even square n = q z , where q = 2', there ezists a generalized

Another way of satisfying the inequality of lemma 5 is to choose highly composite numbers

N = p:'pFz . . . p';", where p l < pz . . . < pk are a sequence of consecutive primes. Now by

Merton's theorem [9, p. 2331:

1 1 n ( l - - ) = L p ln(x) +O(---) 1nZx P S Z

(where the product is over all primes 5 x ) .

Lemmas 4, 5 and the above equation, when combined with theorem 3, yields

(3)

THEOREM 7. Given any pr ime p1, by choosing k = k ( p ) su f ic ien t ly large, and choosing a

sequence of k consecutive primes p 1 , . . . , p k we obtain a generalized And& plane R of order

N = p:' . . . p z k , f o r all ( ~ 1 , . . .ak 2 1 such that A admit3 no A-conics.

On the ubiquity of Dcnniston-type translation 295

Thus we have established, in the previous two theorems, that there do not exist A-conics in

at least one generalized Andrk plane of order 2N, whenever the non-prime integer N is even,

but not a power of 2, or whenever N has sufficiently many prime factors. But at, the other

extreme, when N is a prime power, we have Seen (proposition 4.7) precisely the opposite

occurs: all generalized Andrk systems of prime power order admit A-conics.

We end the section by summarizing the extent to which the index 1x1 provides information

about the non-existence of A-conics. Thus we have seen in theorem 3

1x1 > # ( N ) =+ does not admit A-conics (4)

and we claim that the above bound is sharp, in the sense that replacing 1x1 > # ( N ) in ’ eqn.

(4) by 1x1 2 d ( N ) falsifies the equation in many cases, e.g., whenever the non-prime N = 2‘,

for k 2 2. To see this observe that N = 2k =+ # ( N ) = 2k-’, and now if we let r = 2‘-’

then we can choose 1x1 = r (c.f., lemma 4.1), and admits A-conics by proposition 4.7.

6. CONCLTJDING REMARKS

We have shown that in the class of generalized Andrk planes there often exists certain trans-

lation ovals called A-conics that are natural analogues of translation ovals in Desarguesian

planes. These A-conics, although defined algebraically, have been shown to be geometric

invariants in the non-Hall case, in the sense that any collineation in the translation com-

plement maps A-conics to A-conics. In a sequel we obtain what might be described as

a combinatorial characterization of the planes admitting A-conic by showing that they are

precisely the planes that are obtainable by replacing an “oval cover’’ of a Desarguesian plane

in the sense indicated by Assmus and Key [l, section 51.

Here we have mainly considered the question of when a generalized Andrk plane of order 2 N

admits a A-conic. It is well-known that generalized planes of order 2 N exist precisely when

N is non-prime. We have shown that at least one such AndrC plane of order 2N admits a

A-conic (i.e., a normalized version of a A-conic), but that if N is even and not a power of 2

then, in addition, some Andrk plane does not admit any A-conic; this can also be arranged

for many other N , e.g., when N has sufficiently many distinct prime factors. On the other


hand, if N is a power of a prime then every generalized And& plane of order 2 N admits A-conics.

Besides the value of N , the index 1x1 also has a bearing on whether or not T A admits A-conics. Thus we showed that if 1x1 > 4 ( N ) then AX cannot admit A-conics, while this conclusion cannot be drawn if 1x1 = d ( N ) , since N = 2k implies 1x1 = d ( N ) , and all planes T A of order

2N admit A-conics.

The most important question concerning A-conics that we have not yet considered, concerns the effects of the kern GF(2’) and dimension r on the existence of A-conics, in planes of order n = 2“. When r = 2 then K A admits a A-conic precisely when 3 is even (see corollary 3.4), and if 3 and T are powers of the same prime then a A-conic is always guaranteed (proposition 4.7). But we are as yet unable to decide in general the effects of 3 and r on the existence and non-existence of A-conics.

REFERENCES

1.

2.

3.

4.

5.

6.

7.

8.

9.

E. Assumus and J. D. Key, Translation planes and derivation sets, J . Geometry 37 (1990), 3-16.

R. H. F. Denniston, Some non-Desarguesian translation ovals, ATS Combinatoria 7 (1979), 221-222.

D. A. Foulser, A generalization of Andrh’s systems, Math. Zeitschr. 100 (1967), 380- 395.

D. R. Hughes and F. C. Piper, “Projective Planes,” Springer Graduate Texts in Math- ematics, New York, 1973.

J. W. P. Hirschfeld, “Projective Geometries over Finite Fields,” Oxford University Press, Oxford, 1979.

V. Jha and N. L. Johnson, A characterisation of spreads ovally derived from Desargue- sian spreads, to appear Combinatorica.

G. Korchmkos, Inherited arcs in finite affine planes, J. Comb. Theory Ser. A (1986), 140- 143.

H. Luneburg, “Translation Planes,” Springer-Verlag, New York, 1980.

H. E. Rose, “A course in number theory,” Oxford University Press, New York, 1988.

Combinatorics '90 A. Barlotti ct al. (Editors) 0 1992 Elsevicr Science Publishers B.V. All rights reserved.

TRANSLATION PLANES

AND

RELATED COMBINATORIAL STRUCTURES

Norman L. Johnson

1. INTRODUCTION In Ravello, in 1988, Beutelspacher spoke of "Benimino Segre's point of view"

which is to It. . . characterize beautiful ( = classical structures by their combinatorial properties (see [8] p.22)."

Similarly, we may ask to realize a given combinatorial structure within a well known geometric structure or conversely. Or we might like to know when one combinatorial structure is equivalent to another and how the two structures interrelate.

These latter statements are, of course, simply variations on Segre's point of

The main theme then will be to see how two ostensible unrelated but

I am, of course, intentionally vague as to what might be considered

view but it will be these variations which will help unify and focus this lecture.

interesting geometric or combinatorial objects can be interrelated.

"geometric", "combinatorial" or "interesting". However, since this allows me more flexibility and latitude, I will not attempt a definition.

This personal viewpoint will usually seek to see how translation planes or nets relate to other combinatorial or geometric objects. I suppose one could say that this

291

298 N.L . Johnson

might be a "view from a plane". Certainly, I am making no claims of completeness and anyone in the audience may easily discover that there are many fascinating situations of related interest about which I shall say absolutely nothing.

Since I am interested in seeing how a translation plane might be related to other combinatorial objects, we begin with some discussion on planes.

We recall that a translation plane is an affine plane that admits a transitive group of translations. By the fundamental results of Andre' [l], given a translation plane, there is an associated vector space Vof even dimension 2 d over a skewfield K such that the points of the affine plane are the vectors of Vand the lines of the plane are translates of certain mutually disjoint d-dimensional K- subspaces called the components. The set of components is called the spread of the plane. The original translation group becomes the vector group of V and the skewfield K is called the kernel if there is no containing skewfield L such that the plane and the components are &spaces.

the stabilizer of the associated zero vector by the translation group. The stabilizer subgroup is a subgoup of I'L(2d,K) and is called the translation complement. The intersection of the translation complement with GL(2d,K) is called the linear translation complement.

One viewpoint as to the study of translation planes is to consider them under the umbrella of group representation theory and analyze the possible spread invariant Gmodules for prescribed groups G. And, certain powerful results may be obtained in this manner. However, as we have said we shall restrict ourselves to connections to other combinatorial objects.

The collineation group of a translation plane is thus a semi-direct product of

2. Flocks of quadric sets.

sets and related structures. In Ravello, J.A. Thas [52] gave a survey of recent results of flocks of quadric

It is hoped that the following will not replicate large sections of Professor

We recall that Thas's lecture.

(2.1) a partial flock #',, d E , PtCof a hyperbolic quadric H, elliptic quadric E, or

quadratic cone C i n PG(3,q) respectively is a nonempty set o f t mutually disjoint conics which lie in H, E, or qminus the vertex) respectively.

Furthermore, for d,, 15 t 5 q+1,

Translation planes and related combinatorial structures 299

for f l E 1 5 t S q-ll and

for ptcl 1 3 t 5 q.

A flock is a partial flock for which t is the maximal attainable value in each

The term flock goes back to Dembowski and these objects have been studied respective situation.

extensively by Thas since the early 70’s. Some of the connections between partial flocks and partial spreads and hence with translation planes can be made apparent by the construction due essentially to Thas for flocks but called the Thas-Walker construction as apparently Walker [53] was the first to consider these connections.

The Thas-Walker construction

t Let P be a partial flock of a hyperbolic quadric, elliptic quadric, or quadratic cone in PG(3,q). Let Xt denote the set o f t planes which contain the t conics of Pt. Embedd the S-dimensional projective space in a 5dimensional projective space so

t * that the quadric set in question lies on the hyperbolic(K1ein) quadric. Let (C ) denote the set of planes polar to the planes of Xt. Then the conics of (C ) share exactly 2,0,or 1 points depending on whether the partial flock is hyperbolic, elliptic or conical.

the 54imensional projective space and lines of the 34imensional projective space, we obtain respectively (q-1)t + 2, (q+l)t, or qt + 1 mutually skew lines in PG(3,q) or equivalently this number of mutually disjoint 2dimensional subspaces in the associated 4-dimensional vector space over GF(q).

2 Thus, we obtain a partial spread of order q and cardinality (q-l)t +2, (q+l)t, or qt + 1 respectively from pt.

Now we point out that i t turns out that associated with these partial spreads are certain collineation groups of orders q-1, q+ l , and q respectively acting in such a way so that partial spreads admitting collineation groups of this type conversely provide partial flocks.

Note that partial spreads of order q and degree k are equivalent to translation nets of the same order and degree. Thus, we may speak of the collineation group of a partial spread. We shall normally not make a distinction between the partial spreads in the projective space, the associated set of mutually disjoint subspaces in the corresponding vector space and/or the translation net

t *

Now using the Klein correspondence between points of the Klein quadric in

2

300 N.L. Johnson

associated with either of the former.

(2.2) (Gevaert, Johnson [15], Johnson (see [19], [23], [25])

a rklimensional vector space V4 over GF(q).

(1) If S admits an affine homology group of order q-1 one of whose component orbits together with the axis and coaxis is a regulus(in the projective space) th of a hyperbolic quadric in PG(3,q).

union the zero vector are subspaces then the reverse Thas-Walker comruction applies to produce a partial flock of an elliptic quadric in PG(3,q).

together with the axis is a regulne then the reverse Thas-Walker construction applies to produce a partial flock of a quadratic cone.

mention of the collineation groups by assuming certain configurations of reguli which make up the structure of the partial spread. We shall make more use of this as we go along. Also, recall that partial spreads of q + 1 mutually disjoint 24imensional subspaces are spreads and are equivalent to(and identified with) translation planes of order q and kernel containing K g GF(q) by taking as lines the translates of the elements (called components) of the spread.

Let S denote a partial spread of mutually disjoint 24mensional subspaces of

he reverse Than-Walker construction applies to produce a partial flock

2 (2) If S admits a collineation group of order q -1 whose nontrivial orbits

(3) If S admits an elation group of order q such that some component orbit

Of course, the statement of the above result could be phrased without the

2

2

We first consider

3. Flocks and partial flocks of hyperbolic quadrics. As was mentioned above in (2.2)(1), a flock of a hyperbolic quadric in PG(3,q)

2 is equivalent to a translation plane of order q and kernel containing K g GF(q) which admits a homology group of order q-1 one of whose component orbits together with the axis and coaxis is a regulus in PG(3,K).

GF(q) to admit such homology groups acting in this way are the Desarguesian and 2 2 regular nearfield planes and three of the irregular nearfield planes (of orders 11 ,23 ,

and 59 ). Of course, each of these planes give rise to a flock of a hyperbolic quadric in PG(3,q). The Desarguesian plane produces a flock each of whose planes contain one of the conics in question share a line and is called a linear flock. The regular

2 The only known translation planes of order q and kernel containing K

2

Translation planes and related combinatorial structures 30 1

nearfield planes give rise to the flock of Thas-Walker constructed geometrically by Thas (see [47]). It was originally conjectured that this might be(was) the complete set of flocks of hyperbolic quadrics. However, once the connections with the groups became apparent, the three examples coming from the irregular nearfields were discovered. This was done independently by Bader [2] and the author [19] by more-or-less the same methods and geometrically for order 11 but without the direct association of the irregular nearfields by Baker and Ebert [5].

of flocks of hyperbolic quadrics when translated into the language of translation planes forced the planes to be of a certain type of plane called Bol planes. [33] , [34],and [35] has made an extensive study of Bol planes and it follows that, with a few possible exceptions on order, Bol planes of finite order are nearfield planes. Putting these pieces together, one obtains the following result:

2

Bader and Lunardon [3] notices that certain results of Thas [48 ] on the groups

Kallaher

(3.1) (Thas [47] - [52] , Bader- Lunardon [3].

The flocks of hyperbolic quadrics are precisely the linear flock, the flock of Thas-Walker, and the three irregular flocks which may be obtained from the

2 2 2 irregular nearfield planes of orders 11 ,23 , and 59 .

Note that the above result has the particularly interesting corollary:

2 (3.2) Corollary. If a translation plane of order q and kernel containing K : GF(q)

admits an aftine homology group of order q-1 one of whose component orbits together with the axis and coaxis form a regulus then the plane is a n e d e l d plane.

Now given this result, it might appear that there is nothing more to sudy with regard to flocks of hyperbolic quadrics. However, there are the partial flocks of hyperbolic quadrics and the associated partial spreads which admit homology groups and which correspond via the Thas-Walker correspondence. And , there are numberous problems in translation planes which are associated with this classification result.

Partial hyperbolic flocks

conica of a hyperbolic quadric(or elliptic quadric or quadratic cone) be extended to a partial flock with t+l conics? Similarly, when can a partial spread corresponding to

We now consider a classical type problem: When can a partial flock with t

302 N.L. Johnson

a partial flock be extended to a partial spread of larger degree?

hyperbolic quadric has t = q conics or one less than required for a flock(such a partial flock is said to have deficiency one). Considering the associated partial spread of (q - l ) t + 2 = (q - l )q -+ 2 2-dimensional subspaces within the 44imensional vector space V4, we note that the homology group acting on the vector space has t = q

orbits of components which together with the axis and coaxis each form reguli. It turns out that there are q+ l 24imensional subspaces which are mutually disjoint from (q-l)q components of the original partial spread but which form a cover for the axis and the coaxis of the homology group. One may visual the group not as a homology group but as a Baer group(fixes a Baer subplane pointwise) where the axis of the homology group is considered as a subplane of a new partial spread which extends the original partial spread minus the axis and coaxis. The author exploits this viewpoint in recent work and shows that the assumption of a Baer group or order q-1 is the most natural when considering partial flocks of hyperbolic quadrics.

Consider the essentially maximal type situation when the partial flock of a

(3.3) (Johnson [21], [23],[25]).

4-dimensional vector space V4 over K

order q-1 then there is a partial flock of a hyperbolic quadric with t conics constructed by the t nontrivial orbits of components of S together with the Baer subplane r0 which is fixed pointwise by B

with the zero vector which is B

Let S be a partial spread of mutually disjoint 2-dimensional subspaces of a GF(q). If S admits a Baer group BqVl of

and the unique Baer subplane incident s-1 invariant and shares its parallel classes with r0.

Conversely, a partial flock of t # q+l conics of a hyperbolic quadric in PG(3,q) gives rise to a partial spread Sof degree (t+l)(q-1) + 2 admitting a Baer group of order q-1.

with t+l conics if and only if the partial spread whose parallel classes are precisely those of the Baer subplane pointwise fixed by the Baer group defines a regulus in the associated projective geometry.

q-1

And, the partial flock with t # q+l conics may be extended to a partial flock

So, in particular, a partial flock of a hyperbolic quadric of deficiency one is 2 2 equivalent to a partial spread of order q and degree (q+l)(q-1) + 2 = q + 1

admitting a Baer collineation group of order q-1 or equivalently to a translation


2 plane of order q and kernel containing K y GF(q) which admits a Baer group of order q-1. flock extending the original partial flock of deficiency one.

one in PG(3,q) may be extended to flocks as there are two known counterexamples pointed out by the author in [21] and with R. Pomareda in [30].

It is not always the case that partial flocks of hyperbolic quadrics of deficiency

4. Flocks and partial flocks of elliptic quadrics.

Let PtE be a partial flock o f t conics of an elliptic conic in PG(3,q). By the n

Thas-Walker construction, there is an associated partial spread of order q' and degree (q+l)t.

x.

corresponding to the partial flock is a set o f t disjoint opposite reguli of a set o f t mutually disjoint reguli in X.

(4.1) Partial flocks of elliptic quadrics in PG(3,q) are equivalent to(via the Thas-Walker construction) translation planes of order q and kernel containing GF(q) which are constructed by multiple derivation in a Desarguesian plane.

But, note that the elliptic quadric itself corresponds to a Desarguesian spread

Going thru the Thas-Walker construction shows that the partial spread

Thus, we obtain the following note:

2

If the partial flock is a flock then Orr [39] for q odd and Thas [49] for q even show that the flock is linear(the associated planes containing the conics share a line in PG(3,q)). Phrasing this result using (4.1):

(4.2) (On [39], Thas [49]) 2

Desarguesian affine plane of order q by the derivation of q-1 mutually disjoint derivable nets is Desarguesian.

cannot be extended to flocks(i.e. maximal partial flocks) if and only if there are non 2 Desarguesian translation planes of order q which can be constructed from a

Desarguesian plane C by the multiple derivation of q-2 mutually disjoint derivable nets in C.

Any translation plane of order q which may be constructed from a 2

Notice that there are partial flocks of elliptic quadrics of deficiency one which

This actually occurs. In [39], Orr gives an example of a maximal partial flock

304 N.L. Johnson

of seven conics of an elliptic quadric in PG(3,9). If 7r is any translation plane which is multiply derived from a Desarguesian

plane of order q by the derivation of t mutually disjoint derivable nets then T admits a collineation group of order q + l which essentially defines the replacement nets of opposite reguli by the orbits of Baer subplanes of the reguli.

2

5. Flocks and partial flocks of quadratic cones.

translation planes of order q and kernel contaning K g GF(q) whose spread is the union of q reguli that mutually share a line. Conversely,

We have noted that a flock of a quadratic cone in PG(3,q) produces a 2

(5.1) (Gevaert, Johnson,Thas [IS])

spread is the union of q re@ sharing a line then there is a corresponding flock of a quadratic cone.

reguli mutually sharing a line and it follows that such partial spreads are equivalent to partial flocks of quadratic cones.

2 If r is a translation plane of order q and kernel containing K : GF(q) whose

Actually, (5.1) may be stated more generally for partial spreads comprised o f t

Generalized Quadrangles There is a bonus for studying flocks of quadratic cones or equivalently

translation planes of order q and kernel containing GF(q) admitting an elation group of order q one of whose component orbits together with the axis is a regulus. This is that each such object creates a generalized quadrangle of type (q ,q).

In [36], Kantor gives a construction of (s,t)-generalized quadrangles by the 2 construction of certain groups of order s t and by defining the points and lines of the

quadrangle to be certain cosets of specified subgroups of the group together with certain other symbols satisfying specific incidence conditions. In particular, for

which may be parameterized by two functions f and g acting on GF(q) such that the conditions for the existence of the generalized quadrangle can be phrased in terms of the existence of certain permutation polynomials involving the functions f and g.

In [47], Thas notices that the conditions on these functions f and g mentioned in the above paragraph are equivalent to conditions of polynomial functions F,G on GF(q) which may be used to construct flocks of quadratic cones. That is,

2

2

(q 2 ,q)-generalized quadrangles, a particular group of order q 5 can be determined


(5.2) ( T b 1471) Each flock of a quadratic cone in PG(3,q) constructs a generalized

quadrangle of type (q ,q). Conversely, a generalized quadrangle of type (q ,q) contructed via the coset geometry of Kantor constructs a flock of a quadratic cone.

2 2

It follows from Gevaert and the author [15] that isomorphic flocks of quadratic cones in PG(3,q) produce isomorphic translation planes of the type mentioned above and conversely any two isomorphic translation planes of the prescribed type correspond, in turn, to isomorphic flocks of a quadratic cone. However, no such correspondence is generally valid when comparing isomorphisms of generalized quadrangles and isomorphisms of flocks of quadratic cones. In fact, Payne and Rogers [45] have recently shown that is it possible that two nonisomorphic translation planes can produce isomorphic generalized quadrangles.

Partial flocks and Baer groups

We have mentioned how partial flocks of deficiency one are of particular interest when considering hyperbolic quadrics. For cones, similarly, we have:

(5.3) (Johnson [21])

a Baer collineation group B of order q is equivalent to a partial flock of a quadratic

cone in PG(3,q) of deficiency one. The partial flock may be extended to a flock if and only if the partial spread defined by the pardel classes of the Baer subplane which is pointwise fixed by B is derivable.

2 A translation plane of order q and kernel containing K : GF(q) which admits

9

q Before leaving the subject of conical flocks, we mention the recent construction

of Bader, Lunardon, and Thas by which a given flock of odd order q produces q related flocks. This procedure is called derivation in [4] but is not to be confused with the derivation of the translation plane associated with the flock. The construction method may be easily seen to apply more generally to partial flocks of quadratic cones as observed by the author in [28].

The construction of Bader,Lunardon and Thas

Let q be odd and Z3 E PG(3,q) and contained in Z4 g PG(4,q) in such a way so

306 N.L. Johnson

that there is a quadric Q4 in X4 satisfying the property that Z3 n Q4 is a quadratic

cone Q with vertex po such that pol = X3 where I denotes the polarity of X4

associated with Q4. Now let PtCbe a partial flock of the quadratic cone Q3 in X3

where P c= { Ci I i = 1,2,. . ., t}. Let T~ denote the plane of X3 = Ho which

contains the conic Ci, for i = 1,2, . . ., t. Then

form piL = H ~ , for i = 0,1,2,, . . . , t.

3

t

= po + pi where pi c Q4. Now

Then, for each i = 1,2,. . . , t, {Hi n H . I j = 1,2, . . ,t for j # i} U {ri} defines 1

a partial flock of the quadratic cone Q4 n Hi with vertex pi(see Bader, Lunardon,Thas

[4] and the extension to partial flocks in [28]).

flock with t conics of a quadratic cone in PG(3,q) for q odd. The t partial flocks obtained in this way are called derived partial flocks or derivations.

Thus, there are t partial flocks with t conics constructed from each partial

BLT - S&S

Corresponding to a flock of a quadratic cone of odd order q, there is a set of q+l points in PG(4,q) with the following property: Let 0 be an associated quadric. Given any three points of the set, there does not exist a point of 0 which is perpendicular to all three given points. Such sets are called BLT-sets after Bader, Lunardon, and Thas [4] who prove that such sets characterize conical flocks of odd order(Kantor [38] actually coined the term BLT-sets).

Combining (5.3) with this construction:

(5.4) (Johnson (281)

admits a Baer collineation group of order q. Then there are q-1 associated translation planes which admit a B a a group of order q and which may be constmcted from R by passing to the corresponding partial flock of a quadratic cone of deficiency one, applying the construction of Bader, Lunardon, and Thas, and then associating a translation plane admitting a B u r group of order q with each of the q - 1 derived partial flocks of deficiency one.

2 Let R be a translation plane of order q and kernel containing K ; GF(q) that

Actually, BLT-sets are used by Payne and Thas [46] to show that there are no maximal partial conical flocks of deficiency one and odd order. The same result is


shown by Payne and Thas for even order but with a completely distinct argument.

obtain: Taking the union of the work on Baer groups and the above remarks, we

(5.5) Corollary(Johnson [28], Payne, Thas [Ire]). Flocks of quadratic cone8 of order q are equivalent to translation planes of 2 order q and kernel containing GF(q) which admit a Baer group of order q. The construction of Bader, Lunardon, and Thas may be given a coordinate

setting so that the functions f and g producing the derivations may be determined. However, the new functions are too complex to use in determining whether the constructed flocks, translation planes, or generalized quadrangles are isomorphic. Most of the work in Bader, Lunardon, and Thas [4] and also in the author’s work [28] is devoted to showing that various of the known flocks produce nonisomorphic flocks(also see Payne and Thas [46]).

generalized quadrangles obtained from the construction process of Bader, Lunardon, and Thas are, in fact, always mutually isomorphic to the original generalized quadrangle. spreads may all be embedded in the associated generalized quadrangle providing some insight as to explaining why generalized quadrangles of this type produce spreads and/ or conical flocks. Payne and Rogers [45] also note how isomorphisms between spreads may be detected via certain collineations of the associated generalized quadrangle.

An interesting application of the classification of simple groups may be seen in trying to determine whether semifield or likeable flocks produce nonisomorphic flocks under this construction process. A flock is said to a semifield or likeable flock if and only if the corresponding translation is semifield or likeable respectively. To ovoid confusion with the ordinary derivation process of affine planes, we use the term skeleton of the translation plane to describe the set of q+ l translation planes containing the original plane and the q constructed translation planes.

nonisomorphic conical flocks.

However, Payne and Rogers [45] have recently shown that the potentially new

Furthermore, Kantor [38] has recently shown how the associated

The follow classification result shows that there are potentially many mutually

(5.6) (Johnson [28](for the likeable part), Johon, Lunardon, Wilke [32]) Let F be a flock of a quadratic cone in PG(3,q) for q odd and let ST denote the

308 N.L. Johnson

skeleton of the translation plane R associated with F. (1) Assume that F admits a permutation group in PGL(4,q) which leave8 the

quadratic cone and the space PG(3,q) invariant and acts transitively on the obnics of the flock. Then F is either a semifield flock or a likeable flock.

(2) Assume the conditions of (1). If further, there is a translation plane # R in the skeleton SR of T which is isomorphic to R then r is either the likeable plane of

Walker or a &field plane of Knuth and flock type.

in the 4-dimensional projective space E4 and acting doubly transitively on the BLT The condition of (2) actually produces a permutation group in PI'L(5,q) acting

set of q+l points {pi I i = 0,1,2, ...,q} of the space Z4 where pi corresponds to a

constucted flock for i # o and po is the vertex of the original quadratic cone.

By the classification theorem for simple groups, the socle N is either elementary Abelian or is simple of prescribed type and degree(for example, see Cameron [lo]). It turns out that the subgroup of N which fixes the point po acts as

a quotient group of a collineation group of the original translation plane T. By analysis of the imposed action of the corresponding collineation group of the translation plane, it may be seen that the flock and/or the translation plane may be described completely under these assumptions.

within the skeletons of various of the known translation planes of flock type. Also see Payne and Thas [45] or Bader, Lunardon, Thas [4] where most of the above ideas originate.

But, also note that the above result gives some new flocks and/or planes

6. Nestsofreguli.

is called a t -na t . Note that this is a generalization of the chains of Bruen [9] which are (q+3)/2-nests for q odd.

To see how t-nests might relate to flocks of quadric sets, recall that in a conical flock, the Thas-Walker construction produces a set of q-reguli which share a line. Suppose that the corresponding translation plane is connected to the associated Desarguesian plane on the same affine points by some sort of net replacement. Then we would expect that there is an associated elation group of order q in the Desarguesian plane which becomes the elation group of the translation plane. Let 7r0

be a Baer subplane in the Desarguesian plane which may be embedded in a regulus

a set of t-reguli in PG(3,q) that determine a partial spread of degree t(q+l)/2


net(so that ro is a line of the opposite regulus) and which does not contain the axis of

the elation group E. Then it is at least possible that the E-image set of this regulus is a q-nest.

way. For example, the translation planes of Baker and Ebert[6] constructed using certain q-nests in Desarguesian planes may be seen to correspond to certain flocks of quadratic cones. This was noted by Payne in [42] and [43].

More generally, the construction of t-nests is interesting in its own right. In particular, we are interested in the t-nests that can occur in translation planes and which correspond to replaceable nets which then give rise to corresponding translation planes. The constructions of (q-1) and q- nests have been given by Baker and Ebert([6],(7]) and by Ebert [12] for (q+l)-nests.

t(q+1)/2, find a Baer subplane whose orbit under this group is exactly t(q+1)/2 and hope that the net defined by the Baer subplane and its images has degree t(q+1)/2. In this case, the net becomes a replaceable t-nest.

Since inevitably there is an associated central collineation group in each of the groups described above, it becomes an interesting problem to determine conditions in translation planes where certain assumed collineation groups force the translation plane to be related to a t-nest for its construction.

For example, for q-1 - nests, we have: (6.1) Johnson [26])

2 Let T be a translation plane of order q and kernel containing K g~ GF(q). If T

admits a cyclic collineation group of order q -1 in the linear translation complement which has an orbit of components of length q-1 then R is either Desarguesian, the Hall plane(derived from the Desarguesian), the regular nearfield plane, derived from the reguluar nearfeld plane, constructed from a Desarguesian plane by replacement of a (q-1)-nest of reguli, or derived from the nest replaced plane of (v).

homology group or Baer group of order (q-1)/(2,q-l).

This actually happens and some of the conical flocks may be constructed in ths

The general idea is to find groups in the Desargesian plane of order roughly

2

Furthermore, in each of the situations, the translation plane either admits a

7. Projective planes with exactly one incident point4ine transitivity. 2 Let r1 be a translation plane of order q and kernel GF(q). It was noted by

Ostrom that if rl is considered projectively and dualized then by deleting as a line

310 N.L . Johnson

any point incident with the original line at infinity of r1 then the constructed dual

translation plane 7r2 permitted derivation. That is, there is a set of Baer subplanes of

7r2 with the same general class of parallel classes such that when redefining these as

lines along with the lines of T~ in the parallel classes not involving the Baer

subplanes, another affine plane 7r3 is determined called the plane derived from 7r2.

If the line deleted to form 7r2 as an infinite point (a) of 7r1 is the center of an

elation group of order q in 7r1 which leaves one of the indicated Baer subplanes

invariant, then 7r3 inherits from 7rl a translation group of order q in which there is

exactly one incident point-line transitivity. If 7r1 is not a semifield plane then r3

turns out to be of Lenz-Barlotti class 11-1; 7r3 contains exactly this point-line

transitivity.

3

Now how does this connect with flocks or nests? Well, if there is a flock of a quadratic cone of order q, there is an associated

2 translation plane of order q and kernel GF(q) which admits q - reguli that share a line and which admits an elation group of order q. This group must leave invariant each of the Baer subplanes (lines of the oppositive reguli) of the reguli. By the above remarks there is an associated projective plane of Lenz-Barlotti class 11-1 provided the translation plane is not a semi-field.

If an elation group E of order q acts on a translation plane of order q with kernel containing K 2 GF(q) then E leaves exactly i - q + 1 24imensional Kaubspaces invariant where i = 0,1,2, or q+l(see [29]). We shall call a translation plane of order q and kernel containing K : GF(q) which admits such an elation group E a translation plane of elation type i.

For example, in [40], Ostrom shows that if one starts with the Luneburg-Tits translation planes of order q the process described above produces an affine plane of Lenz-Barlotti class 11-l(exact1y one incident point-line transitivity). There is an elation group in the Luneburg-Tits planes which makes the translation plane of elation type 1.

there are at least i projective planes of Lenz-Barlotti class 11-1 which may be constructed as in the previous paragraphs by various derivations of various affine dual translation planes of the dual of the projective extension of the original

2

2

2

Similarly, it turns out that given a translation plane of elation type of type i,

Translation planes and related cornbinatorial structures 311

translation plane. In [29], the author has made a general study of translation planes of elation

type i and has determined many new projective planes of Lenz-Barlotti class 11-1. Note that any flock of a quadratic cone in PG(3,q) gives rise to a translation plane of elation type q + l which, in turn, produces at least q+l projective planes of Lenz-Barlotti class 11-1. There is a converse construction from such projective planes of Lenz-Barlotti class II-l(cal1ed type q + l - planes).

Summarizing some of the previously mentioned results. we have:

(7.1) (see Johneon [29] for (iv) - the other equivalences are due to the work of Bader, Gevaert, Kantor, Lunardon ,Payne, Thas, and the author)

The following incidence structures are equivalent: (i) flocks of quadratic cones in PG(3,q), (ii) translation planes of order q each of whose spreads is the union of reguli

(iii) translation planes of order q and kernel GF(q) that admit a Baer group

(iv) translation planes of order q and kernel K: GF(q) admitting an elation

(v) genera l id quadrangles of Kantor type (q2,q) and (vi) projective planes of Lenz-Barlotti class 11-1 of order q and type

q+l(provided the corresponding flock is not a semifield flock). (vii) If q is odd then BLT - sets are equivalent to any of the structures listed

above.

2

sharing a line,

of order q,

group one of whose component orbits union the axis is a regulus in PG(3,K).

2

2

2

8. Derivable nets and projective spaces.

Ravello - a short discussion of derivable nets. Finally, let me end my remarks here in Gaeta with that which I began in

In section 7, we have discussed derivation briefly. Of course, the replacement of the lines of the regulus by the lines of the opposite regulus is essentially derivation. However, one does not know a priori that every derivation can be realized in this way. Furthermore, in each of the previous sections, derivation is ubiquituous as there as always reguli lurking about. So, there is a certain symmetry obtained in discussing derivation in general.

Cofman, in 1975 (see the references in [22], [27]) showed that corresponding to every derivable affine plane there is a Sdimensional affine space associated in such a

312 N.L. Johnson

way so that it could be seem that the Baer subplanes involved in the derivation procedure are always Desarguesian.

In fact, Cofman’s ideas give the direction in which a structure theory for arbitrary derivable nets may be found. Actually, an extension to the corresponding projective space with the appropriate interpretation provides the desired structure result.

Let X denote a 3dimensional projective space either finite or infinite. Let N

denote a fixed line of X. Define an associated combinatorial structure R as follows: lines, points, parallel classes and Baer subplanes of R are defined so be the points of 6 - N , lines of X skew to N, planes of X which contain N, and planes of Z which do not contain N respectively with incidence in R that whch is inherited from x.

Then it turns out that R is a derivable net and conversely, given a derivable net, a projective space may be constructed so that the derivable net and the projective space are interconnected exactly as in the previous paragraph.

(8.l)(Cofman(see reference [22]) , Johnson [22], [27])

structuxe may be essentially embedded in the other.

speaks, we shall be content to conclude with this example.

Derivable nets and 34imensional projective spaces are equivalent; each

As this is exactly the type of characterization of which Segre’s point of view


References

J . Andre'. Uber nicht-Desarguessche Ebenen mit Transitiver Translations Gruppe. Math. Z. 60(1954),156-186.

L. Bader. Some new examples of 5ocks of Q (3,q). Geom Ded.(to appear).

L. Bader, G. Lunardon. On the flocks of quadratic cones. Combinatorics '88(to appear).

L. Bader, G. Lunardon, J.A. Thas. Derivation of flocks of quadratic cones. Preprint .

R.D. Baker, G.L. Ebert. A nonlinear flock in the Minkowski plane of order 11. Preprint .

R.D. Baker, G.L Ebert. A new class of translation planes. Preprint.

R.D. Baker, G.L. Ebert. Nests of size (q-1) and another family of translation planes. J. London Math. SOC. (to appear).

A. Beutelspacher. Geometry - from B. Segre's point of view. Variations on a classical theme. Combinatorics '88 (to appear).

A.A. Bruen. Inversive geometry and some translation planes. I. Geom. Ded.

P.J. Cameron. Finite permutation groups and finite simple groups. Bull. London. Math. SOC. 13 (1981), 1-22.

P. Dembowski. Finite Geometries. Springer-Verlag, New York, Inc. 1968.

G.L. Ebert. Spreads admitting regular elliptic covers. Preprint.

G.L. Ebert. Some nonreplaceable nests. Combnatorics '88(to appear).

H. Gevaert, N.L. Johnson. On maximal partial spreads in PG(3,q) of cardinalities q -1 + 1, q -q + 2. ARS Comb., 26 (1988), 191-196.

H. Gevaert, N.L. Johnson. Flocks of quadratic cones, generalized quadrangles, and translation planks. Geom. Ded. 27, no. 3 (1988), 301-317.

H. Gevert, N.L. Johnson, J.A. Thas. Spreads covered by reguli. Simon Stevin. Vol. 62 (1988), 51-62.

V. Jha N.L. Johnson. Nests of reguli and flocks of quadratic cones. Simon St evin (to appear).

N.L. Johnson. Semifield flocks of quadratic cones. Simon Stevin, 61, n o . 3 4 (1987),

N.L. Johnson. Flocks of hyperbolic quadrics and translation planes admitting affine homologies. J . Geom. vol. 35 (1989), 50-73.

+

7(1977), 81-98.

2 2

313-326.

314 N.L. Johnson

1261

1271

1281

1291

1301

1311

N.L. Johnson. Derivation is a polarity. J . Geom., vol. 18,no. 1(1989), 97-102.

N.L. Johnson. Translation planes admitting Baer groups and partial flocks of quadric sets. Simon Stevin, vol. 63, no.2 (1989), 163-167.

N.L. Johnson. Derivable nets and 3-dimensional projective spaces. Abh. d. Math. Sem(Hamburg) 58 (1988), 245-253.

N.L. Johnson. Flocks and partial flocks of quadric sets. Lincoln Conf. AMS, 1987(to appear).

N.L. Johnson. Derivation. Combinatorics '88. Anals of Math. ( to appear).

N.L. Johnson. Problems in translation planes related to flocks of quadric sets. IX-Elam, Santigo '88(to appear).

N.L. Johnson. Nest replaceable translation planes. J . Geom. vol 36( 1989), 49-62.

N.L. Johnson. Derivable nets and 3-dimensional projective spaces. 11. The structure. Archiv d. Math. (to appear).

N.L. Johnson. Derivation of partial flocks of quadratic cones. Rend. d. Roma(to appear). N.L. Johnson. Ovoids and translation planes revisited. Geom Ded. (to appear).

N.L. Johnson, R. Pomareda. A maximal partial flock of deficiency one of the hyperbolic quadric in PG(3,9). Simon Stevin (to appear).

N.L. Johnson R. Pomareda. Andre' planes and nests of regli. Geom Ded.(31(1989), 245-260.

N.L. Johnson, G. Lunardon, F.W. Wilke. Semifield skeletons of conical flocks. J. Geom. ( t o appear).

M.J. Kallaher. Projective planes over Bol quasifield-fields. Math. Z. 109, 53-65( 1969) , 5345 .

M.J. Kallaher. A note on Bol projective planes. Arch. d. Math. 20(1969), 329-333.

M.J. Kallaher. A note on finite Bol quasi-fields. Arch. d. Math. 23(1972),164-166. n

[36] W.M. Kantor. Some generalized quadrangles with parameters (qz,q). Mat. Z. 192( 1986), 45-50.

[37]

[38]

[39]

W.M. Kantor. Generalized quadrangles and translation planes. Alg., Groups, Geom. 3 (1985), 313-322.

W.M. Kantor. Notes on generalized quadrangles, flocks and BLT sets. Preprint.

W.F. Orr. The Miquelian Inversive plane IP(q) and the associated projective planes. Thesis, Univ. Wisconsin, 1973.


[40] T.G. Ostrom. The dual Liineburg planes. Math. 2. 92 (1965), 201-209.

[41]

[42]

S.E. Payne. a garden of generalized quadrangles. Alg., Groups, Geom. 3 (1985),

S.W. Payne. The Thas-Fisher generalized quadrangels. Preprint.

323-354.

[43] S.W. Payne. Spreads, flocks and generalized quadrangles. J. Geom. 33( 1988), 113-128.

[44] S.W. Payne. An essay on skew translation generalized quadrangles. Geomd Ded. (to appear).

[45] S.W. Payne, L.A. Rogers. Local group actions on generalized quadrangles. Preprint .

[46] S.W. Payne, J.A. Thas. Conical flocks, partial flocks, derivation and generalized quadrangles. Preprint.

[47] J.A. Thas. Generalized quadrangles and flocks of cones. Europ. J. Comb. 8(1987), 441-452.

[48]

[49]

[50]

[51]

[52]

[53]

Norman L. Johnson Mathematics Department University of Iowa Iowa City, Iowa 52242

J.A. Thas. Flocks, maximal exterior sets and inversive planes. Preprint.

J.A. Thas. Flocks of finite egglike inversive planes, in "Finite Geometric Structures and their applications. (E. by A. Barlotti), Rome, (1973), 189-191.

J.A. Thas. Flocks of nonsingular ruled quadrics in PG(3,q). Rend. Accad. Naz.Lincei. 59( 1975), 83-85.

J.A. Thas. Some results on quadrics and a new class of partial geometries. Simon Stevin 55(1981), 129-139.

J.A. Thas. Recent resuts on flocks, maximal exterior sets and inversive planes. Combinatorics '88. Anals of Math. (to appear).

M. Walker. On translation planes and their collineation groups. Thesis, Westfield College, Univ. London 1973.


Cornbinatorics '90 A. Barlotti et al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. 317

FINITE REFLECTION GROUPS AND THEIR CORRESPONDING STRUCTURES

Helmut Karzel

Mathematisches Institut, Technische Universitat Miinchen, Arcisstr. 21. D-8000 MUnchen 2, Deutschland

Abstract To each reflection group ( r , D ) we can associate plane and spatial incidence

structures and in the finite case define a geometric order n of (I',D). There are exactly four types of finite reflection groups, regular Euclidean planes, plumb kernel Euclidean planes (in both cases the plane structure is an affine plane), elliptic geometries and center geometries. We will discuss the properties of the corresponding incidence structures and in particular calculate all the parameters of these four different types in dependence of the geometric order n.

1. INTRODUCTION

The notion of a reflection group has its origion in the plane absolute geometry. If we consider for instance the Euclidean o r hyperbolic plane (cf. [ S ] 5 17), then we can associate to each line L exactly one involutory motion 1, called reflectlon In the h e L, which fixes exactly the points of the line L. By a motion y one under- stands a permutation of the point set , which preserves the lines, the order structure and the congruence in the stronger sense, that for any two points a. b we have (a , b) = ( y ( a ) , y(b)) . We know, that the motion group G of an absolute plane is generated by the set D of all reflections in lines, more precisely, each motion can be written as a product of a t most three reflections in lines (cf. [S], (17 .6 ) ) . Furthermore one can prove the following Three Reflectlon Theorem (cf. [S 1, (17.13), (17.14), (18 .8 ) ) .

(1.1) Let A,B,C, X i , X 2 , X 3 be lines with A # B. a) If there is a point p such that p € A f' B, then there is a line D with 6 = x o o c" if and only if p E C. In this case D is uniquely determined and p E D .

b) If there is a line L with L 1 A , B , then there is a line D with 6 A o B o c " if and only if L 1 C. In this case D is uniquely determined and L 1 D.

c) If for each i E ( 1 , 2 , 3 } the product x o g o

one line D such that D = X1 o X2 0 X3.

N N

is involutory, then there is exactly N N N N

These theorems led t o the notion of a reflection group ( c f . e.g. [ S ] 5 19, for the historical background cf. [ 4 1) :

318 H. Karzel

Let r be a group with the neutral element 1, J := { y e rl y 2 = 1 y} the se t of all involutions of r and D C J. For the pair ( r , D ) we can define the following notations: For y € r let T:= { x € D I y x E J}. A subset b C D is called a pencll, if there are a , b E D with a 4 b and b 3 Let b be the se t of all pencils. Two pencils a,b E 23 are called joinable, if a n b 4 0. Let bp := {b E b I V a E b : b n a + 0) be the set of all proper penclls, hence the pencils which are joinable with every other pencil. For a € D let b( a ) := { b € 23 I a € b} and b ( a ) := {b E BPI a E 6 ) . The pair (r,D) is called a refledlon group if

( S l ) D is a system of generators of r ( S 2 ) V b € b , V x , y , z € b : x y z E D

and a refelectlon group with (property) domaln, if furthermore

( = { x € DI a b x E J}).

P

(S3) BPS 0

( S 4 ) V x E D : b ( x ) : = 0 o r IBp(x)I 2 3. P

Let D' := { x E D I l ap(x) l 2 3) { x E D I a p ( x ) + 0).

In this paper ( r ,D) will be always a reflection group with domain. To each reflection group ( r , D ) we can associate plane and spatial incidence structures:

We consider b a s the se t of points and B := { b p ( a ) / aEDI} as the se t of lines P of a plane. Then any two distinct points can be joined by exactly one line. This plane is a substructure of the extended plane (b,@) where s := { b ( a ) l a E D}. Here any two distinct lines of B intersect in exactly one point of 8, but two distinct points of b need not be joinable by a line of a (cf. 5 5).

To obtain the spatial structure we remark that by the reduction theorem (D4= D 2 ) (cf. (2 .7)) , the set C := D 2 forms a subgroup of r of index 1 or 2. This group G has the fibration 8 (6'1 b E a} (cf. (2 .12) ) , which allows us to define the corresponding kinematic space o r group space (D(l") := (C ,S) , where G is the set of points and (31 := {y XI y E C , XE 8} the s e t of lines ( c f .56 ) .

In a lecture given in 1963 in Hamburg (cf. [1],[2],[3]), I showed, that in the case of a reflection group with domain the bundle ((31(p),@(p)) consisting of all lines and planes incident with a fixed point p E C forms a pappian projective plane, and that there is a line preserving bijection between the points of the extended plane (6.9) and the bundle plane ((31(p),@(p)). Therefore the plane of a reflection group can be extended in a unique way t o a pappian projective plane. Then it is easy to prove that r/Z is a subgroup of an orthogonal group. The center Z of r has a t most two elements.

In the first sections ( 5 2 , 3 ) we repeat definitions, notations and results from [ l ]

and [31 including a presentation of the different types of reflection groups (cf. 5 4) . Our particular matter of concern, the finite reflection groups are studied in more detail

Finite refexion groups and their corresponding structures 319

in the last four sections. A coarse classification is obtained by Theorem (7.3). In the first case the corresponding plane is affine, and therefore these structures are called Euclidean planes (cf. 8 8). The Euclidean planes for their part split into the regular and the plumb-kernel geometries. In the second case we have a subdivision in elliptic reflection groups, characterized by the existence of polar triangles (cf. 5 9) . and in the center geometries, where the center Z consists of exactly two elements (cf. 8 10). The parameters correspond as well for the two types of Euclidean planes as for the elliptic and center geometries.

2. PROPERTIES OF REFLECTION GROUPS

In this section let (r, D) be a reflection group. Then we have ( the proofs, which are not given here, can be found e.g. in [l 1 or [51 § 19):

(2.1) V a , b E D with a S b : a , b E z .

(2.2) Let b E 23 and a ,b , c E b . a) If a + b, then b = b) a b c c b

(2.3) V a , b E 2 3 with a S b : Ian61 I 1.

(2.4) V a , b , c E D with a b c E J : a b c E D .

(2.5) V b E 23: 161 t 3.

Proof. Let 9 E 23 and a , b E 9 with a b, hence = 9 by (2.2)a). Then 9E 23 (a),

hence by (S4) lBp(a)1, l!Bp(b)1 2 3. Therefore we may assume b 4 9 and a 1 6, and

there are q , r E %,(a) \ { # } with q 4 c. Since @ , q , r are proper, a E # , q , r and a 4 6, the intersections p:= 9 fl b, q := q n 6, r := r f l b consist in each case of exactly one element and these elements p , q , r are distinct, i.e. 161 2 3.

(2.6) V a € D: lb(a)l 2 3.

Proof. By (S3 ) and (S4) there is a 9 E 23 with a 4 9 and by (2.5 ) there are three

distinct b,c, d E 9 . Then by (2.1) and (2.2 ) ab, ac , a are three distinct pencils, which contain a.

(2.7) (Reduction Theorem) D4 = D2

P P

p - -

By (S1 ) and (2.7 ) we have : 3

(2.8) r = D~ u D ~ , ~ - _ a r, ( r : D ~ ) s 2 .

(2.9) v b E 23: b3 = 6, b2 b n b2 0 and b2 is commutative

(2.10) For each b E 23 and each c1 E r the following assertions are equivalent:

I', b l i b2 s r,

(a) c i ~ b2, (b) bcic D , ( c ) c1 = 1 or c i E D2 and b = 2 .

320 H . Karzel

2 Proof. By (2.9), " ( a ) 3 (b)". "(b) 3 (c)": b a C D 3 c1 E bD C D hence a = 1 or a E 8, i.e. ;; = b. " ( c ) * (a)": Let a 4 1, i.e. b = and a = a b with a ,b E D. Then by (2.1) a , b E = 6 , hence a E b . (2.11)Let a,PEr with a D = P D . Then a = P .

Proof. a D = BD 3 D = a- lPD and then

(2.12) Let C := D2 and 3 := { 6'1 b E 83). Then (G, 8) is a kinematic fibered group, i.e.

(F l ) VF E 8 is a subgroup of G with { l } 4 F 4 G.

- 2

h

a-'P E D2. Hence D = a-'P, and if a-'P 1 7 -

E 8, thus by (2 .2)a) 6 = { a-'P } a contradiction to (2.6).

(F2) u S = G

(F3) VF,,F,ES with F, SF,: F i n F , = { l } (F4) V y E G V F E S : y F y - ' E $ .

Proof. By (2 .8 ) G is a subgroup of r. Since b C D for each b E 8, b2 C C. Thus by (2.9) 8 consists of subgroups of C and (F2) holds by the definition of the pencils. By (2.2 )a) we have (F3) and together with (5.5 ), (F l ) . Finally in order to prove (F4) let F = b2, a := yay ' E D2\{1}, hence and therefore: x E b e a x E J - y a y - i y x y - l E J y z y - ' Z and consequently yb2y-' E 8.

b = with a = a b E D2\(1}. Then -

E 8. For x E D we have by ( S l ) and (2.41, y x y - ' E D yxy- ' E Z . Thus yby- ' =

- h

3. INNER AUTOMORPHISMS OF REFLECTION GROUPS

Let again ( r , D) be a reflection group. For each y E r we denote by - y : r -+ r ; < -+ y <y-' the inner automorphism of the group r and by i? the group of all inner automorphisrns of r. By (2.4) and (Sl ) we have:

(3.1) V y E r : y ( D ) D, i.e. all inner automorphisms of r are automorphisms of the reflection group (T,D) and this implies V y , a E r : T ( b ) = 8, y ( 8 and

y ( c ~ ) = E, y ( z 2 ) F 2 if : = y a y - ' .

) 6 P P -

(3.2) t / y E I' the following statements are equivalent: (a) y = i d (- y E Z : = { < E r l V€,EET: C,t=<<}) (b) id

- ( c ) V P E 8,: Y ( P ) = P.

Pmof. "(c) 3 (b)" . Let a 6 D'; hence there are ~ , b 6 6 with b and a E a,b,

i.e. a = a n b by (2.3). Then u ( a ) = u ( a n 6 ) = ? ( a ) fl y(b) = a 17 b = a . "(b)*(a)". Let d E D \ D ' , Q E S P , a , b E @ with a S b . B y ( S 3 ) a n d ( S 4 ) a , b € D ' ,

hence d + a , b and there are q , r € 6 \ { @ } with a E q and b E C . Since d 4 D',

P

P

Finite reflexion groups and their corresponding structures 32 1

- - - d 4 (@ U q U r). Let b ' := q n and a ' := r n a. Then a ' ,b ' E D', aa ' = a d , bb' = bd E 6, aa' 4 66' and d =&? n s ' . By a ,a ' ,b ,b ' E D' and 71 Dl = id we have y ( a a ' ) = a a ' and 7(=) = r b ' , thus y(d) = ydy- '= d. Consequently TID = id and so by ( S l ) y = i d .

- -- h

(3.3) If y E Z then either y = 1 or yE D.

Pmof. Let y E Z \ { l ) . B y ( 2 . 8 ) , Case 1. y = a b E D . Then y a y b = 1 and for all x E D, ( y x ) = y x = 1. Conse- quently either y x = 1 for one x E D, i.e. y = x E D, o r otherwise a b x = y x E J for all x E D, i.e. y = D a contradiction to ( S 3 ) and (S4) .

Case2. y = a b c E D \ D . Then y s l . If y = 1 , hence y E J a n d s o y E D by(2 .4) . Let 1 4 y2 = a b y c = a b a b . Then by (2.7) y2 E D2\{1} and so b := 7 E 9, and by (2.4) a b a , b a b E D . Thusby(2 .1) a , b E b a n d y c = a b a b c = a b y = a y b = b c b E D , hence c E 6. By (S2) y = a b c E D.

(3.4) IZI 5 2, i.e. Z = (1) o r Z = { l , z } with Z E D .

Proof. Let u,v E Z\{1} and xE D. Then uvE Z and ( u v ) 1 by (3.3). Consequently ( u v x ) =(uv) 'x2=1, i.e. u v x E J fo ra l l xED\{uv) . Thus,if u s v , then GE9 by (3 .3) and == D\{uv) a contradiction to ( S 3 ) and ( S 4 ) .

3 y E D 2 or y E D . 2 2 2 2 2

- 3 2 2

2

2

2

By [S 1 (19.6):

(3.5) V g , x € D : g ( x x ) = G , i.e. t / b ~ % ( g ) : g ( b ) = b .

(3.6) Let b E b and g E D \ b .

(a) If g(b) = b then b C gJ and g(x) x for all x E 6.

(b) If b E S P then b n g J S 0 .

Pmof. a) Let x E 6 . Then x = b f' because g 4 b and by (3.5) g x g = g ( x ) g(b) C' g(G) b f7 b) By a ) we may assume g(b) # 6. Then since b E b h := b n g(b) is uniquely

P' determined and so g h g g ( h ) = g(b) f? b = h 4 g, i.e. g h E J

= x, hence since g + x, g x E J a x E gJ.

h E gJ n 6.

BY C31:

(3.7) For a E D \ Z let a' := a J n D = Z and an a' U {a}. Then

(a) Either a' E 23 (then a' is called the pol of a ) o r a'€ 23 (then an is called

(b) If a' E 9, then Z = {l}.

( c ) If a'€ 9, b € D \ Z and cE a'\Z, then b'c ?8 and a o = c n .

(d) If z E Z\{l}, then a' =

the plumb kernel of a).

is a plumb kernel.

322 H . Karzel

(el If there are a , b E D with a t b, a'€ 23 and )a1 n b'l 2 2, then X J yJ, i.e.

(f) If there are a .b ,c E D with a b c = 1, then x E b for each x 6 D, 1

- - 1 1 x = y f o r a l l x , y E D with x n y t0.

for all x , y E D with x s y and a', b l n c l = 0.

1 1 1 Ix n y I

4. CLASSIFICATION OF REFLECTION GROUPS

The theorems of 85 2 and 3 allow us t o define the following types of reflection groups ( T,D):

Deflnltlon. A reflection group ( r , D ) is called:

centerfree, if z:={cErl v y E r : < y = y r ; } = (11, center geometry, if Z 4 { l } , hence Z { l , z } with Z E D (cf.(3.4 )), elllptlc, if one of the following equivalent conditions holds (cf. (2.8 1): ( e l ) 3 a, b , c E D with a b c = l , ( e 2 ) D 3 = D 2 = r , ( e3 ) D 2 n D 3 + 0 , non-elllptlc, if D2 f' D3 = 0; here D2 is a subgroup of I' of index 2, plumb-kernel (Lotkern) geometry, if there is a pencil b e 23 such that ( x y ) = 1 for all x ,yE6; in this case b is called a plumb-kernel Lotkern, regular, if there are no plumb-kernels, rectangular geometry, if there are a ,b E D with a 4 b, a'€ b and I a l n b l l 2 2.

By (3.7) the plumb-kernel geometries contain the center geometries, and the

2

regular groups contain the elliptic and the rectangular geometries.

By [3] Satz ( 7 . 1 0 ) ~ ) . (7.111, (7.12) we have:

(4.1) For b E b: b is a plumb-kernel a 3 a ,b ,c E b with b # c and a b , a c E J a V a E b \ Z : 6 = a o .

(4.2) If (I',D) is regular, then V b E b

Proof. Let a E 6 , r E b with a 1 r and b := b 17 r . Then a # b, b & and Z(b) = 6. If a(r) r then aba Z ( b ) = Z(b f' &) a(6) n a([) 6 n f b. Hence a b E J since a b 4 1. If ab 4 J, then a([) + r; let x := r n a([). Then a d r, hence

a + x and x = a ( x ) = a x a . Consequently axEJ. Nowlet ~ € 8 (b) \{b,r} . By Z ( b ) S b , - p , a ( t j ) 4 tj and for y := tj n ~ ( L J ) , y = a (y ) = aya 4 x. This implies xy E b and a( G )

xy . By ax, ay E J, x 4 y and ( r , D ) regular, we have by (4.1) a d G. Then c := b n 4 a and c = Z(c ) = aca; hence ac 6 J and b = z. Now let a = ab, y = c d E D r'l J with 0: =?. Then u := acd E b and cd = a u E J. Since (I',D) is regular by (4.11, u b, i.e. c i =y .

3 C I E D ~ ~ J : b =;. P '

P

- 2

Finite reflexion groups and their corresponding structures 323

S. THE RELATED PLANE INCIDENCE STRUCTURES

To each reflection group (r. D) there corresponds the plane n( r, D) := (6 , DI) and the extended plane IIe( I', D) := (8, D). A point b E 6 and a line a E D are called lncldent iff a E 6. The plane n( r, D) is an Incidence space, i.e. any two distinct points of 8 are incident with exactly one line, and each line is incident with a t least two points ( in our case with at least three points ( c f . (S3) , (S4) ) ) .

P

P

In the extended plane Ile( r, D) any two distinct lines a, b are incident with exactly one point, namely a; each line is incident with a t least three points (cf . (2 .6)) and (by (2.5 1) each point is incident with at least three lines. But in general two distinct points need not have a joining line. Obviously the plane n ( r , D ) is a substructure of the extended plane FIe(r.D), and each inner automorphisrn 7 with y E r induces by (3.1) an automorphisrn as well of the plane II( r , D ) as of the extended plane IIe(r, D). Here we have (cf.[ l]):

(5.1) Theorem. Let ,?) := {%(a) I aE D}. Then there is a subset U c p(8) such that (6, ,D U U ) is a pappian projective plane. For each y E r the map 6 - 6 ; b - 7 ( b ) = y b y-' is a collineation of the projective plane (%,a U U ) .

6. THE CORRESPONDING SPATIAL STRUCTURE : THE GROUP SPACE D ( r )

For a reflection group ( r , D ) we define the following operations (cf.[3] 5 9): L e t G := D2 (by ( 2 . 8 ) we have either r = G or G is a subgroup of index 2 in I') and ( rxr ) f :={(a ,p)c rx r l a - ' B E G \ { i ) } . Then

a$) -+ CL n p := a-'p .a-1'

I

2 - v(r) - -. a LI p :=aa-'p

n: {y) - w-) - LI: [

324 H . Karzel

The corresponding group space o r kinemtic space D ( r ) is defined as follows: C is the set of points,

B := rl ( G x C)' = {yb21 y E C, b E b} is the set of lines,

B, := {yb21 y E C, b E bp} the se t of prolective hes,

@ := [D3] = { [ a l l CL E D3} the se t of proper planes.

and

P

The group space D( r) is an incidence space with the additional properties (cf.[ 1 I): d is a subset of the set @ of all planes of the incidence space (C,B), such that

d = { E E d l ] L E B o : L C E } . If E E @ and A , B E I ? ( E ) : = { X E Q I X C E } s u c h t h a t

A € B,, then A n B # 0. For each y E G the maps y p : C -. G; 5 -. y s yr: C -+ C; F; .-, Ey

yp(B0) = ~ ~ ( ' 2 , ) 8, and y & d p ) = yr (d 1 = d

P

P and

are automorphisms of the incidence structure of D ( r ) with

P P'

7. FINITE REFLECTION CROUPS

We will give a complete classification of all finite reflection groups (r, D) and compute all parameters of (r , D) and the corresponding incidence structures.

(7.1) V u E bp, V b E b, V d E D : I 6 I 5 I UI lb(d)l.

Proof. Let a # b , c := a n 6, c ~ b ( c ) \ { a , b } (c f . (S4)) and d E c\{c} . Then d da,b, hence b + %(d); x ---t and b ( d ) -. a ; [ -. [ n u (note u E b !) are injections. Consequently Ibl 5 IS(d)l = lal.

P

By (7.1 ) all proper pencils u 6 b have the same cardinality and we call n := I a I - 1

the geornetrlc order of the finite reflection group ( r , D). By ( S 3,4) there are a, b E b P with a # 6 . Let c := a n 6. Then p : ( a \ { c ) ) x ( b \ { c ) ) -. b \ b ( c ) ; (x ,y) -, xy is a bijection, hence I b \ b ( c ) l = n2 and together with (7.1) we have part a ) of the following statement:

P

-

(7.2 ) (a) 1231 = n2 + n + 1.

(b) If 23, then: b e bp 161 = n + l .

(d) If there is an a E D with 1% ( a ) \ = n + l , then D'= D, ID1 n 2 + n + l , 23 = bp (c) lbpl 5 ID'I ID1 Ibl

P and (23 ,D) is a projective plane. P

Proof. b) Let M := U { b ( x ) l x € 6 ) ; by (7.11, IMI = Ibl . ( l 2 3 ( x ) l - l ) + l = 161 * n + l .

Thus by the definition of "proper" pencils we have: b E M = b Q Ib( = n + l . P c ) Counting the incidences we have by (7.1) ID/( n + 1 ) = c 161 5 ( n + 1) * 161, hence

6€ b

Finite ref exion groups and their corresponding structures 325

ID1 5 lbl,

d) Againby (7.11, b ( a ) = %(a) = { a ,,.. . ,an+l}. Let b E %\%(a) . Since a i € b for

each i € { l , ..., n + l ) , x i : = a i n b exists.Thus I b l = n + l a n d b y b ) b € b P . Conse-

quently 23 = b thus D' = D, and by ( a ) and ( c ) ID1 n2 + n + l . b = 6, implies

that

P P

P' ( b p , D ) is a projective plane.

Now let c E D ' \ Z , a.b E c1 with a 4 b, c E 23 (c) \{%} and d := c n a. We consider the orbit { Y ( C ) ~ x E c ) which is a subset of b (c) . We have to discuss the following cases:

1 1. V xE c \ c : X ( C ) 4 C. Then (I',D) is center free because otherwise for zE Z \ { l } ,

z E c1 and T ( c ) = C , hence z E c by assumption and so c = r 2 since c d Z. But 1 1 then ;(C)= c for all xE c . Suppose there are x,y E c \ C with x 4 y and X ( c )

= y(c). If c E b, hence c1 = a, then xyd E c1 with xyd 4 d, i.e. xyd c and x y d (c ) = sly( c ) = c a contradiction.

If cue b, hence c0 = a b U {c}, then d = c and again xyd = xyc E co with xyc 4 c, i.e. X Y C E c \ { c } , hence xy C ( C ) = X F ( C ) c a contradiction t o our assumption. Thus in both cases I{F(c)l x t c l } l = l c l l i lbp(c)l .

2.3 g E c \ C with g(c) = c. Then by (3.6)a) c c g J n D, hence by (3.7) either c = g l (i.e. c is the pol of g ) o r g EZ.

8 ) c g . i.e. ( r , D ) is regular and the pol of g is a proper pencil. By (4.2) there 1 1 are u,v E c with uv E c2nJ and c = = g . Since g 1 g , uvg 4 J. But (uvg)' =

u(vg)(uv)g u(gv)(vu)g = ugug 1 because vg, uv, ug E J. This implies uvg = 1 and consequently ( r , D ) is elliptic. L e t x,y E c1 with x + y and X( c ) = y( c ) . Then be-

1 e d and so e d c. This gives us uve 1, hence e uv g = xyd. Since d c c = g we have xy = g d E J. On the other hand, uvg = 1 implies by ( 2 . 8 ) : V x E D 3 y , w E D with x = y w . Thereforefor [EB ( x ) and h,kExl: K([)=x([)

P 1 P

1

- - _ _ 1

1

1

_ _ _ causeof d = c n c 1 , e : = x y d E c ' and F ( c ) = x y d ( c ) = x y ( c ) = c . Since x + y ,

P h = k or hk E J. This yields T I C I 1 I = I{T(c)l xE cl}l i IWP(c ) l for all c 6 D'.

0 - b) g zE Z, i.e. (I',D) is a center geometry, and by (3.7)d) zc c is a plumb- 1 kernel with C S co. Then F ( c ) = Z ( c ) = c and F(c) 4 c for all x E c \ ( z } . If F ( c )

= y ( ~ ) for x , y E c \{z ) , then u : = x y c E c u and U ( C ) = X ~ C ( C ) = ~ ~ ( C ) = C , hence either u = c and then x y or xyc u z E Z and then y = x z c . Consequently 1 +$([.'I - 1 ) ~ I{F(c)~ x E cl}I IWp(c)\(z}I and more precisely lcoll 2 . 1 % ( c ) \

_ _ _ - 1

P {a 1.

326 H. Karzel

We have proved the theorem:

(7.3) Let c E D ' \ Z and e E b P ( c ) with e+c ' . 1

(a) If x ( e ) 9 e for all x E c \e, then ( r , D ) is center free and non elliptic and we have I { X ( a ) l x E al}l = la1[ 5 lbp(a) l for all a E D' and a E %,(a)

and further lao[ I; Ibp(a)l if a n € 6.

1 (b) If g( c ) = e for an element g E c \c, then (r, D) is either elliptic or a center

geometry. In the elliptic case we have I{x(a)l x E a }I = T l a I 2 ) b p ( a ) l for all a E D' and a 6 b (a). In the case of a center geometry I { x( a ) I x E a}\

1 I 1

P - 1 - la 0 I and lao[ 12 . l bp (a ) \ {s}l for all a E D ' \Z and u E b p ( a ) \ { z } .

(7.4) If ( r , D ) is regular, c E D and b E b,, then:

(a) l ~ ~ ( c ) l l c l l

(b) If b 4 c', then 1bn e l / = 1

( c ) If b = c' then (I',D) is elliptic.

Proof. a) By (4.2) for each c E D' and each r E b & c ) , r2 n J is a single element

a and we have r = G, hence x := C L C E F n c . Therefore 123 (c ) l 5 lcll.

b ) i s a consequence of (3.6), (3.7) and (4.2).

c ) Since b E B there are a , b E b with a b € J by (4.2) and so a b c = 1.

1 P

P

By (7.3) we have a classification of finite reflection groups. In the next sections

we will discuss the different classes in more detail.

8 FINITE EUCLIDEAN PLANES

In this section let (r , D) be a finite reflection group such that the assumption a ) of (7.3) is valid. We have to distinguish between the regular case and the case of a center free plunb-kernel geometry.

If ( r , D ) is regular, then by (7.3)a) and (7.41, la 1 I = 16 (a ) l for all a E D ' and P

( r , D ) is not elliptic. That means, if a E D', therefore b E D'. element y E D', and since xy E J,

ments a , b E D with a 9 b.

b E D and a b J , then 8 E b and

x1 n b is a single

This gives us D = D'. For any two ele-

f\ b1

P By (7.4)b),c), for each b E b and each x E D, P

x E D'. E b and a b 4 J, the map b ( a ) -, BP(b) : r -. P P

is an injection. This implies ) b p ( x ) [ = lBp(y)1 = Ix' 1 I lyll for all x ,y E D . We set


q := 133 (x) l for x E D. Now let a , b E D with a b E J. Since ( r , D ) is not elliptic, 1 the map 23 -. 23 ( a ) x b P ( b ) ; r - - (G,fi) with x : = F 0 a , b* is an

injection. Thus for all P

c E D, i.e. q = lcll < n + l by (7.2). Since 23 = u{23p(x)I x E a} and la( n + l

we have (%,I = ( n + l ) ( q - 1 ) + 1 = nq + q - n and because of (7.11, q = 123 (x)l I 123(x)I

= n + l . Thus n(q-1) + q 5 q 2 a n(q-1) 5 q ( q - 1 ) n .s q < n + l , i.e. n q and therefore 123p(=n2. Also IDI-n=(Bpl.(n+1)=n2.(n+l), i.e. ID( = n ( n + l ) . Conse- quently (23,,D) is an affine plane of order n.

P y := tj P P

.s q2. Further, since (r, D) is not elliptic, c l E 23\23

P

P

Now let (r, D) be a plumb- kernel geometry. Since (r, D) is centerfree, each a E D determines a plumb-kernel a'. and for a ,b E D we have "ao = b' @ (ab)2 1". Therefore a' n c' = 0 for (ac) ' 4 1, i.e. a' E 33\23,, by (7.1) la'[ n, 123p(a)l i n and this implies

( a ) V ~ E B ~ : b2n J=0.

Obviously, if a E D', b E D with ( ab )2 4 1 (i.e. a' 4 b'), then (W,(a)l 5 lb'l and

by (7.3)a) (a'1 I l%,(a)( 5 Ib'l. This yields:

( 8 ) v x,y E D, v a E D': q := lx'l = ly'l = IBp(a)l.

For a , b E D with (ab)' 4 1, i.e. a' 4 b', the map

23, + a' x b'; r .+ (x ,y) with x := & a', y := r n b' is injective, hence by (p),

( Y ) lBpl 5 q2.

For b E 6, we have bp u { bp(x)I x E b} and so by ( 8 ) :

( 8 )

n q, lbpl= n ,

1 + ( l a P ( x ) l - 1) . Ibl 1 + ( q - l ) ( n + l ) . Again from ( y ) and ( 6 ) we deduce 2 D = D', ID1 = n(n.1) and that (23 *D) is an affine plane.

P

gives us the most parts of the following Altogether and with [ l ] this

(8.1) Theorem. Let (I',D) be a finite reflection group such that there are c E D \ Z and c E 23 with c E e and x ( c ) 4 c for all x E c \ c and let n be the geometric order of ( r ,D) . Then:

1 P

(a) ( r , D ) is centerfree and non elliptic.

(b) D = D' ( c ) (BP, {23p(x)I x E D}) is a finite pappian affine plane of order n.

(d) D ( r ) = ( D 2 , Q ) is a (3- l ) - s l i t space where Q, = {yb'l y E D2, b E bp} is the

se t of all projective lines, 8, = {yb'l yE D2, b E 23\23,) the set of all affine

328 H . Karzel

lines, '2 = 8, U 8, and Q = {[a] I CL E D3} is the s e t of all planes which contain a t least one projective line.

P

(e) The group space D ( r ) has the following parameters: ID21= n3 + n 2 = 4 2

( f ) If ( r , D ) is regular, then ( r , D ) is a rectangular geometry and we obtain the

projective closure of n( r, D) (6 ,D ') in the following way: b is the set of P all points of the projective plane, b\b is the line of infinity, and each b ( x )

with X E D is extended by the pencil 5, where y E D\{x) with y l = XI.

(g) If (r, D) is not regular, i.e. (r, D) is a centerfree plumb-kernel geometry, then

( b , { ' s p ( x ) ~ x E D } u { x a ( x € D } ) with ?& P ( X I : = % P ( x ) U { x u } istheprojective

closure.

I L I = ~ + I for L E Q , , I M I = ~ for M E % ' , 1 ' 2 , 1 = n , l a , l = ( n + t ) ( n 2 + n ) = ( n + 1 ) n.

P P

Remark. Each (3 - 1) - 8lk space can be gained as a trace space from a 3-dimensional

projective space by omitting one line together with their points. The planes corre-

sponding to these reflection groups are called E u c l l d w planes. There is a one to one

correspondence between these finite Euclidean planes and the class of all quadratic

field extensions ( L , K ) where K is finite. The characteristic of K is 2 if and only

if (r, D) is a plumb-kernel geometry.

9 FINITE ELLIPTIC REFLECTION GROUPS

If we assume the condition b ) of (7.3), there are two subclasses, the finite elliptic

reflection groups and the finite center geometries. In this section let (T ,D) be elliptic (cf. (7.3)). Then r = G = D3 = D2. Therefore J = D by (2.4), and the map -:

D -, b: x -, x̂ is an injection by (3 .7) f ) with the polarity property "y E < x E 7". By (4 .2) bp C 6 . Let D := {x E DI x̂ E bp} and Du := D \ D By (7 .2)b) Dp = {x E DI 1x1 = n+l} . For a E D. y E r and A c r let y (a ) := ?(a) = yay- ' and let A(a) := {y(a) l Y E A} be the orbit of a with respect to the inner automor-

phisms of the se t A.

P P'

(9.1) Let ( r , D ) be elliptic and a E D Then: P'

ID1 - n (a) Ir(a)l . (b) If Z fl Dp =# 0, then 1; II Dpl = la n DUI

= a Du for b E a Dp, u €2 Du.

F

and a ( b ) = Z 0 Dp, a ( u ) -


( c ) ID1 = 2 . IDpl+n, IDU[ = IDp[ + n and D r ( a ) is one orbit.

(d) "a n D 4 0 a D = D'", and then 16 0 DUI = 16 n D 1 = 9 for all b E DU, P P

P

IDPI =*, n n-1 IDU[ =w and ID1 = n 2 .

(el 2 n D P 4 0.

Proof. a) Let y E r \ { l} . If y 4 J, then there exists a b E 2 n 7, because a E Dp.

We have y b € J and y b ( a ) = y(a). Consequently r ( a ) D(a). Let x,y E D \ a with x 4 y and assume x ( a ) y(a). Then 4 a, and for c := 2 n G ( a E D !) and

d := x y c we have d 4 c, d E G, hence d 4 a and d ( a ) xyc(a) = x y ( a ) a. There- fore by ( 3 . 6 ) d = a, hence xy = a c E J, i.e. y = xac . This shows us: If x = a, then

y = c E a a contradiction to x,y 4 2. If x E D \ ( G u {a}), y E D and c := a n &?, then

'*x 4 y A x ( a ) = y(a )

Therefore

b), C ) By a ) each orbit r ( a ) , a E D has the same length, and since r(a) c Dp we

have ID I= 1 . if X is the number of orbits. By ( 7 . 2 ) ~ ) and (7.2)a),

IDPI i ID1 5 l + n + n 2 , hence (X-2)IDI i X.n. Since for a E D', 1 % (a)l 2 3 we have

1 + 3 n ID'I i (DI, thus (X-2) 3 < X, i.e. X < 3. So we have to discuss the two

cases:

X = 2, i.e. ID1 = ID I + n

X = 1, i.e. ID1 = 21D I + n .

Both cases yield that (8 , D ) is not a projective plane, and so by (7 .2) , b ( x ) \ 8 ( x )

4 0 for each x E D, implying x̂ n DU 4 0 for x E D

P

y = x a c " . D - + 1 ) - IDI-n by (7 .2) .

2 I r ( a ) l = ID(a)l = 1 + 1 I h P

P

P

P

P

P P

P'

Now let u E 2 n DU and b E a n D Then 2 ( u ) c a 0 Du, a ( b ) c a n D and P 2

P' for x ,y ,c E 2, x ( c ) = x c x = y(c ) = ycy implies 1 = x c ( x y c ) y = x c c y x y = ( x y ) ,

hence x y o r xy = a. Thus Ia(u)l = la(b)l = - 121 = -. This gives us b). If

we assume X = 2, hence IDuI ID1 \ ID I n, then a n Dp 4 0 since la1 = n + l .

Let m := 7.

we have Ibi Dul m and 6i n g. n DU = 0 for i, j E { 1 , . . , , m ) with i 4 j because

a E Dp. Therefore w2= m2 i IDU[ = n, hence (n-1)2 5 0 which implies the

contradiction n = 1. Thus only X = 1, i.e. ID1 = 2 * ID l + n is possible. P

d) "*"Bye). D p c D ' . B y b ) " " = I G n D p I = / G n D u I 2 f o r a l l x E ; ; ' n D P' and

and k, := 17 P'

k 1 := IF Dpl

1 n + l 2 2

P n + l By b) ;ODp consists of m distinct elements b,, . . . , bm, and

J

DuI are constant for all y E 2 n Du. Since a E D

330 H. Karzel

DU = (x^ n Du) (7 fl Du), hence IDuI = (+)L+ 9 . k,. Now y m D , P x € K n D

k := IF fl Dp(

c ) implies 7 n + l ( - n + l + k 2 ) = y ( 2 + T + k l - l , n - 1 thus n + l + 2 k 2 = n + l +2k, ,

i.e. k, = k,. Together with c), this shows us tha t f o r y E D' n D,,

n - 1 + 2 k ( n + l ) 4 = IF nDuI is constant with 2 k I; n, because < n + 1 and ID I (

Now assume D # D' and le t b E D \ D ' , i.e. 6 C D, and 161 s n. Then D 0 1;; n Dpl x E 6). and if for x E 6, x̂ fl D 4 0, then 1; D I = k, since x E D,.

Consequently ID I S ( 6 1 . k ~ n . k , h e n c e ( n - t ) ' + 2 k ( n + l ) ~ 4 n k , i.e. ( n - t ) ( n - l - Z k )

S O , thus n - l s 2 k s . n . Since n i s o d d , w e h a v e k = ? , l D p l = W and 161=n.

But the map 6 -, 6: x -, bx is an involution with no fixed points. Thus 1g1 is even,

which is a contradiction. Therefore we have D = D', thus 17 D I = IF 0 DUI f o r P al l yE DU. Counting the incidences between D and DU we obtain ID I - n+i IDuI . k = ( I D I + n ) . k,

[ ( n - 1 l 2 + 2 k ( n + l ) ] . ( n . 1 ) [ (n-1I2 + 2 k ( n + i ) + 4n]2k , a

(n-11, + 2 k ( n + l ) ( n + l + 2k)Zk * ( n - 1 l 2 = 4k2 3 k = u.

1 P

P

P P

P

P P 2 hence P

This implies 2 I D ~ I = w, ID,I = w, I D I = n2.

e) Let us assume tha t there is an a E D

D C D \ D ' . Let b E z, hence b E D ' n DU. By ( S 4 ) there is a C E D

and c # a . Then (2 UC) c DU and la U C I 2 n + l ; thus

( * I IDu[ 2 2 n + l .

Since g ( b ) C a, n + l and la (b) l there is a d E &'\a(b), and a l so

z ( d ) c a, Iz (d) l q, so tha t 2 Z ( b ) U a ( d ) . Let k, := IX n DpI for x E Z(b)

and k, := 1; n D 1 for x E a ( d ) . Since in both cases x E DU, we have k. s. d.

If k := k, = k, then by c ) each x E D, n D' is incident with k elements y of D P

(i.e. xyE J) and each a E D

T h i s g i v e s u s : ( I D U I - n ) ( n + l ) = I D I . ( n + l ) = I D U n D ' l a k s . I D U l . k s IDU/ -?=. P

ID,,l(n+l - q) 5 n(n.1) * IDU/ ( n + 3 ) 5 2 n ( n + l ) =+ ID,I 5 2n < 2n a

contradiction to (*).

If k, 4 k,, then again by c), the set D, n D' spl i ts into the t w o disjoint subse ts

with Z C DU, i.e. a 1 D'. Then by c )

with b E C P

P P

P 1 2

is incident with n + l elements of DU n D'. P

Finite reflexion groups and their corresponding structures 33 I

D: := {x E DUI 1, fl D I = kl} and D; := {x E DUI 1, n D I =k2} . Counting the P P

incidences between elements of D ( IDU[ - n ) ( n + l ) = lDpl * ( n + l ) ID:[ k, + ID;[. k, s (ID:I + ID",) 9 s lDUl . T . n - 1

Thus also IDuI < 2n a contradiction to (*).

and D, n D' we obtain here: P

Now we can s ta te

(9.2) Theorem. Let ( r , D ) be a finite elliptic reflection group with the geometric order

n. Let bu := {:I x EDu} and let be := % \ ( a p U a,) = %\{?I x E D} be

the set of ends. Then -

(a) D ' = D = J , I D I = n 2 a n d f o r e a c h X E D ,

(b) Dp and D

( c ) I D ~ I = 1bPt= -J--&, n n-1

x E B .

(and consequently also b U P and 23,) are orbits with respect to r,

i.e. if a € D and bED,,, then D = r ( a ) and DU = r ( b ) . P P

ID,I = = w, IB,I = n + 1.

(d) Va E Dp: B(a) fl be 0; V b E D,: I N b ) n bel 2, VK,O E 6,: K fl g 4 0,

and be is an orbit with respect to r. - -2 -2 2 (e) Let a E D

( f ) iri= 1 +

(g) For each c E be let T(c) := { c } u {x^ l x E c} . Then (%,a := { b ( x ) l xE D}

b E D U , c E B e ; then ( a l = ( a I = n + l , 1g1=lb I = n - l , lc l=lc I = n . P'

. n + 9 . ( n - 2 ) + ( n + l ) ( n - l ) = n(n2-1) .

u I T ( [ ) / [ E be}) is a finite pappian projective plane of order n.

(h) D( I') = ( r = D2, '2) is a 2-porous space which can be obtained from a 3- dimensional projective space by omitting Q, { y a 2 1 y E r, a E D ~ } , resp. B, = { y e 2 [ y E r, c E be}, resp.

B 2 = { yg21 y E r,bED,} is the set of projective, resp. affine, resp. 2-lines and

B 53, u Bl u B,. eP := { [ y ] l y E I"} is the set of all planes which contain

projective lines, and also which are not tangent to Q. The map r -, eP; y - [ y 1 is a polarity which can be extended on the surrounding projective space.

(1) The group space D ( r ) has the parameters Irl = ID2[ = IB,I = ( n + l ) n ( n - l ) ,

the points of a ruled quadric Q.

332 H. Karzel

Proof. b) By ( 9 . 1 ) ~ ) we have only to show that DU = r ( b ) for b E DU. L e t y E DU.

Since D = D' there are a , x E D with b E a and y E 2. By ( 7 . 6 ) ~ ) there is a y E r such that y(a) = x, hence y ( a ) = 2. Therefore we may assume b,y E a. By (9.1 )e)

and b ) there is an element ~ € 2 with c ( b ) = y. Thus D u = T(b).

c ) b = b,

lbul = IDu[ =

d),e) For each c

By (9 . l )b) and e ) , 121 = I{ (7 .1 )andso b ( a )

by (7.1) Ib(b) n bel 2. Let r E be. By (7.2)b),

between D, and be we obtain (c f .c ) ) 2 . thus lrl = n for all c E be.

P

6, lj be and 1231 n2+ n + 1 (cf. (7.2)a), 1% I = ID I = w, P P

(cf. (7.6)d)) imply lbel = n + l .

D we have { x e c } = {GI c E G } = b ( c ) n (ap u bu). n + l x E ;}I = 2 7 n + l , hence %(a) = {;I x E a} by

be = 0. By ( 9 . l ) d ) , e ) 161 = I { ? [ x E b}1 = 2 .% = n - 1 hence

lrl i n. By counting the indices

(%,I . n ( n + l ) n ,

-

= 2 . IDuI

Naw we consider the set M := { { ~ , g ] I F,LJ E be, r 4 0). Then (MI = ( "; ) and

each { ~ . g } E M is incident with a t most one x E D,. On the other hand each x E Du

is incident with exactly one element of M. Hence [MI . 1 = ( n;l ), i.e. r n 4 4 0 for all [ , g E be. Now let { ~ , 4 ] E M, a := n g and b E 2. Then

b ( a ) = a, hence b({r ,g) ) = {r ,g ) and b(b^) = 6. Since a E b ̂ 4 ~ , g we have b ( r ) = g

and b ( g ) = I, i.e. be is an orbit.

f ) By l- = G = D2, (2.12) and c) , e) we have:

Irl= 1 I 1 . ( n . 1 - i ) + IbuI ( n - i - i ) + Ibel. (n-1) 1 + + %P 2 + ( n + l ) ( n - I ) .

g) Let u,b E 23 with u 4 b and u n b = O . In order to show that there is a line in 8 incident with u and 6 , we have only to consider the cases a,b E 8, and u E bu,

b E be. Let first u = 2, b = b ̂ with a , b E D, and c := x. Then c E be, for otherwise there is a c E D with c = ;, i.e. c a = ac, c b = bc € J and so c E 2 () 6 = u 06.

But c E be and a , b E c tells us u , b E T(c) and the line T(c) is unique. Now let

u = 2 E b,, b E be and S := u { b ( x ) 1 x E b}. Then T(b) E 8, T(6) 0 S = {b} and

I T ( b ) I = 1 + 1 6 1 = l + n by e ) and I S I = l + l b l . n = l + n 2 . Since l b l = n 2 + n + l (cf. (7.2)a)) and since by assumption u = 2 4 S we have u E T(b) .

Next we have to consider two distinct lines and we can restrict ourselves to the

cases a(;), T ( e ) and T(b ) , T(c) . If a E e then %(a ) n T(e) = ( e ) . If a 4 c then

1 a IDu/

+ ( n + 1 ) n ( n - 2 1 +


by above b := a n e is a single element and %(a) By d ) there is an a E b (7 e , hence a E T(b) n T(e). Thus (b,(Y) is a projective plane of order n and by [ l 1 (b),(Y) is pappian.

T(c) = ( 6 ) .

10 FINITE CENTER GEOMETRIES

Finally we have to study the center geometry (r, D). Let Z = {l , z} be the center of

r and le t a E 23, with z 4 a. Then be := {xo= 1 xE a } = b ( z ) is the set of all

plumb kernels, G := D 2 s r, Now let

a ,x ,y E a with x S y . Since xyz # 1 and z 4 a we have 1 4 ( x y z ) = ( x Y ) ~ . There-

fore n + l = I Q I = la2[ is odd, and

(1) u = {;(a) = x a x ( x E a } and so be = {;(ao) = (xax)'( xE a} are orbits with

Consequently either 8, C b D' \ {z } l

are constant for all u 6 D \ {z}, and k := 16 ( x ) \ {xu} I is constant for all x E a.

Since lb,l = Ib (z ) l = n + l we have:

( 2 ) ID1 = 1 + ( n + l ) P , P s n .

(3 ) lb,\bel= l + ( k - l ) ( n + l ) .

( 4 ) I D ~ \ { Z } I = P ' ( ~ + I ) , ~ 1 s ~ .

(r: G ) = 2 and u v w # 1 for all u,v,w E D . 2

2 respect as well to a a s to a . or 6, 0 b =0, P := lu'l - 1 and P' := luo P P

P

By ( 3 ) k is constant 2 2 for all x E D'\ {z} and by (7.3)b) we obtain:

( 5 ) 3 p E N : p ( P + 1 ) = 2 . k

If we consider a,bE a with a # b, then for each F E b p ( b ) \ {b'}, x := r n a' exists

and x E a' n D ' \ ( z ) . Furthermore for f , g E b (b)\{b'} with F # g we have x :=

F 0 an # y := g 0 a'. This implies k s P' s P and together with (S),

( 6 ) P = 2 k - 1 s n

Since n is even we have P < n, thus lu'1 P+1 < n + l for all u E D\ {z) , and so by

(7.2) b ) ua 4 bp, i.e. Se n b = 0, and so z 4 D'. By counting the incidences bet- P ween b and D' we obtain:

Ibpl. (n+l)(:)(l+(k-l)(n+l))(n+l)= ID'I . k ( f ) P ' ( n + l ) . k,

hence 1 + ( k - l ) ( n + l ) = P'. k.

= 2(k-1) , i.e. v E (1.21. v = 1 implies n = k. hence by (6) the contradiction P = 2n - 1 s n.

P p 1, i.e.

P

Therefore n = v k for v E N and v(k-1) = P I - 1 i I - 1

334 H. Karzel

(6) v = 2 yields n = 2k, e l = 2k-1 = n - 1 P (i.e.

and (4) :

( 7 ) D = D ' U { z } , ID' I=n2-1 , l b p l = k ( 2 k - 1 ) = ( ; ) .

For 23, := b \ (bp u be) we obtain:

( 8 ) (bU( = n2 + n + i - (;) - (n.1) =

Since la 61 = 1 for each a E 23 and each b E b, and 1bP(x)I = k for each x E D'

we have k(n-1) 1 % I = k . 161,

1~j=i+n.lb~l+(n-l).lb~l+(n-2).lrR,J i + n ( ; ) + ( n - i ) ( n + i ) + ( n - 2 ) ( " ; ' ) , hence:

( 9 ) ICI = ( n - i ) n ( n + i ) , Irl 2 . IGI = 2 ( n - i ) n ( n + t ) .

= n for all r E b e ) , hence by ( 2 )

P hence 161 n-1. This gives us :

P

We obtain here the same parameters as in the elliptic case and ( b , D ) can be extended to a projective plane in a similar way. For each a" E be let T(a") := { a"} u { r E b U I [ n a " = O } and I t :={T(a ' ) la 'Ebe} . Then ( 6 , D U Z) i sapro jec t ive plane, such that lb,(x)l = n-k 23 ( x ) for each xE D'.

I P I We have proved the most parts of

(10.1) Theorem. Let ( r , D ) be a finite center geometry with the center Z { l , z )

and the geometric order n. Let be := { x" = k? I x E D \ Z} the se t of all plunb-kernels

and bu := % \ ( B e u bp). Then:

a) D = D' u (21, J = D' 0 zD', ID1 = n2 and for each x E D', x" = x̂ U {x},

I W P ( X ) l = lb,(x)l !$ Ibe(x)I = 1.

b) D', { z } ?

C ) 18~1 ( be , bp, bu are orbits with respect to r.

), IB,I = I ~ , I = n + 1.

d) V a E D': b ( a ) f l be = {a"} = { Z ) , V b E b,: I{r E be[ F n b = 011 = 2 . 2 2 e) Let a € 3,. b E bu, r e be: then IuI = la I = n + l , 161 = lb21 = n-1, I c l = I C I = n.

f ) ICI = n ( n 2 - i ) , iri = 2 . [GI = 2 n ( n 2 - i ) .

g) For each a E D', hence each an E be let T(a") := {a'} u {r E bU( f n a' = 0)

and let @ := 1b(x)1 x E D} u { T(a")l a E D'}; then (B,@) is a finite pappian projective plane of order n.

h) Let 9, := {ya21 Y E G, a E BPI, B, := {yn21 y E C, n E be}, B2 .= {yb21 y E C, b c bu} and B := B, u 8, u P,. Then ( G . 2 ) is a 2-porous space which can be obtained as a trace space from a 3- dimensional projective space by omitting the points


of a ruled quadric. t := {[y]1 y E r \ G = D3} is the se t of all planes which P contain projective lines, i.e. lines from Qo. The map G -, @,; y -, [ z y ] ( 5 E GI z y < E D} can be extended to a polarity of the surrounding projective space. The group space D ( r ) has the parameters ICI = I @ 1 = n(n2-1), = (;)', I Q ~ I = (n+1 l2 (n -1 ) , IQ,I ( n61)2.

1) P

Proof. b) Let a , b E D' u E bp(a), b E b p ( b ) and c E a n 6. By (1) there are x E a,

y E b with c x a x and ycy = b, hence F ( a ) = y x a x y = b, i.e. D' is an orbit. Now let a,b E b u $ b and c := a n b. Then { x ( a ) l x E c } C b p ( c ) and by

( 7 . 3 ) b ) ~ { x ( a ) ~ x E ~ ) ~ = ~ / c " ~ = ~ ( I + l ) = k = l b P (c)l hence b E b P ( c ) = Ix (a ) IxEz} . Consequently 23 is an orbit.

For a,b E b, with a $ b we may assume c := a n b 1 0 with c E D' since D' is an

orbit and y(bu) 23, for all y E r. Then M := { x ( a ) l x E z} C bu(c) and for x,yEc

with x ( a ) = y ( a ) we have u := zxy E c and u ( a ) a. If x $ y , then u $ z and by

(3.6) either u E a (i.e. u = a n co = c ) o r u ( v ) v for all v E a, i.e. a u" a contradiction; hence y = xzc . Consequently IMI - Ic I = k = b ( c ) and so b E bu(c) = M,

i.e. b, is an orbit.

d),e) Let b E 6,. Then bp = u {bp(x) l x E b}, hence by c ) and a ) (; ) 1 % I P = 161 . $ and so 161 = n-1 . Since I{x"I x E b}l 161 n-1, and lael n + l , there are exactly two plumbkernels [ .g with b n [ = b g = 0. For e E be, [ e l I+ 1 = n.

g) Let a,b E b with a n b = 0. Theneither a,b E b, o r a E b,, b E be o r a E be,

b E 23,. If a = a" E be, then T(a") 3 a,b. Now let a,b E bu. Then M := {[ E %,,I

[ n b S O } = u {bu(x ) l X E b } , a B M and [MI = 1 + 1 b 1 . (2 - 1 ) = 1 + k ( n - l ) ( n - 2 ) ,

hence IWU\MI = 2(n-1). By d ) there are c", d" E be, co # do with b n c" =

b n do = 0, hence b = T(c") T(do). Since bU = u {bu(x)l x E c"} u (T(c")\{c"})

we have IT(c")\{c"}l = lbul - lbu(x) l . (lc"1 - 1 ) (';I) - . (n -1 ) = n. There-

fore IT(c")l = n + l and lM 6 (T(c") \{c") ) u (T(do)\{do})l = 1 + ( n-i ) + 2(n-1)

= ( ) = IbuI. thus ( M U T(co) U T(do) ) \{co U d"} = b,, and so either u E T(co) o r a E T(do). With that any two distinct points of b are joinable by exactly one line of 8.

Now since 181 ID1 + lbel = n 2 + n + 1 ("Ga) 161 and Ib(x) l = Ib(z) l IT(x0)l

n + 1 for all x E D', the incidence space (b,@) is a projective plane of order n.

P'

P

0

1 0 2 U

n

336 H . Karzel

REFERENCES

[ 13 KARZEL, H.: Gruppentheoretische Begrtindung metrischer Geometrien. Vorlesungs-

[2] KARZEL, H.: Spazi Cinematici e Geometria di Riflessione. Appunti redatti de E. ausarbeitung von G. GRAUMANN, Hamburg 1963

ZIZIOLI. Sem. di Geom. Combinatorie n. 30 - Settembre 1980, 1st. Mat. Univ. di Roma, p.1-11.5

[31 KARZEL, H., and KROLL, H.-J.: Spiegelungsgruppen und ihre zugehorigen Inzidenz- strukturen. Mtinchen 1977. Unveroffentlichtes Manuskript.

[ 4 ] KARZEL, H., and KROLL, H.-J.: Geschichte der Geometrie seit Hilbert. Darmstadt 1988

1.51 KARZEL, H., SaRENSEN, K., and WINDELBERG, D.: Einfuhrung in die Geome- trie. Gottingen 1973

Combinatorics ’90 A. Barlotti et at. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. 337

On Nonisomorphic BIBD with Identical Parameters

Wen-Fong Ke

Department of Mathematics, University of Arizona, Tucson, Arizona, 85721, ‘USA

Abstract In this paper, we assign to every BIBD a sequence and use it to compare BIBD’s with

the same parameters. We then explore examples of nonisomorphic BIBD’s constructed from Ferrero pairs with isomorphic regular automorphism groups. A class of BIBD’s with special sequences is also provided in the last section of this paper.

1. INTRODUCTION

Given a finite group ( N , +) and a regular group of automorphisms @ of N ( ( N , @) is referred to as a Ferrero pair,) one can always construct a balanced incomplete block design, abbreviated as BIBD, ( N , B ; , E), where 0; = { a u + b I a, b E N , u # 0}, and $u + b = {q5(u) + b I q5 E a} [5]. This BIBD has parameters (v, b , r , E , A): 1) = INl, k = I@[, b = v (v - l ) / k , T = v - 1, and X = 6 - 1. Since the parameters depend only on the orders of N and a, another Ferrero pair ( N , @ ‘ ) with = I + ’ [ will give us another BIBD ( N , B;, , E) having the identical parameters as ( N , BG, E ) has. It is natural for one to wonder about the difference between them. Modisett 191 gives an example showing a geometric difference between two designs constructed this way with identical parameters. In his example with N = Z29 @ Z29, he found two regular groups of automorphisms of N , one of them is cyclic of order 8, the other isomorphic to a dihedral group of order 8; one of the designs is circular but the other is not, where circular means that three distinct points of N belong to at most one block.

One notices that the two regular groups of automorphisms in Modisett’s example are not isomorphic. But can one always expect to get isomorphic designs from isomorphic regular groups of automorphisms of N ? We’ll show by infinitely many examples in the next section that this, in general, is not the case.

2. ISOMORPHIC OR NOT?

There is a situation when two isomorphic regular groups of automorphisms @ and a‘ of a finite group N give rise to isomorphic BIBD’s, and that is

Theorem 1. If 0 and a‘ are conjugate, then ( N , B ; , E ) and ( N , D ; , , E ) are isomorphic.

Proof Assume that 0’ = g0g- ’ for some g E Aut (N) . We claim that g serves as an isomorphism between ( N , ,Sg, E) and ( N , B;, , E). Since g is already a bijection from N to N , it remains to show that g takes a block in B; to a block in D;,. So let

338 W.-F. Ke

$a + b E B; , where a E N' and b E N . For any y E a,

g ( y ( a ) + b ) = 9 0 y(a) + s(b) = 9 0 cp 0 s - ' ( s ( a ) ) + d b ) l

which is in W g ( a ) + g ( b ) . Therefore, g(Oa+ b ) = @'g(a)+g(b) E B&. Hence the result. For any BIBD ( N , B , E), and for any three distinct points z, y, z E N , we'll denote

by [I, y, z ] the number of blocks in B containing {z, y, z } . Given a Ferrero ilair ( N , a) and ( N , a'), we define a sequence S$ = (no, nl , 712,. . .) by letting

n k = 1 { { ~ , y } c N * Iz#y,[O,z,yl=kin(NiB;iE)}I i

for all k, k = 0,1,2, . . . , where N' = N \ (0). We remark here that in ( N , B G , E ) . [z, y, z ] = [0, y - 2, z - z] for all distinct triple {z, y, z } C_ N [5 ] . The following lemma enables us to use these sequences as a method of telling differences between BIBD's obtained from two Ferrero pairs ( N , a) and ( N , 0').

Lemma 2 . If two designs ( N , B $ , E ) and ( N , B & , E ) are isomorphac, thcn the s e - quences S$ and S$, associated t o ( N , a) and ( N , a') are rdentrcal.

Proof. Let B be an isomorphism between ( N , BG, E) and ( N , B&, E) . Talie a fixed positive integer s and let W, = { { a , b } s N * I a # b,[O,a,b] = a in ( A ' , B $ , E ) ) and 14'; = { { a , b } N' I a # b, [O,a,b] = s in ( N , B & , E ) } . We want to show that

L e t x = { {B (a ) ,B (b ) } I { a , b } E W,} a n d E = - ( { B ( a ) - 6 ( 0 ) , 6 ( b ) - 6 ( 0 ) ) I { c I , ~ } E Wd}. It is easy to see that IW,l = 1w.l = 1W.l. Since

IW8l = IWll.

10, q a ) - e(o), V ) - e(o)l = [W), q a ) , f3(b)l = [O, a, bl = s,

- we have

Consider 8-' and change the roles of W, and IV:, we can get IWjl 5 IWsl. Hence IIV,l = IWlI, and the result.

From Lemma 2, if the sequences S$ and S$, are not identical, then ( N , B$, E) and ( N , B&, E) cannot be isomorphic either. Note that in Modisett's example, we have for one design, nk = 0 for k 2 2, but for the other one, there is an nk # 0 with k 2 2.

Consider now a finite field F with IF1 = q = p', p an odd prime, r 2 1, and let N = F @ F be a 2-dimensional vector space over F . Fix a mapping f : F + F with f(0) = 0 and flp E AutF', where F* = F \ (0). Then for any a E F' \ {l}, the mapping af : N + N defined by a f ( z , y ) = (az,f(a)y) is a fixed point free automorphism of N . For if a f ( z , y ) = (3, y), then az = z and f(a)y = y. Since a # 0, we have z = y = 0. It is also easy to see that if ipf = {a, I a E F'}, then a, is in fact a regular group of automorphisms of N . Moreover, 0, is isomorphic to F'. This technique of constructing @ j from iP and f came to the author's attention indirectly from A. P. J. van der Walt.

We shall use f E Aut F' in the sense that f l p E Aut ( F ' , .) and f(0) = 0 tliruugliout the rest of this section. The notations F2 and F" will be used as F2 = {(x, 1 ~ ) I .r, y E F } and F*2 = {(z,y) I z , y 6 F'} . Also, let 0 = (0,O) denote the identity eleiiient of N . We observe first a fact about these ( N , B$, , E)'s.

C W;, and so IW,l = 1x1 = 5 IW:l.

On nonisomorphic BlBD with identical parameters 339

Lemma 3. If f E Aut F' then ( N , B ; , , E) and (N,B$,-l , E ) are isomorphic.

Consequently, S;, = S;,-l.

Proof. Let 1c, : N + N be defined by $(z ,y ) = (y,~), for all (z,y) E A'. Then ZC, is a bijection from N to N . So if we can show that it sends a block of B;, to a

block of B;,-l , then we have found ourselves an isomorphism between ( N , B;, , E ) and (N,BG,-l ,E). Take a block B = @ j ( z , y ) + ( 2 , ~ ' ) = {(ax + z',f(a)y + y') I a E F ' )

in B;,, where (I, y) , (2, y') E N and (z, y ) # 0. Then (y, 2) # 0 and

1c,(B) = { ( f ( a ) y + y',az + 4 I a E F * )

= {(f(a)y,az) I a E F' ) + (Y',"') = { ( f ( a ) y , f-' ( f ( a ) b ) I a E F') -I- ( Y ' , 4 = { ( b y , f - ' ( b ) 4 I b E F * } + (Y',"') = @f- I(Y, 2) + (Y', 4,

which is indeed a block in B; . So the proof is complete. I-'

Now we are ready for our first class of examples.

Theorem 4. For f E A u t F , the Ferrero pair ( N , @ j ) yields ( N , B $ , , E ) so that each [o,x,y] is either 0 or q - 3, for all x, y E N', x # y.

Proof. The only blocks in ( N , B a , , E ) containing 0 = (0,O) E N are those of the form @ ~ ( - x ) + x , x E N' (cf. [5]). Suppose that b = ( b l , b z ) , c = (c1 ,cz ) E IV', b # C ,

and [ O , b, c ] # 0. Let T.T.' = {x E N * I b, c E @j(-x) + x}. Since 11V1 = [0, b. c ] as one can easily see, our goal is to show that IWI = q - 3.

Notice that if x = (z1,22) E W , there are a, a' E F' \ { 1) such that b = a!( -x) + x and c = a'f(-x)+x, or (b l , b , ) = (-a21 +zI,-f(a)zz +zz) and (c1 ,cz) = ( - u ' T ~ + 21 , -f(a')zz + 2 2 ) . From these we get

bl = (1 - u ) q ; { c1 = (1 - a')z1,

and

(3.1)

(2.2)

Since b # c, we cannot have a = a'. Case 1) There exists an x = (z1,22) E W such that z1 = 0. Then, from (?.I) ,

bl = c1 = 0 , and so if x' = (z\,zk) E W then x\ = (1 - a)-'bl = (1 - a')-'cl = 0. Since b and c are nonzero elements of N , neither b2 nor c2 can be 0. From (2.2) we have

bzc;' = (1 - f(a))(l - f(a'))-'. (2.3)

Conversely, each pair of solution (a,.') for (2.3), { a , u ' } F* \ {l}, in turns, determines an x = ( 0 , ~ ~ ) E N' with 2 2 = (1 - f(a))- 'bz = (1 - f (a ' ) ) - l c2 such that

340 W.-F. Ke

x E W . Since f E Aut F , every a E F' \ (1) will yield an unique a' E F' \ {l}, a' = 1 - ((1 - a)f-l(b; 'cz)) , such that (a,a') is an solution for (2.3), except when 1 - f(a) = bzc;', which yields f(a') = 0, or a' = 0. Therefore IWJ = q - 3, or equivalently, [0, b, c] = q - 3.

Case 2) There exists an x = (21, 2 2 ) E W such that 2 2 = 0. In this case, bz = c2 = 0 and so every x' = (xi, 2:) in W would have 21 = 0. Now neither b1 nor c1 can be zero, so we have, from (2.1), that

61 c;' = (1 - a)( 1 - a')- ' . (3.4)

On the other hand, each solution (a ,a ' ) of (2.4) determines an x = (z1,O) with 21 = (1 - a)-'b1 = (1 - a')-'c1 such that x E W . Again the explicit solutions ( a , a ' ) E F" for (2.4) can be obtained by varying a through F', and get for each a an unique a', u' = 1 - (1 - a)-'b;'cl, except when a = 1 and 1 - a = b l c f ' , the latter will yield a' = 0. Therefore, [0, b, c] = IWI = q - 3.

Case 3) x = (z1,zz) E W implies 0 # { z l , ~ } . Let x = ( z 1 , z z ) E TY. F r o m (2.1) and (2.2), we get

(2 .5)

( 2 . G )

b1c;' = (1 - a)(l - u y ;

b2c;' = (1 - f(a))(l - f(a'))-',

f ( b l C , ' ) = f((1- a)(l - a ' ) - ] ) = (1 - f(a))(l - f (a ' ) ) - l = bzc;'.

and so

If ( z , z ' ) E F*' is a solution for (2.5), i.e., blc;' = (1 - ;)(1 - =' ) - I , then blc;' = f(b1c;') = f((1 - z)(l - z ' ) - ' ) = (1 - f(.z))(I - f(z'))-', i.e., ( 2 , ~ ' ) is also a solution for (2.6). Therefore, IWI equals to the number of solutions (a ,a ' ) E F" for ( 2 . 5 ) . A s discussed in Case 2), there are q - 3 solutions for (2.5). Hence IWI = q - 3 and so [0, b, c] = q - 3. This completes the proof.

So, if f E Aut F , then the sequence Sz, = (no, 121,. . .) for the Ferrero pair ( N , 4,) has the property that n k = 0 except for k E ( 0 , q - 3). There are q2 - 1 blocks in ( N , Bi, , E) containing 0, and each has q - 1 elements. Each of these q2 - 1 blocks

containing 0 also contains ('i ') pairs X, y, with I(O,x,y}I = 3. So, tlieie are

(' ') (q2 - 1) occurrences of such triple {O,x,y} in blocks. Hence we have

1 2

n9-3(q - 3) = ( q 2 2 ) (qZ - 1) = - (q - 2)(q - 3)($ - l),

and so n9-3 = -(q-2)(q2 1 - 1). On the other hand, there is a total of (" 7 ') distinct 2

1 2

triples containing 0 in N , so there are ( q2 ; 1 ) - ;(q - 2)((y - 1) = - (q - 1 )?( q -t- I ) q

of them belong to no block at all. In other words, no = i ( q - l ) z ( q + l ) q

On nonisomorphic BIBD with identical parameters 34 1

Suppose now that g E Aut F' but g @ Aut F. We'll show that the BIBD ( N , B g 9 , E ) is not isomorphic to an ( N , f3$, , E) obtained from an f E Aut F. Since Qg and @ f

are both isomorphic to F', these BIBD's will serve as examples where nonisomorphic designs are obtained from isomorphic regular groups of automorphisms as we have promised.

By Lemma 2, we can prove that ( N , B$,, E) $ ( N , B$, , E) by showing that Si, # S;, . By Theorem 4, all we have to do is to show that there is some TZ; # 0, with b # 0 and I; # q - 3, in S;, = (nb,n;, . . .). The first step toward this is

Lemma 5. I n ( N , B $ , , E ) , there i s an s E F* such that [(O,O), (1,1), (s , - l ) ] # 0.

Proof. Suppose that {(O,O), (1,l)) C a , ( - x ) + x , where x = ( 2 1 , ~ ~ ) E Ar*. Then there is an a E F' such that 1 = (1 - a)x1 and 1 = (1 - g(a))z2, so -1 = (1 - ( 2 - g ( a ) ) ) z * . Set 8 = (1 - g-'(2 - g ( a ) ) ) q , then {(O,O),(l,l),(s,-l)} G @g(-x) + x. Therefore, [(O,O), (0, I), (s, -1)) 2 1 as required.

For every y E F', define #(y) to be the number of blocks B E B g g such that {(O,O), (1, l), (y, -1)) C B. Lemma 5 says that there is some y E F' with #(y) # 0.

Leinma 6. I n ( N , B; , , E), #(y) = q - 3.

Proof. Suppose that y E F' with #(y) # 0, and x = ( 2 1 , ~ ) E N' such that {(O,O),(l,l),(y,-1)) C @ , ( - x ) + x . I f a , a ' E F 'a resuchtha t ( l , l ) = a g ( - x ) + x and (y , -1) = ab(-x)+x, then 1 = (l-a)rl and 1 = ( l -g(a) )xZ, and alsoy = (1-a')a.l and -1 = (1 - g ( u ' ) ) ~ . Note that q, 2 2 # 0. Hence

IEF'

y = (1 - a')(l - a)-1 { -1 = (1 - g(d))(l - g(u))- l . ( 2 . 7 )

The latter equation is equivalent to g(a) + g(a') = 2 . Remember that a # a' and

Conversely, suppose that a,a' E F* \ {l), a # a', such that g ( a ) + g ( a ' ) = 2. Set 2 1 = (1 - a)-', 2 2 = (1 - g ( u ) ) - l , and y = (1 - a ' ) q = (1 - a')(l - .)-I. Then {(O,O),(l,l),(y,-1)) C @ , ( - x ) + x , where x = ( 1 1 ~ 2 2 ) . Therefore,

1 # {%a' ) .

c #(Y) = l t ( % 4 I %a' E F'\ t l ) ,g (a) + 9 ( 4 = 211. IEF'

For every a E F' \ {l,g-'(2)}, let a' = g-'(2 - g ( a ) ) , then g(a ) + g ( a ' ) = 2. Hence

l { ( a , 4 Ia ,a 'E F ' \ t l } , g ( a ) + g ( a ' ) = 2 ) 1 = q - 3 ,

We remark here that, by (2.7), if #(y) # 0, then #(y) = #(y-'). In F', y = y-' happens only when y = f l . So if we can find y E F' such that y # fl and #(y) # 0, then since y # y-' and #(y) = #(y-') 2 1, we can conclude that 0 # #(y) 5 - q - 3 ~

2

342 W.-F. Ke

q - 3, and that nb(,) # 0. But #(1) # 0 only happens when 1 = (1 - a ) ( l - d)- ' or a = a' which is not in our consideration. On the other hand, #(-l) # 0 can only happen when -1 = (1 - a')(l - a ) - ] , or equivalently, a + a' = 2. So the next step toward our goal is to show that there are some a,a' E F' \ (1) such that (1, + a' # 2 but g ( a ) + g ( a ' ) = 2. To do this, we need some help from

Theorem 7. L e t K be a f in i te field of character no t 2 and let p : I< + A' with p ( 0 ) = 0. Suppose tha t p l ~ . E Aut h?. T h e n the following s ta t emen t s are equident: 1 ) p E Aut K ; 2) a + b = 1 in K impl ies p(a) + p ( b ) = 1; 3) p ( a ) + p ( b ) = 1 in K impl ies a + b = 1.

Proof. 1) + 2 ) and 1) =$ 3) are obvious. We note that p(-1) = -1 since ~ ( - 1 ) ~ = ~ ( - 1 ~ ) = p ( 1 ) = 1 and p(-1) # 1. 2) + 1). Suppose that a + b = A, k # 0. Then k- la + k - 'b = 1. By the hypothesis,

p ( k - ' a ) + p(k - ' b ) = 1. But then p ( k ) - ' p ( a ) + p(k ) - ' p (b ) = 1, and so p ( a ) + p ( b ) =

If a+b = 0, then a = -b, and so p(a) = p ( - b ) = -p(b) , or p(a )+p(b ) = 0 = p ( a + b ) . Hence p E Aut h'.

3) =+ 1). By 2) it suffices to show that p(a ) + p ( b ) = 1 whenever a + b = 1. Suppose that pm = 1 ~ . If a + b = 1 with either a = 0 or b = 0 , then it is easy to see t,hat p(a) + p ( b ) = 1. Suppose that a + b = 1 and 0 @ { a , b } . Then 1 = a + b = p-'(p(a)) + p- ' (p (b ) ) . By the hypothesis, p-'(a) + p-'(b) = 1. Again, 1 = p - ' ( n ) + p - ' ( b ) = p ( p - * ( u ) ) + p(p- ' (b) ) , and so p-'(a) + p-2 (b ) = 1. We can repeat this argument, over and over to reach the conclusion that p ( a ) + p ( b ) = p-(m-1)(a)+p-(m-')(6) = 1. Hence the result.

Collorary. Lct I< be a f inite field of character no t 2 and let p : --+ I< uiil.!~~ p ( 0 ) = 0. S u p p o ~ e that ~ I K . E Aut I<*. If f o r all a , b E F , p(a) + p ( 6 ) = 2 implies a + b = 2 , t h e n p E Aut I<.

P ( k ) = + b).

Proof. Suppose p(a) + p( b) = 1. Then

2 = + +(b) = p ( p - ' ( 2 M a ) + p ( p - ' ( 2 ) ) p ( b )

= p ( p - ' ( 2 ) 4 + P(P--'(2)b).

By the hypothesis, p- ' (2 )a + p- ' (2)b = 2. Take a = 0, b = 1, we get p - I ( 2 ) = 'z or p ( 2 ) = 2. Therefore, 2a + 2b = 2 and a + b = 1. By Theorem 7, p E Aut Ii.

In our consideration, g is not an automorphism of F , so this corollary guarantees us a pair ( a , ~ ' ) , {a ,a ' } E F with a +a' # 2 but g ( a ) + g ( a ' ) = 2. If 0 $ { a , a ' } , then the remark before Theorem 7 says that there are some nonzero nk's in S$# with k $! (0, q - 3). Suppose that 0 occurs in {a , a ' } . We may assume that a = 0, and so we have a' # 2 and g(a') = 2 since g(0) = 0. Let b = g ( 2 - a') . Then g- ' (b ) + g-'(3) = ( 2 - a ' ) + a' = 2 , but b + 2 # 2 since b = g ( 2 - a ' ) # 0. Thus, some nonzero R ; ' S

occur in S$ with k 6 ( 0 , q - 3). By Lemma 3, we have some desired nonzero ia i . 's

occurring in S$, and these k's are neither 0 nor q - 3. Therefore, we have proved

Theorem 8. Le t F be a f in i te field of order q, q = p' 2 3 f o r s o m e odd p r i m e p , and integer T , T > 0. Le t f, g be t w o mappings f r o m F t o F wi th f(0) = 0 = y(0). I f f E Aut F , but g E Aut F' \ Aut F , then ( N , Bi, , E) and ( N , B i g , E ) are no t iuomorphic.

9 - 1

On nonisomorphic BIBD with identical parameters 343

c 2

3

C2 + C + 2 = 0

C3 + 2C2 + 1 = 0

2

6

3. EXAMPLES

l(mod 4), and if ( is a primitive element of F , then the mapping g : F + F defined by g ( ( ’ ) = for all k E N, and g ( 0 ) = 0, is always in Aut F’ but not in Aut F . Because if we take i E N such that 6’ = 2-l , then C’ + c’ = 1 but g ( ( ’ ) + g ( ( ’ ) = 2(-1)’(-’ = (-1)i4 = f 4 # 1. By Theorem 7, we cannot have g E A d F . But note that sometimes we still c a n have g @ Aut F for some q = p‘, p E {3,5} , and r > 1.

Table 1 contains some SG, for various F , which are calculated by running a software package called “CAYLEY” on a SUN workstation. “CAYLEY” was specially designed for abstract algebra by J. J. Cannon [2].

a) We point out here that if F,p ,q are as in Theorem 8, with p > 5, y

SG, (7272, 5328, 1296, 0, 0, 0, 0, 0, 0, 0,

132, 0, . . . ) (24576, 10752, 1152, 4608, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 240, 0,. . . )

(94176 80064, 5184, 10368 0 0 4032,

552, 0, . . . ) (126074, 105456, 0, 32448, 0, 234, 91, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 325, 0,. . . )

(184632,112896,14112,32928,0, 7056, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0 , 0, 0, 0, 0, 0, 756, 0, . . . )

(542400, 652800, 129600,67200, 12000, 4800, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,1560, 0,. . . )

0, 0, d, 0, 0, 0, 0, 0, 0, 0 , 4 b, b, 0, 0,

Table Q 13

17

25

27

29

41

b) We are going to supply more examples of S: arising from special ( N , @)Is which really draw our attention. Still we consider a finite field ( F , +, .) of order y = p” for some odd prime p , and integer r > 0, such that q > 9. Let c be a primitive element of F . Define 4 : F + F via 4(z) = Cz, for all 2 in F . It is easy to see that 9 is a fixed point free automorphism of (F , +). If fact, 4 generates a regular group of automorpliisms of ( F , +), which is isomorphic to ( F * , .). For every divisor k of q - 1, we get a Ferrero pair (F,@k), where @ k is the subgroup of 0 generated by 4‘, and l@kl = (y - l ) / k . From (F , @ k ) we obtain a BIBD ( F , a;, , E). We shall put our attention on the case when k = 2. We have

Theorem 9. lfk = 2, then SG2 = ( O , . . . ,O,n,,n,+1,O,. . .), where ns and ?] ,+I are nonzero positive integers, and

i f p E 1 (mod 4); s = { ! ( q - 7) i f p = 3 (mod 4).

Moreover, n, and n,+l satisfy the equations

i ( q - 9)

+ na+l = $(q - l ) ( q - 2); s + n,+1 ( 8 + 1) = +(q - l ) ( q - 3)(q - 5).

344 W.-F. K e

Q 17 19 23 29 31 37

Before proving this theorem, we list some special cases in Table 2 to illustrate the theorem.

%* (0,0,24,96,0 ,...) (O,O,O, 108,45,0,. . .) (O,O,O,O, 165,66,0,. . .) (0,0,0,0,0,84,294,0,. . .) (0,0,0,0,0,0,315,120,0,. . .) (0.0,0.0,0,0.0,144,486,0, . . . )

Proof of the theorem. Let a, b be two distinct elements of F*, and suppose that. [O,a,b] # 0. Set W = {(x,y) E F'' I ab-l = (1 - x2)(1 - y2)- ' } and Af = { c E F' I a, b E @2(-c) + c}, so that [0, a, b] = [MI. The idea is to set up a correspondence between W and M, then, by counting the number of elements in W , we are able to get IMI and [0, a, b] .

L e t c E M . Thentherearern,nE {1,2 ,..., -- (' - 2, 1) such that 2

a = c - $2"(c) = (1 - (2")c; b = c - $*"(c) = (1 - C'")C.

Let (" = x and C" = y and rewrite (3.1) to get a = (1 - x 2 ) c and b = (1 - y2)c, so that

ab-' = (1 - x 2 ) ( 1 - y2)-' , (3.2)

or (x ,y) E W . Conversely, suppose (z,y) E W and set c = (1 - x 2 ) - ' a = (1 - yz)- 'b. Then

{O,a,b} Notice that for any distinct nonzero a', b' in F', the above argument shows that

[O,a', b'] is not zero if and only if the set W' = {(z,y) E F*2 I a'b'-' = (1 - r2)(1 - y2)- l} is not empty.

Since (1 - z2 ) - ' a = (1-xf2)-'a if and only if x2 = xf2, and (1 -y2)-'b = (1 -y")-'b if and only if y2 = yf2, and x2 = zz for any z in F' has two solutions for x in F', we have that 41MI = IWI. Let w = ~ b - ' and rewrite (3.2) as

@z(-c) + c, or c E M.

W y2 = 1. -22 - - 1

1-w 1 - w (3.3)

Then W = W' \ (WI U W2 u Ws), where W' = {(z,y) E FZ I (x ,y) is a solution for (3.3)}, WI = {(x,~) E F2 I x2 = y2 = l}, W2 = {(x,y) E F2 I x 2 = 1 - w ) , and lv, = {(x,y) E F2 I y2 = (w - l)/w}. This is because a, b, x and y are nonzero, but

On nonisomorphic BIBD with identical parameters 345

x2 = y2 = 1 implies a = b = 0, x2 = 1 - w implies y = 0, and y2 = (1 - w)/w implies I = 0. Now IW1l always equals 4, and IW2l (resp. IW31) is either 2 or 0 accordiiig to whether 1 - w (resp. (w - l ) /w) is a square in F o r not.

By Lemma 6.24 of [8], IW'I = q - q(-l)r](-)q(-), where r] is the honio- 1 -W

1 - w 1 - w morphism from F' to {I, -1) defined by

'I(') = { yl, otherwise.

Therefore,

if x is a square;

1 w - 1 z ( q - r](-1) - 8), if ~ ( 1 - w) = v(-) = 1;

W

(3.4) w - 1

1; 1 w - 1 z ( q - r](-I) - 4), if ~ ( 1 - w) = q ( - ) = -1. 1 W

1 [O,Q,b]=-IWI= 4 f(q+q(-1)-6), ifr](l-w)r](?) = -

But -1 is a square in Fq if and only if p G l(mod 4), or equivalently, r ] ( -1) = 1 if and only if -1 is a square in Fq. From (3.4), we get

(1) if p = l(mod 4) then

w - 1 if r](1 - w ) = r](- ) = 1; W

z ( q - 5 ) = f ( q - 9) + 1, otherwise; [O,Q,bI =

(2) if p 3(mod 4) then

So that s = f(q - 5 ) or f (q - 7) according to the situation whether p l(mod 4) or p

The above argument also assure us that for any distinct nonzero u ' , b' in F , the set W' = {(z,y) E F" I a'b'-' = (1 - x2)(1 - y2)-'} is not empty. As we noted before, this implies [O,a',b'] # 0. Since there are 1 = $ ( q - l ) ( q - 2) distinct triples in (N,23i2, E) containing 0, we have 1 = n8 + n,+1. Secondly, there are q - 1 blocks containing 0, and each has ( q - 1)/2 elements. Each of these q - 1 blocks contains

0 also contains ((' -1)/2) pairs x, y with l{O,x,y}I = 3; consequently, there are

3(mod 4). This concludes the first part of the proof.

I' = ((' -1)'2) (q - 1) = i ( q - l ) ( q - 3)(q - 5 ) occurrences of such triple {O,x, y} in

blocks, and so we get I' = n, . s + n,+1 - (s + 1). The proof is now complete.

4. Remarks. The author believes that the sequence $4 for a Ferrero pair ( N , @ ) contains some information for the BIBD ( N , D $ , E ) that we don't know now. On the

346 W.-F. Ke

other hand, there are a various ways that one can construct BIBD’s from a given Ferrero pair ( N , @). For example, ( N , Do, E) and ( N , B,, E) are sometimes BIBD’s (See [l], [4].) We can also define sequences S+ and SG for ( N , B + , E) and ( N , S,, E) , respectively, if they are BIBD’s. Naturally, it is the relations between the sequences and the BIBD’s that we would like to know.

5. Acknowledgement

Arizona for his guidance on doing the research and writing up this paper. The author wants to express his thanks to Professor J.R. Clay in the University of

6. References 1

2

3

4

5

G

7

8 9

10

Betsch, G., and J.R. Clay, Block designs from Frobenius groups and planar near- rings, Proc. Conf. Finite Groups (Park City, Utah), Academic Press, 197G, 473- 502. Cannon, J.J., An introduction to the group theory language, Cayley, Conipnta- tiond Group Theory, M.D. Atkinson, ed., Academic Press, London, 19S4,145-163. Clay, J.R., Generating balanced incomplete block designs from planar near-rings. J . Algebra 22 (1972) 319-331. Clay, J.R., More balanced incomplete block designs from Frobenius groul)5, Dib- Crete Math. 59 (1986) 229-234. Clay, J.R., Circular block designs from planar near-rings, Ann Discrete Afatli 37

Clay, J.R., Tactical configurations from a planar nearring can also generate 1x11- anced incomplete block designs, J . Geometry 32 (1988) 13-20. Clay, J.R., and H.J. Karzel, Tactical configurations derived from groups liaviiig 2 1

group of fixed point free automorphisms, J . Geometry 27 (1986) 60-GS. Lidl, R., and H. Niederreiter, Finite Fields, Addison-Wesley, Massachusetts, 19S3. Modisett, M.C., A characterization of the circularity of certain balanced incomplete block designs, Doctoral dissertation, Univ. of Arizona, Tucson, Arizona, 1988. Modisett, M.C., A characterization of the circularity of certain balanced inconiplete block designs, Utilitas Mathematica 35 (1989) 83-94.

(1988) 95-106.


Incidence loops and their geometry

Mario Marchi

Dipartimento di Matematica UniversitA Cattolica, via Trieste 17, I 25121 Brescia, Italia

Introduction

As it is known an incidence group ( P , L, .) is a group (P , . ) together with a structure ( P , L ) of an incidence space such that both structures are compatible, i.e. for any a E P the map a* : P + P ; I + az is an automorphism of ( P , L). If furthermore ( P , L , . ) is fibered, i.e. each line 1, E L(1) := { X E L : 1 E X } is a subgroup, then by A // B :* A- 'A = B-'B we have a parallelism relation on L. Thus (P , . ) is isomorphic to a group of translations, acting regularly on P.

The incidence group ( P , L, .) is called &sided if also each map *a : P + P , I + za , for any a E P , is contained in A u t ( P , L). A h i d e d fibered incidence group is a kinematic

space and for each a E P and X E L(1) we have the property aXa-' E L(1). The notion of incidence group can be generalized by weaking the assumptions concerning the algebraic structure of P ; so we have the notion of incidence groupoid,

incidence loop and kinematic loop. The first steps in this direction have been taken by H.Karze1 '76 [ 5 ] , H.Wahling '77 [15], G.Kist '80 [9]. More recently there appeared the papers [6] by H.Karze1 and G.Kist and [18] by E.Zizioli where many examples of kinematic loops were derived from alternative kinematic algebras ( A , K).

The general properties of incidence loops ( P , C, .) are studied in this Note in $1, where is also stressed the connection between incidence loops and the existence of a set C of collineations acting transitively on the set P of points of the incidence space

(P ,C) . The case of an incidence space with parallelism ( P , L , / / ) is studied in $2

348 M. Marchi

where conditions are stated so that C is a set of translations. §3 is devoted to %sided

incidence loops and kinematic loops. In $4 and 5 the case is considered of an incidence space with parallelism (P, C, //) which permits a regular set C of translations. Then we can define a family 3 of subsets of P (which we shall call fibers) behaving like the lines of C (proposition (4.2)).

For the set 3 U C we can prove (proposition (4.4)) certain configurational propositions (which we shall denote by ( K l ) (K2)) which are ageneralization of both the Desargues

configuration and the so-called Prismenaxioms.

Furthermore we will prove that C is a group if and only if a further configuration theorem (K3) is fulfilled (cf. propositions (4.5) and (5.1)).

Additional configurations are also studied in the propositions (4.6) and (4.7) which are resp. a generalization of the so-called Scherensatz and of the Pappus afine

configuration.

To end, $6 and 7 are devoted to the study of different examples of incidence loops.

1. Incidence loops and collineation sets

By an incidence loop we mean a triple (P ,C, . ) where the following conditions are fulfilled:

11. (P, C) is an incidence space i.e.: i) V a , b E P, with a # b, 3 ! L E C with a , b E L ;

we shall write a,b := L; ii) V X E C : 1x1 2 2. In the following P will be called the set of points and L: the set of lines.

Al. (P,.) is a loop such that V a E P

is an automorphism of (P, C) i.e. v a E P, v L E C:

a'(L) := a L E C.

The unitary element of this loop will be denoted by 1. Thus

(2) C := {a' : a E P } A u t ( P , C )

is a regular set of colldneataons of (P, C ) and the identity map "Ed" belongs to C. Vice

Incidence bops and their geometry 349

versa we know the following connection between regular permutation sets and loops

(cf. e.g. [12], [18]):

1.1 Let ( P , L ) be a n incidence space, C a collineation set acting regularly o n P with

id E C and let 1 E P be a distinguished point. If for any a E P, ,'a*" denotes the

collineation of C uniquely determined by a*(1) = a, then ( P , .) with a b := a'(b) i s

a loop and 1 is the neutral clement. Thus ( P , L , . ) is a n incidence loop.

On incidence loops we know the following general facts (cf. [ll] and [18]):

1.2 Let ('PIC, .) be a n incidence loop and C the set o j collineations defined in (2).

(i) (P,.) is a group if and only i f C is a group.

(ii) T h e following conditions (a), (b ) and also (c), ( d ) are equivalent:

(a) V a,P E C : N p(1) = 1 + P a(1) = 1;

(b) V a E P 3 a-1 E P with a . a-1 = u-l . a = 1;

(c) c = c-'; (d) (left inverse property):

(L.I.P.) V a E P 3 a-1 E P such that V b E P : a-'(ab) = b.

(iii) V z , y E P, V A E C(l) 3 B E L(1) such that

(3) 4 Y A ) = (.Y) B .

An incidence loop ('P, L, .) is called fibered incidence loop if the following additional condition is fulfilled:

A2. V M E L(1) := { L E C : 1 E L } the subset ( M , .) is a subloop of (P, .).

Thus C( 1) is called a parti t ion of P in subloops. Furthermore we have:

(4) L: = { u M : M E L(1), a E P} .

350 M . Marchi

2. Fibered loops and incidence spaces with parallelism

An incidence space (P, C) is called an incadence space with parallelism ( P , C , / / ) if there is an equivalence relation / / defined on C, called parallelism, such that the following condition (named Euclidean aziom) is fulfilled:

12. V x E P, V L E C: 3 ! M E C with z E M , L / / M .

In the following we shall write (z / / L ) := M .

If ( P , C , //) is an incidence space with parallelism then 6 E Aut(P, C) is called a dilatation if for all X E C : 6(X) / / X. We shall denote by Dil(P, C, / / I the group of dilatations and by 7 & Dil(P, C, / / ) the set consisting of the identity and all fixed point free dilatations; 7 will be called the set of translations.

In [ll] and [18] the following facts are proved.

2.1 Let ( P , C, .) be a fibered incidence loop and let ”//” be defined on C as follows:

( 5 ) X / / Y : H ~ A E C ( ~ ) , x , y ~ P : x = x A and Y = y A . Then:

i ) ”//’I is a parallelism if and only if

(6) ii) if ”//” is a parallelism then C & Dil(P, C, //) H

( 7 )

V A E C(l),V a E A,V z E P : (xu) A = zA ;

V A E C(l),V z , y E P : (xy) A = z(yA) .

Now let (P ,C , . ) be a fibered incidence loop with the property (6) and let ”//” be the corresponding parallelism as defined in (2.1). Then

(8) Since id E D i l ( P , C , / / ) is the only dilatation which fixes more than one point, we

have C n Dil(P,C, I / ) c 7 and C c 7 if (7) is valid. From now on we assume ( 7 )

for (P ,C , 3). In the case that (P,.) is even a group, hence (6) and (7) are valid, then ( P , C, .) is a fibereti incidence group and C is a group of translations of (P, C, 11). If ( P I C , .) fulfils the condition (6) then we call ( P , C, .) a fibered incadence loop with

parallelism and we assume that (PI C, .) is provided with the parallelism / / given by (2.1). Furthermore in [18] is proved that

V a , b E P , V M E C(1) : ( b / / a M ) = bM .

Incidence loops and their geometry 35 1

2.2 If ( P , L, //, .) is a f inite fibered incidence loop with parallelism, fulfilling condition

(7) and IC(l)l > 1, t hen ( P , . ) is a group.

In the section 6, we shall see that for all kinematic Moufang loops (G, L, .) which are derived from alternative kinematic algebras the law (7) is valid if and only if (G,.) is a group. (cf. [18], IS). On the other hand an example of a proper fibered incidence loop with parallelism fulfilling condition ( 7 ) will be given in the section 7.

Let us now return to the general case. In [12] the following proposition is proved:

2.3 Let ( P , C , //, .) be a fibered incidence loop with parallelism fulfilling condition (7).

Let M E L ( l ) , x E P \ M , 6 E M' o Ma and E E Ma o Ma o M'. T h e n

a) If 6 ( x ) = x t h e n 6 = i d .

b) T h e following s tatements are equivalent:

i) (Me)- ' = Ma;

ii) if 6(1) = 1 t hen 6 = i d ;

iii) M' o Ma i s fixed point free.

c) T h e following s tatements are equivalent:

i) ( M , .) is a group;

ii) iffc(1) = 1, t hen E = i d ;

iii) M' o M' o M' is fixed point free;

iv) V a , b E M,V x E P : ( a b ) x = a(bx ) .

3. h i d e d incidence loops and kinematic loops

An incidence loop ( P , C, .) is called ?hided i f

A3. for each a E P the map

(9)

is contained in Aut(P, C). ( p + p 5 -+ xa

352 M. Marchi

3.1 Let ( P , L, .) be a h i d e d incidence loop. Then:

(i) V A E L(1), V I, y E P 3 C E L(1) : ( A I ) y = C ( z y ) ;

(ii) L = { a x : a E P , X E L(1)) = {Yb : b E P , Y E L(1)).

Let us remark that in a fibered incidence loop where for every X E L( 1) one has 1x1 = 2, the subloops ( M , . ) , for M E L(1), are subgroups and furthermore the incidence loop is 2-sided. In order to avoid this case henceforth we shall assume

(10) V X E L : I X I 1 3 . A kinematic loop ( P , L, .) is a 2-sided fibered incidence loop, i.e. L(1) consists of subloops. If furthermore the condition (6) is fulfilled, then the incidence space ( P , L) can be provided by (2.1) with a parallelism / / and if also

(11) V A E L(1), V a E A, V I E P : A(ax) = As

is valid, then

(12) X / /r Y :* 3 A € L(1),3 e , y € P : X = A s and Y = Ay

defines a second parallelism, called right-parallelism. The problem is open if, under these conditions, (P, L, //, /I,.) is a double space, i.e. V A , B E L if a E A,b E B then

4. Configurations in fibered incidence loops with parallelism

Henceforward ( P , L, //, .) will always denote a fibered incidence loop where the conditions (7) and (10) are fulfilled and where ”//” is defined as in (2.1). For any M E L( 1) and for any I E P let us consider the set of points

M * ( s ) := {a* (s ) : a* E M ’ } = {ax : a E M } = Ma:

and the family of subsets of P

3 ( M ) := { M * ( I ) : I E P} = {MI : I E P} .

Incidence loops and their geometry 353

Of course, since 1 E M , z E M z .

F ( M ) will be called a partition of the point set P if and only if we have:

(13) V I , Y E P : M * ( z ) n M ' ( y ) # 0 + M'(z) = M*(y) .

4.1 If M E L(1) then:

v u E M , v z E P \ M : ( a / / G) nMz = {UZ}

Proof. Since by (8) ( a // =) = a(=), we have: ax E [ (a / / 1,nMzI. Let us now assume y E [ (a / / G) n M z ] . Then there exists m E M such that y =: mx hence y E a(=) n m ( G ) which implies (a / / F) = ( m // F), i.e. m E ( a // c) n M .

But since z @ M , ( a / / =) n M = a, hence m = a, i.e. y = ax . 0

4.2 For any M E L( 1) the following conditions are equivalent.

(14)

(15)

(16)

(17)

(18) of maps, is a group.

F ( M ) is a partition of P.

V a E M , V z E P : M z = M ( a z ) .

( M , . ) is a subgroup of (P, . ) .

V a , b E M , V z E P : a(bz) = (ab)z.

The couple ( M ' , o ) , where M' := {a' : a E M } C C and "0 " is the composition

Proof.

(14) + (15). Since az = 1. ( a x ) E Mz n M ( a s ) , we have MI = M ( a z ) .

(15) + (14). Let a , b E M with az = by E (Ma: n My). Then M z = M ( a z ) =

= M(by) = My.

(16) e (17). By (2.3,~). (17) + (15). Obvious. (15) + (17). Forany a , b ~ M a n d z E P \ M w e h a v e b y ( 7 ) a ( b z ) E ( a b ) . F = = (ab // c) and by (4.1) (ab // F ) n M z = { ( a b ) z } , furthermore by (15) a(bz ) E

E M ( b z ) = M z and thus a(bz) E (ab // p) n M z ; this implies (ab)z = a(bz ) i.e. (ab)'(z) = a*ob'(z) for all E P\M. Consequently (ab)' = a'ob' since a', b',(ab)*

are dilatations, and RO for any y E P, (ab)y = a( by).

354 M. Marchi

(17) e (18). By (17) ( M ' , o ) is a semigroup isomorphic to ( M , . ) . Since ( M , .) is a

loop, ( M e , 0 ) and ( M , .) are groups, and clearly (18) implies (16). 0

By (2 .3 ,~) and (4.2) we have:

4.3 Let be M E C(1); i f M' is a group then, 3 ( M ) is a partition of P and for

m E M,x E P we have:

m(Mz)=Mx .

If for each M E L( 1) the family of subsets 3 ( M ) is a partition of P then by (4.2) (14) and (16), the set L( 1) consists of subgroups and also (15) is valid for all M E L( 1). Therefore by [16] (5.2) the tripe1 ( P , F , n ) defined by

(19) 3:= U { 3 ( M ) : M E C(1)) = {Mx : M E L(l),Z E P}

is an incidence space with parallelism. The "lines" of 3 will be called strings and if a , b E P with a # b then zb E 3 will denote the uniquely determined string joining a and b. For F E 3 and x E P let (x n F) E 3 such that x E (x F) and (x n F ) jj F. Clearly if M E C(1) and x , y E P then ( y n Mx) = My. For the remainder we assume that L(1) consists of subgroups and we shall be concerned with both incidence structures C and 3 defined on P.

It follows immediately from the definition:

It makes sense now to investigate the geometrical configurations defined on P by the set L: U 3. Since for each x E P , G is a subgroup there is exactly one 2-l E P with x + I-' = I-' . x = 1 and we ha.ve:


4.4 The following configurational propositions hold.

i) V a, b,c E P distinct and t E P with c = ta:

- (21) a, b n ayc = { a } H ( c // a,b) n ayc = { c } ;

configuration ( K l ) :

-- (23) t b E ( c // a,b) n ( b n c c ) = ( t a // a,b) n ( b // a , t a ) ;

- (24) a , b n C c = { a } H ( c / / a , ) n ( b n ~ c ) = { tb} ,

01" - h -- a , b n a , t a = { a } ~ ( t a / / a , b ) n ( b / / a , t a ) = { t b } ,

in particular

- - (25) {I} = 1, b n 1, c @ (c . l,b) n (z. b ) = { c . b} .

ii) Configuration (K2):

n F i ) = {a , } for any i , j E {1,2,3},i # j ) from a1,a2 / / bl,bz,al,a3 // blrb3 it

t/ Fl , F2, F3 E F, Fl n F2 n F3 distinct, and V air bi E Fi( i = 1,2,3) (with (m n

follows -

// b2,b3 .

Proof. (i) Let s , t E P with b = as and c = ta . Since a # b,c we have s , t # 1 and so S := 1, s, T := l,t and a , b = a S = (as), aTc = T a .

Then by (4.3) and (7): t ( 3 n aye) = t . ( a S n T a ) = [ ( t a ) S ] n T a = cS n T a =

= ( c // a,b) n ayc. Hence (21). Now: a E a,b n aTc = a S n T a , b E [a,b n ( b / / .Ti)] = [bS n Tb]. For t' E T \ (1) we have: t'a E a S H a,t'a = a S = ( t ' a )S = t ' ( a S ) = t ' ( b S ) = b,t'b H t 'b E bS. This gives us (22). Since b E a S = a,b implies t b E ta,tb = cS, we have ( c // a,b) fl ( b / / ayc) = c S f l Tb.

Hence (23).

- -

h

h

356 M . Marchi

- Now we show (24). By (23) we have ( t a / / a,b) = ta,tb. Therefore by (21) and (22) respect, { a } = a,b n U ~ U e { t a } = ( t a f f a,b) n uTa = ta, tb n ta ,a H

H ( t b // t a ,a ) n t u , t b = { t b } .

(ii) Because of (21) for any j = 2 ,3 we have: { b l } = mf7 F1, and { b j } = b l , bj n Fj.

Let a1 # b l , u E P with bl = uu1 and U := F. Then a l , bl = Ual = F1 / / Fz =

= Uaz // F3 = Ua3 and thus by (24) u az = b2,u a3 = b3. This implies u(-) =

- h

A h - -

h h

h

- = bz,b3 / I a2,a3 0

Now a third configuration may be considered which involves deeper properties for C.

4.5 If ( P , .) i s a group then the following configuration ( K 3 ) holds:

V L 1 , L z , L3 E C, L 1 f f Lz f f L3, distinct and V a , , b, E L , with ( u z j n L , ) = { a , }

f o r a n y i , j ~ { 1 , 2 , 3 } , i # j ) and A h h h

a 3 2 / I bl ,b2 7 a 3 3 / I bl ,b3 7

it follows h h

a z 3 // bZrb3 .

- Proof. For z E {2 ,3} let ui E P with a, = ui a1 and U , := 1,ui. Then by our

assumption and (24) , b, = ui b l . Let u := u3 u z l ; since (P,.) is a group, uaZ =

= u3al = a3 and ub2 = u3bl = b3. Thus a z 3 f f b z , b3. 0 h h

Let us now return to the general case.

4.6 The following configurational proposition ( K 4 ) holds.

Let E , F E F be distinct with E that:

F . Let a,, b,(z = 1,2,3,4) be distinct points such

i) al ,a3~bl ,b3 E E;aZ,a4rbZrb4 E F , ii) V j = 2 , 4 : l m n El = la3,ajn El = 1,

iii> a1,a2 // b i , b z , a l , a a // b i , b 4 , a a , / / / m, then: G K / / b3,b4 .

- - -


Proof. By the assumption (ii) we have V J' = 2,4: by (22) { a j } = al,aj n F =

= a3,aj n F , by (21) { b l } = mfl E , {b3} = b2, b3 n E , and then again by ( 2 2 )

{ b j } = bl ,bj n F and { b z } = b z , b 3 n F. Now let m E P such that ma1 = bl . Then

by (24) for j E {2,4}, { a ] } =- n E =m n al-ul implies { m a j } =

= ( m a 1 // al,aj) n ( a , // al-al) = (bl / / al,aj) n F = m n F = { b j } i.e. m a j = b j , and now in the same way ma3 = b3.

But m .

- - -

h

- - = m u 3 , ma4 = b3, b4 implies a3,a4 // b3, b4. 0

The configuration (K4) is a generalization of the so-called Scherensatz which is fulfilled in Desarguesian f i n e planes (cf. e.g. [8]).

4.7 Let M E C(1).

a) If the group ( M , .) is commutative, then the following configurational proposition

(K5) holds.

Let E , F E 3 be distinct with E

distinct points such that:

F M . I f a l , a z , a 3 E E ; bl,bz,bs E F are

i) lal,bzn El = I f in El = laz,nnEl= 1, - -- -

ii) a1,bz / / a2,b1, a2,b3 // a3,b2,

then - - al,b3 // a3rbl I

b) If there are E , F E 3 distinct such that E

following conditions are valid:

i) V 2 E F : a , n E = { a } ,

ii) V a2, a3 E E,V b l , bz , b3 E F with a , b2 // a2, bl and a2, b3 // a3, b2 : a , b3 / / a3, bl ,

then ( M , .) is commutative.

F M and a E E such that the

- - - -- -

--- Proof. (a). By the assumption (i) and (21) each of the lines a l , b2, a l , b3, a z , b l ,

u2, b3, a3, bz intersects each of the strings E and F in exactly one point. Now let

m1,mz E P such that a1 = m l a 2 , and a3 = m2a2. Then E / / l ,mi , hence mi E M

and, by (24), b2 = m l b l and bz = m2b3. Further by ( 1 6 ) a2 = m l l a l and so

a3 = m 2 ( m 1 1 a l ) = ( m 2 m ; ' ) a l = ( m l ' m 2 ) a l . On the other hand, again by ( l 6 ) ,

( m l l m 2 ) b ~ = m T ' ( m z b 3 ) = rnl 'bz = bl , i.e. a ~ , b3 // a3, bl.

-- ----

- -

358 M. Marchi

Configuration (K5) is the analogous of the Pappus a f i n e configuration. For a generalization of this proposition in kinematic spaces, cf. [7].

5. A characterization of incidence loops which are incidence groups

Till now we have studied the general case of a fibered incidence loop with parallelism ( P , L, //, .). If in addition (P, .) is a group then (P, L, .) is a fibered incidence group (cf. [3]). Furthermore ( P , L, a ) is a two-sided incidence group if and only if 3 = C. In this case (P, L, //, .) is called a kinematic space. The two parallelisms / / and // are now defined in the same line set L. / / I := // is called left parallelism and /Ir := // right parallelism (see e.g. [4], [14]). In this case the proposition (K1) studied in $4 is exactly the axiom characterizing a

Doppelraum in [7], and (K2), (K3) are the Prismenaxioms of the same Note. When 3 = L, the condition // = // describes exactly the case of the so-called central trans-

lations (cf. e.g. [lo]); i.e. translations 7 # id such that V z, y E P : z, ~ ( z ) / / y, ~ ( y ) .

In this case for any M E L(1) and z E P , M z E L and configurations (K2) and (K3)

coincide.

In $4 we have studied some configurations which are valid in any fibered incidence loop with parallelism ( P , L, //, .) only assuming conditions (7) and (17): configurations K1, K2, K4 and, if ( M , . ) is commutative, K5. The further configuration (K3) was

proved under the stronger condition that ( P , - ) is a group. We shall prove now that this condition is also necessary. In this way we have a geometrical characterization of such incidence loops which are incidence groups.

h

h

h

--

5.1 Let ( P , L, //, .) be a fibered incidence loop with paralleliam fulfilling the conditions


(7) and (10) and for each M E L(1) let (Id,.) be a subgroup of P ( c f . 4 .2) . Then

the loop (PI .) is a group i f and only i f the conjguration (K3) ( c f . 4 .5) holds.

Proof. By (4.5) we have only to prove that under the assumption of (K3), (P, .) is a group. For u , v E 7' \ { 1) let A(u, v ) := {z E P : u(vz) = (uv) z } be the associator of u , v. We have to show A(u, v ) = P. We will proceed with the following steps.

i) If 1, u , v are collinear, then A(u, v ) = P b y (4.2), ( 1 6 ) and (1 7).

ii) If l , u ,v are not collinear, then A(u,v) 2 R := P \ (V U W ) where V :=

w := 1 , u v .

Proof. By (25) we have:

and -

V u , b ~ P \ { l } w i t h b $ ~ : { a b } = ( u ~ ~ ) f l ( ~ a b ) a n d u b $ ( l , U ~ ) .

Therefore { uv} = ( u V fl Uv) and W # U, V . Then for any z E P \ (V U W ) we have {VX} = [(u .=) n VZ] g (G u v) and {(uw) X} = [ ( ( u v ) .=) n W Z ] g

_-_ - ( F U W ) . Then 0,2)2 // G // uv,(uv)z, z z z V , x , ( u ~ ) z // W , hence by

(K3) v- V ~ V / / 1 , u // vma. Therefore (ti.). E ((vz) // v j v ) and by (8) ( u v ) z E (uv) .= = ((uv) / / 0,2)2. Thus together with (23) {u(vz), (ti.).} C

n fi. (vz). Now respectively by (24), (22) and (24): .2: @ V $ { 1) = F n + G n z T z = {z} + 0,2)L n v j v =

= {v} + 1 ( u v ) . l , z n l , u . ( v z ) l = l henceu(vz)=(uv)z.

iii) If l , u , v are not collinear, then A ( u , v ) = P.

Proof. Since u* o v* and ( u v)* are dilatations which by (ii) coincide on R, and IRI > 1 we have (uv). = u* o v*, hence A(u, v ) = P.

Ah-- h

6

[((uv) / / m) fl ((vx) / / v z v ) ] = (uv) .

- -

6. Fibered incidence loops derived from algebras

In [18], 56, the following example is studied.

Let ( A , K ) be an algebra with identity 1. For T c A let T* := T \ (0). If u E A let u* (z ) := ar and * u ( z ) := zu for each z E A .

Then E := { u E A : d , * u E S y r n ~ } is the set of units. Let U be a subset of E such that (U, .) is a loop and I i * c U . Since K* is a central subgroup of the loop (U , .)

we can form the factor loop G := U / K * (cf. [2] p.61).

360 M . Marchi

If we consider the projective space (P, 0) with the set P := A * / I i * of points and the set 0 := { ( K a + K b ) * / K * : a, b E A* with Ka # K b } of lines corresponding to the vector space ( A , K ) , then we can provide G with a structure C of an incidence space by taking the trace-space: C := { X n G : X E 0, IX n GI 2 2).

For any u E E the maps u*,* u are K-linear permutations of the vector space ( A , K )

and hence iI* : P + P ; K*x -t K*ux,*iI : P + P; K*x -+ K*XU axe collineations of the projective space (P,o) . Thus for any g = K*u E G (u E U ) the maps g* : G -+ G;x + go and * g : G -t G ; x -t xg are automorphisms of (G ,L) .

Therefore (G,C, . ) is a 2-sided incidence loop, fulfilling the condition (3) and the properties ( i ) and (ii) of proposition (3.1), with L = {a X : a E G,X E S} where S := { [ ( K + K a ) n U ] / K * : a E U \ K } = L(1). Now we assume further that ( A , K ) is kinematic, i.e. a2 E K + K a for each a E A.

Then if a E U \ A', the set K + Ka is a commutative and associative subalgebra of ( A , K ) hence X := ( ( K + K a ) n U ] / K * E C(1) is a commutative subgroup of the loop (G, *). The following proposition is proved in [6], $7 and [18], $6.

6.1 Let (A, I i ) , U , G , C be defined as above. Then:

a) (G,C,.) is a %sided incidence loop;

b) If ( A , Ii) is kinematic t hen (G ,C , . ) is a kinematic loop and all fibers X E C(1)

are commutat ive subgroups;

c) If ( A , Ii) is kinematic and if ( U , . ) f u l f l s the right inverse property (i .e. v a E E U 3 a-l E U such that V b E U : ( b a ) a-l = b holds) t h e n ( 6 ) is valid;

d) If (A, K ) is kinematic and alternative (i .e. V a , b E A : a ( a b ) = a2b, (ab )b = ab')

then:

(a) U := E is already a loop,

(p ) ( G , L, .) is a k inematie Moufang loop,

(7) the condition ( 6 ) is fulfilled,

( 6 ) (G , .) is a group if and only if ( A , K ) is associative,

( E ) the condition (7) as valid if and only if ( A , K ) is associative.

Incidence loops and their geometry 36 1

7. A proper fibered incidence loop with a regular set of translations

In this last section we present an incidence space with parallelism ( P , C, //) which admits different translation sets C acting regularly on the set P of points. When C is a group, then by (1.1) and (1.2, i) P can be turned in a group such that (P, L , .) becomes a fibered incidence group. If C is not a group but nevertheless acting regularly on P , ( P , L , a ) becomes a (proper) fibered incidence loop, fulfilling the condition (7) since C is a set of translations. This example is studied in [13], 53; for other remarks one can see also [l] and [15]. I n t h e g r o u p P : = Z 2 : = { a : = ( a l r a 2 ) : a 1 , a 2 E Z}(witha+b:=(al+pl,az+Pz))

let ya = (ya1,ycy2), for y E 2, and [G.C.D. (a)] be the greatest common divisor of a1 and a2. Then C(0) := (2 - a : a E P \ {(O,O)) and [G.C.D. (a)] = 1) is a fibration of (P,+), and so ( P , + , C , / / ) with C := { a + X : a E P , X E L(0) ) and " A / / B :e A - A = B - B" for A, B E L, is a commutative kinematic space. Our aim is to provide the space ( P , C, //) with another transitive set 7 of translations. Let PE := {a := (a1,a2) E P : a I , a 2 E 22) ; Po := { a := (a1,a2) E P : a 1 , a Z E

E 2 2 + 1);PM := { a := ((Y1,02) E P : + (Y2 E 2 2 + 1). For any a E P we define the dilatations

While for a # 0, a+ is fixed-point-free, a* has a fixed point (namely: (1/2 a1,1/2a2)) if and only if a E PE. Therefore 7 := P z U Po+ u P$ U Po' U P h is a transitive set of translations of P. For a , b E P we have the following rules:

hence:

362 M. Marchi

From this we deduce easily the following statements. i) (Q, := P i u P$ u P h , 0) is a group, regular on P; if we denote V a E P : a* := a+

iff a E PE U PO, a* := a' iff a E P M , and V a , b E P : a * b := .*(a), then ( P , C, *)

is a (non commutative) incidence group, which is fibered but not kinematic. For if I E P \ {0} let [z] := and a E [z] \ (0) with [G.C.D. (a)] = 1. Then eithers

a E Po or a E P M . Let [a]E := PE n [a] , [a10 := Po n [a] , [a]M := PM n [a]; then: if a E PO we have [a] = [ U ] E u [a10 and if a E PM we have [a] = [a]E u [ U ] M . In any case one can verify that [a]* o [a]* = [a]' and ( [a]* ) - ' = [a]* and therefore ( [ a ] , * )

is a subgroup of (?,*). Furthermore if a E PM hence [a] = [a]E U [a]M; then if z E P \ [a] for any h E [a] we have: h * z := h+(z ) = h + z E ( z // [ a ] ) iff h E [ a ] ~

a.nd h * z := h ' ( z ) = h - z E (( -2) // [ a ] ) iff h E [a]M; therefore [a] * z @ C and thus ( P , *) is not 2-sided.

ii) Q 2 := P i U Po' U P h is a set of translations, regular on P , which gives rise to an incidence loop ( P , C, *) by denoting for any Q. E P : a* := a+ iff a E PE, a* := a' iff a E Po U P M , and then for any a , b E P : a * b := a*(b). In fact (72 is not closed with respect to the composition of mappings because P h o P h = P,'UP$ g Qz, P;oPh =

= P& g Q, and ( P , *) is not associative. For let a, b E PM with a - b E PO; then a*ob* = a'ob' = ( a - b ) + = (a ' (b))+ = (a*b)+ # (a*b)' = (a-b)' = (a -b ) ' , which implies a*(b*c) = a*[b*(c)] = a-b+c # (a*b)*c = (a*b)*(c) = (a-b) ' (c ) = a-b-c.

Therefore ( P , C, *) is a proper incidence loop, but where for elements of P , U PM

the operation * is associative since there the operations * and o of (i) coincide. Therefore for a E PM with [G.C.D. (a)] = 1 (hence [a] = [a]EU [ a ] M ) the pair ( [ a ] , *)

is a subgroup of ( P , t).

For a E PO with [G.C.D.(a)] =1, hence [a] = [ a ] ~ U [ a ] ~ , we obtain [ a ] , * [a] =

= [a]E + [a] E [a] and [a10 * [a] = [a]:([a]) = [a10 - [a] = [a] , hence [a] * [a] = [a]. Now (aa)*o(pa)* = (aa)+o(pa)+ = ((a+p)a)+ = ((a+,B)u)* if a, p E 22, (aa)*o(pa)* =

= (aa)' o (pa)' = ( ( a - /?)a)+ = ( ( a - p ) a)* if a, p E 2 2 + 1, (&a)* o (pa)* =

= (aa)' o (pa)+ = ( ( a - P)a)* = ( (a - P)a)* if a E 2 2 + 1,p E 22, since then a - p E 2 2 + 1, (aa )* o (pa)* = (aa)+ o (pa) . = ( ( a + p)a)* = ( ( a + P)a)* if a E 22, p E 22 + 1 and ( (aa )* (pa))* = ((aa)+ (pa))* = ( ( a + p) a)* if a E 2 2 , ( (aa)*(pa))* = ((aa)*(pa))* = ( ( a - P)a)* if a E 22 + 1.


This shows ( ( a a ) * (pa))* = (aa )* o (pa ) * , i.e. ( [ a ] , * ] is a semigroup and moreover ( ( a a ) * (pa) ) * I = (aa) * ( ( p u ) * z) for all a, p E 2 and I E P, hence [a] fulfils (16). Further if a E 2 then ( a a ) * ((-a).) = 0 and if a E 2 2 + 1 then (aa) * ( a a ) = 0. So ([a], *) is a group. Now let A E L(0) be arbitrary and L, y E P. Since I* E 7 we have (I * y) * A =

= (I * y)*(A) // A,I * (y * A ) = s*(y*(A)) // A and (I * y)*(O) = I * y = z*(y*(O)). Consequently (I * y) * A = I * (y * A), i.e. (7) is valid for (P,L,*). The given parallelism / / and the parallelism obtained via (2.1) are the same. With this we

have the result:

(7.1) There are fibered incidence loops ( P , L, . ) such that (7) and (16) are valid but

where (P, .) is not a group.

Acknowledgment.

This Research is supported partially by the Italian Ministry of University and Scien- tific and Technological Research (M.U.R.S.T.) (40% and 60% grants) and by G.N.S.A.G.A. of C.N.R.

The Author likes to thank prof. H.Karze1 for fruitful discussions and comments.

References

[l] BILIOTTI, M.: Sulle strutture di traslazione. Boll. Un. Mat. It. (5) 15-A (1978), 667-677.

(21 BRUCK, R.H.: A Survey of binary Svstems. Springer Verlag, Berlin-Gottingen-Hei- delberg 1958.

[3] KARZEL H.: Zweiseitine Inzidenznruppen. Abh. Math. Sem. Univ. Hamburg, 29 (1965), 118-136.

[4] KARZEL H.: Kinematic Spaces. Symposia Mathematica, Istituto Naz. di Alta Matematica, 11 (1973), 413-439.

[5] KARZEL H.: Relations between Incidence Loops and normal Quasi-fields. Journal

of Geometry 1 (1976), 9-10.

364 M . Marchi

[6] KARZEL, H. and KIST, G.P.: Kinematic algebras and their geometries. Rings and D.REIDEL, Dordrecht (1985), Geometry, Nato AS1 Series; Series C, Vol. 160.

437-509. [7] KARZEL H., KROLL H.-J., SORENSEN K.: Invariante Gruppenpartitionen und

Doppelraume. J. reine angew. Math., 262/263 (1973), 153-157. [8] KARZEL H., SORENSEN K., WINDELBERG D.: Einfuhrung in die Geometrie.

Vandenhoeck and Ruprecht, Gottingen, 1973. [9] KIST G.P.: Theorie der verallgemeinerten kinematischen Raume. Habilita-

tionsschrift, T U Munchen (1980). [lo] MARCHI M.: S-spazi e loro problematica. Quaderni Seminario Geornetrie Combi-

natorie, n.1 (1977). 1st. Mat. Univ. di Roma. [ll] MARCHI M.: On a new class of Geometrical Hvperstructures. Rivista di Matema-

tica pura e applicata, 1 (1990), 93-111. [12] MARCHI, M.: Configurations in Incidence Loops. Journal of Combinatorics, Infor-

mation & System Sciences, 15 (1990), 287-300. [13] MARCHI, M.: A configurational Characterization of Incidence Loops which are

Incidence Groups. Journal of Geometry, 41 (1991), 124-132. [14] MARCHI M., PERELLI CIPPO C.: Su una particolare classe di S-spazi. Rendiconti

Sem. Mat. Brescia, 4 (1979), 3-42. [15] PERMUTTI, R.: Geometria affine su un anello. Atti Acc. Naz. Lincei, Memorie 81

(1968), 259-287. [16] REINMIEDL B.: Verallaemeinerte kinematische Raume. Beitr. zur Geom. und Alg.

der TU Munchen 19 TUM-M9009 (1990), 1-96. [17] W ~ H L I N G H.: Proiective Inzidenzgruppoide und Fastalgebren. Journal of Geome-

try 9 (1977), 109-126. [18] ZIZIOLI, E.: Fibered Incidence Loops and Kinematic Loops. Journal of Geometry, - 30 (1987), 144-156.


Families of Menon Difference Sets

D. B. Meisner

Mathematics Department, Royal Holloway and Bedford New College, University of Lon-

don, Egham, Surrey, TW20 OEX, United Kingdom

Abstract

In their paper “Perfect Binary Arrays” [12], Jedwab, Mitchell, Piper and Wild give

an iterative construction of an infinite family of Menon difference sets. Their results are generalized and further infinite families of Menon difference sets are found.

1 Menon Difference Sets

Let G be a multiplicative group with identity element 1 . Let D = {gl , g2, ...,gk} be

a subset of G. Then D is a ( v , k, A)-difference set if it satisfies the following properties:

(i) for each g E G \ (1) there are exactly A pairs gi,gj of elements of D such that

g = 9i9j1

g = grlgm.

(ii) for each g E G \ (1) there are exactly X pairs gl,gm of elements of D such that

The expression gigJ:’ is known as the difference between the elements g;,gj . The

numbers (v , k, A) are known as the parameters of the difference set. There are exactly

k(k - 1) non-identity differences from the set D so the parameters must satisfy the

relation k(k - 1) = A(v - 1). Bruck [2] has shown that if D is a subset of G which satisfies property (i) then

D also satisfies property (ii). Thus only property (i) needs to be considered to check

whether or not a given subset of G is a difference set.

For a general background on difference sets the reader is referred to Lander [15].

Let D be a (v,k,A)-difference set in the group G. Then the subset G \ D, the

complement of D in G, is also adifference set of G with parameters (v,v-k,v-2k+A).

A ( v , k, A)-difference set D shares the parameters v and n = k - A with its complement

in G. The parameter n is known as the order of D.

366 D.B. Meisner

Menon difference sets are difference sets whose parameters satisfy v = 4n. Clearly the complement of a Menon difference set is also a Menon difference set. The Bruck-

Ryser-Chowla theorem (Baumert [l]) states that if ‘u is even then n must be a square. It

follows that the parameters of a Menon difference set are either v = 4t2,k = 2t2 + t , X =

t2 + t or the parameters of the complement, (4t2 , 2t2 - t , t2 - t), for some integer t. Only

one of these cases needs to be considered since a Menon difference set with parameters

of the one kind gives rise to a Menon difference set with parameters of the other kind

by complementation.

Example 1 The trivial difference set {z ,z2,z9} and its complement are Menon difference sets in the cyclic group C4 = ( ~ ( 2 ~ = 1). The subset {z,y,cy} of the Klein

4-group, K4 = (zlylz2 = y2 = l), is a Menon difference set.

The only known Menon difference sets in a cyclic group are the trivial ones in C4

given in example 1. Turyn [22] has proved that no cyclic group of order 4t2 with t even

contains a Menon difference set and has conjectured that no other cyclic group contains a Menon difference set.

Example 2 Jedwab, Mitchell, Piper and Wild [12] give an iterative construction for Menon difference sets in the rank 2 abelian groups C2+ x C21+1 , C,i+a x C21 , C3.21+1 x

C~.2i+trC3.2i+a x C3.21, (I 2 l), where C, is the cyclic group of order 8 . These are

constructed from difference sets in the relevant groups of order 16 and 36 listed in

Kibler [13].

The existence of Menon difference sets in the groups C21+1 x C21+1,C2i.+a x C21, (I 2 l), was first proved by Davis [3] (see also Dillon [lo]). The existence problem for

Menon difference sets in abelian 2-groups has recently been settled. Turyn’s exponent

bound [22] states that if an abelian group G of order 2”+‘ contains a Menon difference

set then the exponent of G is at most 2*+2. Kraemer [14] has shown that any abelian

2-group which satisfies Turyn’s exponent bound contains a Menon difference set. For more results on the existence of Menon difference sets the reader is referred to the recent work of Davis [3-81, Liebler [16], McFarland [17-191, Pott [21] and Turyn [23].

Example 3 The set M below is a (36,15,6)-difference set in the non-abelian group

De x Cz x Cs, where Da is the dihedral group of order 6 generated by a, b with a’ =

b2 = 1,ab = ba2, and C2,C3 are the cyclic groups of order 2,3 generated by c ,d with

Families of Menon difference sets 361

cz = dS = 1 respectively. Kibler [13] has listed all the difference sets in this group.

M = {1,a,aZ,d,ad,azd,b,abd,aZb6,c,cd,cd2,bc,azbcd,abc6}.

Result 1 gives a construction, due to Menon [20], for combining two Menon differ-

ence sets to create a Menon difference set in a larger group. Menon difference sets are

the only difference sets for which Menon’s construction works.

Result 1 Let D1 be a Menon difference set in the group G1 and DZ a Menon difference set in thegroup Gz. Then (D1 x Dz) U ( (GI \ 01) x (Gz \ Dz)) is a Menon difference set in the group G I x Gz.

Result 2 gives a relation between Menon difference sets and Hadamard matrices. Menon difference sets are sometimes referred to in the literature as Hadamard difference

sets or H-sets.

Result 2 Let M be a Menon difference set in a group G. Let H be the matrix of order v with rows and columns indexed by the points of G whose (g;,gj) entry is 1 if

gj E (Mg;) and -1 otherwise. Then H is a Hadamard matrix.

Let D1 be a Menon difference set in the group G1 and Dz a Menon difference

set in the group Gz. The Hadamard matrix which corresponds to the difference set

(D1 x Dz) U ((GI \ 0 1 ) x (Gz \ Dz)) of GI x Gz is the Kronecker product of the two Hadamard matrices corresponding to D1 and Dz by Result 2.

Let G be a group containing a Menon difference set. Then it is possible to construct

a Menon difference set in the group G x C4 by applying Result 1 with Gz replaced by

C4 and Dz replaced by the Menon difference set {2,22123} in Ca. Similarly, if Dz is

replaced by the Menon difference set {z,y, zy} in K4 then a Menon difference set in the group G x K4 can be constructed.

In the sequel a different method is given for constructing Menon difference sets in

groups of order 41GI from Menon difference sets in the group G.

2 Construction of Menon Difference Sets

Suppose that a Menon difference set MI exists in a group GI. In this section a

construction for a Menon difference set Ml+1 in a group Gl+1 of order 41Gll is given.

This construction uses MI and another set with properties yet to be defined. It will also

be shown how the application of Menon’s construction, which forms Menon difference

368 D.B. Meisner

sets in the groups GI x Z4 and G I x K4 from M i , can be expressed in terms of the construction given.

Let G1+1 be a group containing a subgroup Hi of index 2. Let Mi+1 be a subset of G1+1, A1 = M1+1 n Hi and BI = T - ~ ( M L + ~ nTH,) where T E Gi+1 \ H I . Then Ai and BI are subsets of H I and M1+1 = A1 U T B ~ .

Lemma 1 Let GI+1 be a group of order 4t2 containing a subgroup Hi of index 2

and Mi+1 a Menon difference set in G1+1 of size (2t' - 2). Let A1 = Mi+' n Hi and BI = T - ' ( M I + ~ n T H O where T E GI+1 \ H I . Then either = (t' - t ) , IBlI = t Z or

(Ail = t', lBll = (t' - t ) .

Proof Count pairs ( g i , g j ) , 9; # g j , in Mi+i such that gigJ:' E Hi. Set lAil = a

and lBil = (2t' - t - a). The difference g;gJT1 E E I l if and only if { g ; , g j } A1 or

{ ~ - l g i , ~ - ' g j } 5 BI and there are exactly X = (t2 - t ) pairs ( g i , g j ) c M1+1 such that g;g71 = h Vh E HI \ (1). so

a(a - 1) + (2t' - t - a)(2t' - t - a - 1) = (t' - t)(2t' - 1;

u' - ( 2 2 - 1)" + tyt2 - 2) = 0

Hence a = t 2 or a = (t' - t ) .

Only one of the cases IAiI = t2 or lAil = (t' - t ) needs to be considered since if the Menon difference set Mi+1 has lA1l = t' then the Menon difference set rM1+1 has [All = (t2 - t ) .

9-19; = gj.

The number of pairs ( g i , g j ) 5 Mi+1 such that gigJ:' = g is IMi+1 ng-'MI+lI since

Lemma 2 Let G1+1 be a group containing a subgroup H I of index 2 and Ml+1 a

subset of G1+1. Let A1 = MI+I n H I and BI = ~ - ' ( M i + 1 fl T H ~ ) where T E Gi+1 \ H I .

For every h E Hi

(i) IMi+1 n h-'M1+11 = IAl n h-'All+ IBI nT-lh-lTBiI

(ii) n h-lT-lMl+ll = [ A l n h-lBII + [ A I n T ~ T B ~ I .

Proof (i) Hi has index 2 so T H I = GI+I \ H I . Now A1 C Hi and TBI 2 T H I so

Families of Menon diference sets 369

Combining lemmas 1 and 2 gives the following theorem.

Theorem 1 Let GI+' be a group of order 4t2 containing a subgroup H I of index 2.

Let T E GI+' \ H I . Let M1+1 be a subset of GI+' with size (2t2 - t ) . Let A1 = MI+' n H I

and B1 = T - ' ( M I + ~ n .Hi) so that Mi+' = A1 U rB1. Then Mi+' is a Menon difference set if and only if

(i) [All = t2 or (All = (t2 - t)

(ii) IAi n h-'Ail + IBl n T - ' ~ - ' T B ~ I = ( t 2 - t ) V h E H I \ (1)

(iii) IA1 n h-lBlJ + [A1 n rhrBlI = ( t2 - t) Vh E H I .

Now suppose that the centre of H1 contains an element x of order 2.

Lemma 3 Let Hi be a group whose centre contains an element x of order 2. Let Al

be a subset of H I such that xA1 = A1 and B1 a subset of Hl such that xB1 = H1\ B I .

Then IAl n h-'BlI = Vh E H I .

370

Proof

D.B. Meisner

Let K be a normal subgroup of a group G. A subset R of G is said to be a relative

difference set of G for X with respect to K if for each g E G \ K there are exactly X

pairs g;,gj of elements of R such that g = g ; g i l and exactly X pairs gr,gm of elements

of R such that g = g i l g m . For a general account of relative difference sets the reader is

referred to Elliot and Butson [ll]. However it should be noted that the definition above

is the one used by Jedwab, Mitchell, Piper and Wild (121 which differs from the standard

definition since it allows for pairs g;,gj of elements of R such that gig,:' E K \ (1). Relative difference sets containing such pairs are used in section 3.

factor group & with elements hf = {h,hz} . G1 has order w. Lemma 4 Let H I be a group whose centre contains an element z of order 2. Let G1

be the factor group with elements hf = {h ,hz} . Let A1 be a subset of H I such

that zA1 = A1 and B1 a subset of HI such that zBi C H I \ B1. Let A;' = {af la E Ai}

and Bf = { b f l b E Bl} . Then, for h E H I ,

Since z is in the centre of Hi the subgroup {l,z} is normal in Hi. Let GI be the

(i) J A ~ n (h f ) - lAf l= ~ I A , n h - 1 ~ ~ ~ = fpl n ( z h ) - l ~ I ~

( i i ) IB;' n ( h f ) - l ~ f l = pZ n h - 1 ~ ~ ~ + I B ~ n ( z ~ ) - ~ B ~ I

Proof (i) For each pair (u{,u!) in A{ such that u{(u!)-' = hf in GI there exist

exactly two pairs ( u : , u j ) and ( z u : , z u j ) in A1 whose difference is h and exactly two pairs

(za: ,a j ) and (a: ,zaj ) in A1 whose difference is ha: where a: = a; or a: = za; depending

on whether aiay' = h or a;aF1 = ha. Similarly given a pair ( a ; , a j ) in A1 such that

a;aF' = h there exists another pair (zailzaj) in Ai such that ( z ~ ) ( z u j ) - ' = h and

two pairs ( z a i , a j ) and ( a ; , z a j ) in A1 whose difference is hz all giving the pair (u { ,u ; )

in Af such that u{(u;)-I = hf.

(ii) For each pair (b!,b!) in Bf such that b{(b;)-l = hf in GI exactly one of the pairs

(b i ,b j ) , (zb; ,b j ) , ( b i , ~ b j ) , ( z b i , ~ b j ) in B1 gives rise to the difference h or hz. Similarly

Families of Menon difference sets 37 1

for each pair (bi, b j ) in Bi such that either = h or = hz there exists one pair

0

Theorem 2 gives a construction for a Menon difference set in G1+1 from a Menon

( b f , b!) in Bf such that b{(b{)-' = hf.

difference set in G I and a relative difference set in H I with respect to {I,.}.

Theorem 2 Let Hi be a group of order 82' whose centre contains an element x of order 2. Let GI be the factor group & with elements hf = {h, hx}. Suppose that

there exist a (4t2, 2t2 - t, t2 - t)-Menon difference set Mi in GI and a relative difference

set Bi in Hi with respect to ( 1 , ~ ) such that xB1 = Hi \ Bi. Set Ai = {ulaf E Mi}. Let GI+1 be any group in which Hi is a subgroup of index 2 and a is central. Let

T E Gl+l \Hi . Then Mi+1 = A1 U T B ~ is a (16t2,8t2 - 2t,4t2 - 2t)-Menon difference set in GI+1.

Proof IMI+1 I = 2lMilt- I Theorem 1 (i) is satisfied and by Lemma 3 Theorem 1 (iii) is satisfied. We have

= 8t2 - 2t and the assumptions of Theorem 1 are satisfied.

so by Lemma 4 (i)

Also IAi n zAil = IAI n Ail = ]Ail.

Now IBi n zBil = 0 so Bi is a relative difference set with respect to {1,z} for A' where

A'(8t2 - 2) = 4t2(4t2 - 1). SO A' = 2t2.

Hence IAI f l h-'AIl+ IBi T - ~ ~ - ~ T B ~ [ = (t2 - t) Vh E Hi \ (1). Theorem 1 (ii) is also

0 satisfied and so Mi+1 is a Menon difference set of GI+*.

Example 4 Let MO be the Menon difference set in D6 x C2 x Cs given in example

3. Recall that C2 was generated by c and that Cs was generated by d. Let C, be

generated by by e and c6 by g. Note that De x Cz x Cs is isomorphic to the factor

group and that D6 x C, x Cs is a subgroup of D6 x C4 x C6 of index 2. Let A. be the subset of De x C, x Cs obtained from Mo by setting A0 = M i U e2Mi where M i is a list of coset representatives of the elements of Mo (given by replacing c with e ) .

A. = { l , a , a', d , ad, a2d, b, abd, a2bd2, e , ed, ed2, be, a2bed,abed2, e2 , ae2,

a2e2,e2d,ue2d,u2e2d, be2,abe2d,a2be2d2, es ,esd ,e3d2, be3,a2besd,abeSd2}.

312 D.B. Meisner

Set Bo to be the subset

Bo = (l,a,a',d,ad,a'd,b,abd,a'bdZ, e,ed,eda,abe,bed,a2be~,e'd',be2d,

he'd', be', be3d2,aes,ae'd,ae'd2,abe21abe2d2,abe'd,abe'd2,

ae'd', a'e'd', a2eS , a' e' d, a2e'd2, a' be', a'be'd, a'be' , a' be' d}

of Da x C4 x Cs. Bo is a relative difference set with respect to {l,e2} such that

e2 Bo = De x C4 x Cs \ Bo. Then by Theorem 2 Menon difference sets exist in all groups for which D6 x C, x Cs is a subgroup of index 2 and e2 is a central element.

For example the subset MI of Da x C4 x C6 given by MI = A; U gBt, where A6

and BL are A0 and Bo expressed as subsets of D6 x C4 x Ce (by replacing d with 9') is

a Menon difference set.

M I = (1, a, a' , g2, ag', a'g', b, abg2 , a2bg4, e, eg' , eg4, be, a' beg', abeg4 , e', ae2 , a2e2, e2g2,aeZg2,a2e2g2, beZ,abe2g2,a2beZg4, e', e'g', e'g4, be', a2be'g2,

abesg4,g,ag,a2g,g',ags,a2g', bg,abgS1a2bg5,eg,egs,eg5,abeg, beg',a2beg5,

e2g5, be'g', be2g5, beSg, be'g', ae3g,ae'g', ae'g5,abe2g,abe2g5,abe3g3, abe3g5,

ae'g5,a2e2g5,a'e'g1a'esgsl a'esg5,a2be2g, aZbe'g',aZbesg, a2be'g3}

Lemma 5 is a converse to theorem 2 .

Lemma 6 Let G1+1 be a group of order 16t2 which contains a subgroup HI of index 2. Suppose that HI contains an element x of order 2 which is in the centre of G I + ~ and let

T E G1+1\HI. Let G I be the factorgroup f i with elements hf = (h,hz}. Let M1+1 be

a subset of G1+1 such that A1 = Ml+1 n H I satisfies ,412: = A1 and BI = ~-l(M1+1 nTHI)

satisfies zB1 = HI \ B1. I f M1+1 is a difference set in G1+1 then MI = (afla E A I } is a

Menon difference set in G I = fi and B1 is a relative difference set in HI with respect

to (I,.}. In this case MI+^ is a Menon difference set.

Proof By Lemma 3 Ml+1 is a (16t2,k,A)-difference set for X = [All . So the order of

M1+1 is (k - A) = lMi+ll- lAll = 1B1I =

IAlnh-'AIl = IAlnh-'(zAI)I = IAln(xh)-'(A1)I Vh E HI and x~ = m, so by Lemma

and M1+1 is a Menon difference set.

2 (9,


Since xBi = Hi \ Bi the set Bf = {bf Ib E Bi} is the group GI. So

pf n ( g ) - l ~ f l = pfl= 4t2 vg E G ~ .

By Lemma 4 ( i i )

4t2 = pi n h - l q + pi n ( z h ) - l ~ ~ I = 2 p 1 n h - l ~ ~ ( .

SO IBI n h-'BiI = 2t2 Vh E Hi \ (1, x} and B1 is a relative difference set with respect to {l,z} for 2t2.

By Lemma 2 ( i )

IAi n h-'Ail = X - 2t2 Vh E Hi \ {l ,~}.

By Lemma4 (i) IMin(hf)-lMiI = !jIAiflh-'AI(. Hence Mi isa(4t2, i, *)-difference 0 set which is a Menon difference set as required.

In the following sections it is shown how the construction process of Theorem 2 can

be repeated recursively in certain cases.

Lemma 6 gives a construction for a relative difference set BI in the group Hi =

GI x (z1z2 = 1) with respect to {( l , l ) , ( l , z )} such that (1,z)Bi = Hi \ BI when there

exists a Menon difference set in the group GI. In particular it shows that Menon's

construction for Menon difference sets in the groups Gi x C, and GI x K4 from Menon

difference sets in the group GI can be considered to be a special case of the construction

given in Theorem 2.

Lemma 6 Let H I be the group GI x C2 where C2 is the cyclic group of order 2

generated by x and GI a group of order v = 4t2 containing a Menon difference set Mi.

Then the subset BI = (Mi x (1)) U ((GI \ Mi) x {z}) of Hi is a relative difference set with respect to {(1,1),(1,x)} for 2t2 satisfying Bi(1,z) = H I \ Bi.

Proof We note first that IBI n (1,z)BlI = 0 since

Since MI is a Menon difference set for ( v , k , X ) its complement GI \ Mi is a Menon

314 D.B. Meisner

So B1 is a relative difference set of HI with respect to ( 1 , ~ ) for 21'. 0

Since GI x C2 is a subgroup of index 2 in both GI x 2 4 and GI x K4 application of

Theorem 2 and Lemma 6 gives Menon's construction for Menon difference sets in the groups GI x C4 and GI x K4 from Menon difference sets in the group GI.

3 Construct ing Relative Difference Se ts

Suppose that G I + ~ is a group which satisfies the conditions of Theorem 2. That is

to say that G I + ~ has order 16t2 and the following properties. G I + ~ contains a subgroup

HI of index 2 and H I contains an element 2 of order 2 which is in the centre of GI+1. Then, by Theorem 2, it is possible to construct a Menon difference set MI+^ from a

Menon difference set MI in the factor group GI = f i and a relative difference set BI of HI with respect to ( 1 , ~ ) with the property that BIZ = BI \ HI. This process may

be repeated using MI+^ to form Menon difference sets in larger groups if appropriate relative difference sets can be found. Theorem 3 gives a method for constructing such

relative difference sets which is similar to the construction of Menon difference sets given

in theorem 2.

T h e o r e m 3 Let G I + ~ be a group of order 16t2 whose centre contains two elements

2 and y of order 2. Let HI be the factor group HI = a with elements given by

gf = {g,gg} for g E G. Suppose that Hi contains a relative difference set Bi for 2t2


with respect to the normal subgroup { l f ,z f} such that xfBl = HI \ BI and that the group G I + ~ contains a relative difference set F1+1 for 4t2 with respect to the normal

subgroup{l,x,y,xy} such that zFi+1 = yF1+1 = G1+1 \Fl+1. Let HI+1 be anygroupin

which G I + ~ is a subgroup ofindex 2 and for which both z and y are central elements. Let

s E HI+^ \ G I + ~ . Let E I + ~ be the subset of Gi+1 defined byE1+1 = {g E G~+l lg f E B I } .

Then the subset B I + ~ = El+1 U sFi+1 is a relative difference set in H1+1 for 8t2 with respect to {l,z} such that xBI+1 = HI+^ \ BI+I.

Proof Since Hl+l = G I + ~ U sG,+1 and x is in the centre of H1+1

zBI+1 = z(EI+l u 84+1) = Z E i + l u Z s f i + l

= (GI+I \ &+I) U 5(G1+1 \ 4+1) HI+^ \ (&+I U 3Fi+1) = H I + I \ &+I.

Lemma 2 applies so for every g E G I + ~

Also IEI+11 = 8t2 SO I . I P I + ~ n (sg)-'Bi+1 I = 8t2 V g E G I + ~ .

Since JBI n zfBll = 0 BI is a relative difference set for A' with respect to { l f , z f}

where A'(JRI - 2) = ?(

By Lemma 4(i) for every gf E Hi \ { l f , z f}

- 1). So A' = 9.

Also

Now

n ~ 4 + ~ 1 = 14+1 n YFI+lI = 0, and n Fi+ll = Ifi+l n ~ f i + ~ l = I f i + l l

so Fi+1 is a relative difference set for A" where A"(IGi1 - 4) = (IFI+11)2 - 21fi+11. So A" = 4tZ.

316 D.B. Meisner

Hence by Lemma 2

IBI+1 n (9)-1B1+11 = I&+i n (g)-'El+i( -k 14+i n 8-1(9)-1SF~+i1

= 82' Vg E GI \ (1 ,~ ) .

So Bi+1 is a relative difference set of H1+1 with respect to ( 1 , ~ ) for 8t' as required

Lemma 8 is a converse to theorem 3.

Lemma 8 Let Hl+l be a group of order 32t2 with the following properties. The centre of HI+^ contains two elements x and y of order 2 and HI+^ contains a subgroup G I + ~

of index 2 which contains both x and y. Let s E HI+^ \ G I + ~ . Let Hi be the factor group a with elements given by gf = (g,gy}. Let B I + ~ be a subset of HI+^ and set

&+I = Bl+lnGI+l a n d 4 + i = 8-1(BI+in3G1+i)- Ify&+i = ~%+i , z&+i = GI+I\EI+I and yF1+1 = G1+1 \ Fl+1, x 4 + 1 = G1+1 \ F I + ~ then B1+1 is a relative difference set with respect to (1, x} only if the subset B1 = (ef le E Ei+1} of Hi is a relative difference set with respect to (If, zf} and F1+1 is a relative difference set of G I + ~ with respect to

{1,%Y, ZY).

Proof Since xEl+1 = G I + ~ \ El+1 and xfl+l = G1+1 \ F1+1,

IGlt1 I 1E1+11 = 1f'l+11 = - = 8t'.

2

Using Lemma 2 and the fact that ys = sy, Bl+1 is a relative difference set with respect to (1,z) for A, where

Also

So by Lemma 2 for every g E GI+I \ ( l ,x}

Now consider FLl = {gf 1g E 4+1}.

Since yF1+1 = GI+I \ 4+1, F f , , = HI. Hence IFLl i7 h-lFif+ll = [HI[ = 81' Vh E HI. By lemma 4(4,


So Fi+1 is a relative difference set of G I + ~ with respect to (l,z,y,zy} for 4t2.

Hence I&+i ng-'&+lI = 4t2 vg E Gi+i \ {l,z,y,zy}. By lemma 4 ( i ) , IBI n (gf)-lBll = iIEl+1 n g-'El+1(. Hence B1 is a relative difference

0 set of H I with respect to {lf,zf} for 2t2.

In section 4 it is shown how this construction for relative difference sets can be repeated recursively in some families of groups. In these cases, using the construction

for Menon difference sets given by Theorem 2, it is possible to find infinite families of

Menon difference sets.

4 Inflnite Families of Menon Difference Sets

Let M be a Menon difference set in the group G. Recursive application of Menon's

construction produces Menon difference sets in any group of the form G x G4 x G4 x . . . x

G4, where G4 is one of the groups Z4 or K4 of order 4. Also, using Lemma 6, it is possible to form a relative difference set B in any group of the form G x G4 x G4 x . . . x G4 x C2,

where Cz is the cyclic group (1, z} of order 2, such that Bc = G x G4 x G4 x . . . x

G4 x C2 \ B. By Theorem 2 it is possible to find a Menon difference set in any group

for which G x G4 x G4 x . . . x G4 x Cz is a subgroup of index 2.

The notion of repeatedly applying the construction of theorem 2 has applications to

groups other than the Ones given above. Repeated construction of Menon difference sets

requires the repeated construction of the appropriate relative difference sets. Theorem 3 gives a method for this construction. However this requires two relative difference sets.

Lemma 9 gives a correspondence between the existence of the two relative difference sets needed for Theorem 3 in groups which satisfy an extra condition.

b

Lemma 9 Let G I + ~ be a group whose centre contains two elements x and y of order

2. Let HI be the factor group & with elements given by gf = {g,gy}. If there is an

isomorphism a from H1 to the factor group such that (zf)" = {z,y} then the

relative difference set BI of HI with respect to {lf,zf} such that B p f = HI \ BI exists

if and only if the relative difference set D1+1 of G1+1 with respect to {l,z,y,zy} exists.

Proof First note that since a is an isomorphism from Hi to a, BI is a relative

difference set of H I with respect to {lf,zf} such that zfB1 = H1 \ B1 if and only if BP

is a relative difference set of with respect to {(lf)",(zf)"} = {{l,zy},{z,y}}

such that {z,y}Bf = \ BP.

378 D.B. Meisner

If D1+1 exists set B,O = {{g,gzy}lg E D1+1} and apply Lemma 4 (i).

If Bi exists set D1+1 = (BP)' U (B,O)'zy where (B?)' is a list of coset representatives of

0

Lemma 10 gives an example of groups for which the conditions of lemma 9 are

elements of BP in GI+] and apply Lemma 4 (i).

satisfied.

Lemma 10 Let G1+1 be the group F x Cz. x Czt, where F is any group and CZ.,

Cat are the cyclic groups of order 2s generated by u and of order 22 generated by r

respectively. Let (a , t ) denote the greatest common divisor of s and t. Without loss of generality suppose that t/(a,t) is odd. Set y to be (l,l,rt) andz to be (l,r',l). Then

there is an isomorphism from Hi =

which maps {z,zy} to {z,y}.

Proof The element

= F x Ca. x 0 1 to the group

has order 28, and the element

has order t. Also (a) n (4) = (l,l,l).

Hence the mapping a from Hi to given by

a ( { ( f , v", T ~ ) , (f, u", r*+')}) = {(f, u"+l%, r b ) , (f, u"+f i+ ' , T*+$)} , f E 8'

is an isomorphism from Hi onto which maps {z,zy} to {z,y}. 0

Theorems 2 and 3 and Lemma 9 prove Theorem 4.

Theorem 4 Let {Gill E N} be a faamily ofgroups indexed by the set N ofnon-negative integers with the following properties. Each group GI+], 1 E N, contains a subroup H I

of index 2. Hi contains an element x i of order 2 which is in the centre of GI+]. The factor group &J is isomorphic to Gi. Each group GI is a subgroup of Hi of index

2. For 1 2 1 GI contains an element yl of order 2 which is in the centre of H i . The factor group & is isomorphic both to HI-1 with {zi,zlyl} mapped to z1-1 and to

the factor group & with (z1,ziyi) mapped to {zi,y~}.

If there exist a Menon difference set MO of Go and a relative difference set Bo of Ho with

respect to {l,zo} such that zoBo = HO \ Bo then there exist Menon difference sets Ml


in all the groups GI and there exist relative difference sets BI in the groups H I with respect to (1,zl) such that zlBl = H I \ BI. Hence there exist Menon difference sets in

all groups for which H I ( I E N) is a subgroup of index 2 and 21 is a central element.

The corollary to theorem 4 is proved by lemma 10.

Corollary Let {GIII E N} be the family of groups given by GI = F x C21, x C21t

where F is any group and C21,, C2,, are the cyclic groups of order 2's and 2't generated

by v1 and wl respectively. Let ( 5 , t) be the greatest common divisor of s and t. Suppose, withoutlossofgenerality, that h i s o d d . SetHI to ~ ~ F X C ~ I + ~ , X C ~ I ~ , ~ I = (l,v;h,l) and yl = ( l , l ,~$~- '~) . If there exist a Menon difference set MO in Go = F x C. x Ct

and a relative difference set Bo in HO = F x Cz, x Ct with respect to {(l,l ,l),zo}

such that zoBo = Ho \ Bo then there exist relative difference sets B1 with respect to

{ ( l , l , l ) , z ~ } such that z l B ~ = H I \ Bl in all of the groups H I and Menon difference sets in all the groups GI and all other groups for which Hl is a subgroup of index 2 and z[

is a central element, (I E N).

The Menon difference sets in the abelian groups C2t x C21 and Cs.al+l x C3.2~+t,

given in [12], are constructed using this method.

Example 5 Set GI to be De x C,I+, x C3.21 in the corollary to theorem 4 with MO the Menon difference set of D6 x C2 x Cs and Bo the relative difference set of De x C, x C3

both given in example 4. Menon difference sets exist in all the groups GI and all other

groups with D6 x C2t+2 x C3.21 a subgroup of index 2.

Acknowledgement

search Council studentship.

This research was supported by a Science and Engineering Re-

References

I am grateful to the referee who suggested the inclusion of many of the papers listed here.

[l] Baumert L. D., Cyclic Difference Sets, Lecture Notes in Mathematics 182, Springer-

[2] Bruck R. H., Difference Sets in a Finite Group, Trans. Amer. Math. SOC. 78

[3] Davis J., Difference Sets in Abelian a-Groups, Ph.D Thesis, University of Virginia,

Verlag (1971).

(1955) 464-481.

(1987).

380 D.B. Meisner

[4] Davis J., Difference Sets in Non-Abelian 2-Groups1 Coding Theory and Design Theory Part II, ed. D. Ray-Chaudhuri, pp. 65-69. Springer (1990) .

[5] Davis J., Difference Sets in Abelian 2-Groups) J. Comb. Th., to appear. [6] Davis J., A Result on Dillon’s Conjecture in Difference Sets, J. Comb.Th., to

[7] Davis J., A Generalization of Kraemer’s Result on Difference Sets, J. Comb.Th.,

[8] Davis J., A Note on Non-Abelian (64,28,12)-Difference Sets, Ars Comb., to appear. [9] Dillon J. F., Variations on a Scheme of McFarland for Non-Cyclic Difference Sets,

appear.

to appear.

J. Comb.Th. (A) 40 (1985) 9-21. [lo] Dillon J. F., Difference Sets in 2-Groups) Contemp. Math., 111 (1990) 65-72 [ll] Elliot J. E. H. and Butson A. T., Relative Difference Sets, Illinois J. Math. 10

[12] Jedwab J., Mitchell C., Piper F. and Wild P., Perfect Binary Arrays, Technical

(131 Kibler R. E., A Summary of Non-Cyclic Difference Sets, Ic < 20, J. Comb. Th. (A)

[14] Kraemer R. G., Proof of a Conjecture on Hadamard a-Groups, to appear. [15] Lander E.S., Symmetric Designs : An Algebraic Approach, London Math. SOC.

[16] Liebler R.A., On Difference Sets in Certain 2-groups, to appear. [17] McFarland R.L., Necessary Conditions for Hadamard Difference Sets, Coding The-

ory and Design Theory Part II, ed. D. Ray-Chaudhuri, pp 257-272. Springer

[18] McFarland R.L., Difference Sets in Abelian Groups of order 4p2, Mitt. Math. Sem.

[19] McFarland R.L., Sub-Difference Sets of Hadamard Difference Sets, J. Comb.Th.

[20] Menon P. K., On Difference Sets Whose Parameters Satisfy a Certain Relation,

[21] Pott A., A Generalization of a Construction of Lenz, Proc. R.C. Bose Memorial

[22] Turyn R. J., Character Sums and Difference Sets, Pacific J. Math. 15 (1965)

[23] Turyn R. J., A Special Class of Williamson Matrices and Difference Sets, J. Comb.

(1966) 517-531.

Memo HPL-ISC-TM-89-019, Hewlett-Packard Labs., Bristol (1989).

25 (1978) 62-67.

Lecture Notes 72, Cambridge University Press (1983).

(1990).

Giessen 192 (1989) 1-70

(A) 54 (1990) 112-122.

Proc. Amer. Math. SOC. 13 (1962). 739-745.

Conf. on Comb. Math. and its Applications, Calcutta 1988, to appear.

319-346.

Th. (A) 36 (1984) 111-115.

Combinatorics '90 A. Barlotti el al. (Editors) 0 1992 Elsevier Science Publishcrs B.V. All rights reserved. 38 1

On the characterization problem for finite linear spaces

N.Melone

Dipartimento di Matematica e Applicazioni "R.Caccioppoli" , Via Mezzocannone 8, 801 34-Napoli, Italy

Abstract

satisfying suitable algebraic relations among the parameters are surveyed. In this paper some results and problems on the characterization of finite linear spaces

1. INTRODUCTION

The well known characterization of projective geometry due to Veblen-Young [26] and the M.Hall's fundamental paper on projective planes [ 111 can be considered as the motivations for the study of the incidence structures of points and lines of geometric objects in combinatorial geometry and, in particular, of the fundamental structure of linear spaces.

A linear space (LS) is a pair IL = ( P,.2 ), consisting of a nonempty set P of elements (the points) and a family & of proper subsets of P (the lines), such that

every pair of distinct points belong to a unique line every line contains at least two distinct points there exist at least two distinct lines

Afinite linear space (FLS) is a linear space with a finite number of points (and hence of lines). Several arithmetical characters can be associated to every FLS IL = ( P,d ), the parameters. We focus our attention on the following parameters:

- thenumber v ofpoints, - thenumber b of lines, - the degree rx of a point X , i.e. the number of lines through X, - the degree k j of a line R , i.e. the number of points on R , - the order q of IL , i.e. q = max( rx - 1 I X E P ) , - the number i(2) of lines intersecting a line R.

The incidence matrix of an FLS IL = ( P,g ) is the (0,l)-matrix A = I I a J II whose rows and columns are indexed by fixed orders on the points and lines, with a x J = 1 if X E R and a x J = O otherwise.

An automorphism of IL is any bijective map on the points which preserves lines. Obviously, the set Aut(IL) of automorphisms of IL is a group, the (full) automorphism group of IL.

2 . EXAMPLES

PROJECTIVE SPACES. It is essentially due to Veblen-Young [26] the characterization of

382 N . Melone

projective spaces as the linear spaces for which the Veblen-Young axiom holds:

any line meeting at distinct points two sides of a triangle meets the third side too .

In particular, projective planes are the linear spaces with painvise intersecting lines.

LINEAR SPACES EMBEDDED INTO A PROJECTIVE SPACE IP. These are the complements of distinguished subsets of points of IP. In particular, affine spaces, affine planes and the (linearly) t-punctured projective planes, i.e. LS obtained from a projective plane by deleting t (collinear) points.

NEAR-PENCIL O N v POINTS. The FLS's on v points with a line of degree v-1 and all the remaining lines of degree two .

(h,k)-CROSS. The FLS's on h+k - 1 points with a line of degree h , a line of degree k and all the other lines of degree two. In particular, the (3,4)-cross is also called the Lin cross.

COMPLETE GRAPHS Kv ON v VERTICES. The FLS's on v points with all the lines of degree two.

2-(v,k,l) DESIGNS. These are the FLS's on v points with constant line degree k . SUBSPACES OF LINEAR SPACES. A subspace of a linear space IL = ( P,& ) consists

of a subset H of P containing every line R of IL which intersects H in at least two points. Clearly, the subset H, with respect to the set &H of all the lines of IL contained in it, is a linear space.

THE INFLATION AND DEFLATION PROCESSES. These processes give rise to new linear spaces obtained from old ones either by inflating a line into a subspace or by deflating a subspace into a line. For example, the Fano quasi-plane is the FLS obtained from the Fano plane by inflating a line into K3 and, viceversa, one gets the Fano plane by deflating the subspace K3 of the Fano quasi-plane into a line (see the picture) .

Fano plane Fano quasi-plane

3 . SOME ALGEBRAIC RELATIONS AMONG THE PARAMETERS

An useful tool in studying the combinatorial geometry of a finite linear space IL = ( P,d ) is to determine algebraic relations among the parameters. We list some basic relations(see,[29])

On the characterization problem for finite linear spaces 383

b I v(v - 1)/2 , and the equality holds iff IL = K2 (3.3)

(3.4)

(3.5)

(3.1) and (3.2) are easily obtained by counting in two different ways the pairs (X, 1) with X E R and the mples (X,Y,R) with X,Y=1, respectively. (3.3) easily follows from (3.2), since k i 2 2 . (3.4) translates the obvious assertion that the number of unordered pairs of lines is greather than or equal to the number of unordered pairs of intersecting lines. The relation (3.5) is obvious.

Proposition 3.1. Let IL = ( P,& ) be a finite linear space. We have b 2 v .

Proof. It suffices to prove that rgR A = v , where A = II a x 1 I1 is the incidence matrix of

IL . Let Rx be the row corresponding to the point X . Since BX Rx = 0 it follows that, X z P

each line R , the Rth component of the vector B R must be zero, i.e. Q1 = Ax Hence, with respect a fixed point Y, P , we have 0 = $ Q l = ry By + sign. Since Q1 = 0 for any line 1 , it follows that By = 0 for every point Y .

taking into account that the lines through Y cover Bx .Thus, all the numbers By have constant

Proposition 3.2 (Stanton-Kalbfleisch [20]). If R is a line of an FLS IL = ( P,& ) , then i(1) 2 kR2 (v - l q ) / ( V - 1) f

4. THE CHARACTERIZATION PROBLEM

M.Hall proved in [111 that every FLS can be embedded into a projective plane.This result is obtained by means of the free extension process , consisting , essentially, in adding a new point to each pair of non-intersecting lines and a new line of degree two to every pair of non collinear points. Such a plane is always infinite. Therefore, the following question naturally arises.

The embedding problem . Can any FLS be embedded into afinite projective plane?

algebraic relations among them describe the structure of an FLS .

Characterization problem . Derive information on the structure of an FLS either (i) from algebraic relations among the parameters, or (ii) from suitable configurations of points and lines, or (iii) from transitivity conditions on the automorphism group.

To attack this problem, it is interesting to investigate how the parameters and suitable

Precisely (see [29] ), the following general problem can be formulated.

The first result in this direction is the fundamental theorem on finite linear spaces due to de

384 N . Melone

Bruijn and Erdos [3] and Hanani [121.

Theorem 4.1. Let IL be a finite linear space on v points and b lines. We have b 2 v , and the equality holds if and only if IL is either a projective plane or a near-pencil.

A history of the fundamental theorem and a list of proofs is presented in [29] . Proposition 3.1 gives an easy algebraic proof of the inequality b 2 v . Hanani [13] (see also Varga 1251 ) stated a relation between the parameters v and i(R) from which Theorem 3.1 can easily be deduced.

Theorem 4.2. If R is a line of maximum degree of an FLS IL , then i(1) 2 v , and the equality holds if and only if IL is either a projective plane or a near-pencil.

A corollary of Theorem 4.1 is the following.

Proposition 4.3 ([16]) . Let IL = ( P,d? ) be an FLS and denote by d?, a family of pairwise intersecting lines with at least three non concurrent lines. We have b, = Idol S v , and the equality holds if and only if IL is a projective plane or a near-pencil.

Proof. Since the lines in d?, are pairwise intersecting, the incidence structure IL* ( the dual of IL), whose points and lines are the lines in d?, and the pencils of sgch lines respectively, is an FLS . If v* and b* denote the number of points and lines of IL , then 1 < b* I v and fqm Theorem 4.1 it follows b, = v* 2 b* S v . Let uj suppose bo = v . Hence, v* = b* and IL is a near-pencil, b, - 1 = v - 1 lines of d?, have a common point and the remaining line of 8, misses this Qoint. Since these lines are pairwise intersecting, it is easy to see that IL is a near-pencil. If IL is a projective plane of order q , then v = bo = q2 + q + 1 and each line of d?, has at least q + 1 points. Hence it easily follows that IL is a projective plane of order q (the dual of IL* ).

is either a near-pencil or a projective plane. If IL

Theorem 4.1 , 4.2 and Proposition 4.3 show that the finite projective planes (or the near-pencils) are characterized as the FLS's satisfying one of the following extremal conditions.

The minimum number of lines, i.e. b = v . The minimum number of lines intersecting any line R of maximum degree, i.e. i( 1) = v . The maximum number of pairwise intersecting lines , i.e. b, = v . Therefore, the following specialization of the characterization problem for FLS's naturally

arises.

Problem. Characterize the FLS's satisfying one of the following relations

b = v + x

i ( R ) = v + y

b, = V - z

for suitable non negative integers x , y , z .


5. RESULTS ON PROBLEM (A)

The first step in problem (A) , i.e. the case x = 0 , is solved by Theorem 4.1 .

b = v + 1. The answer to this case has been given by Bridges [I] with algebraic arguments. Combinatorial proofs can be found in [ 21 , [ 211 . Theorem 5.1. For an FLS IL the equality b = v + 1 holds if and only if IL is either a 1-punctured projective plane or the 2-punctured Fano plane.

b = v + 2 . This case has been solved by de Witte [30] . Theorem 5.2. For an FLS IL the equality b = v + 2 holds if and only if IL is either a 2-punctured projective plane or one of the following structures. (i) TheLin cross (ii) The affine plane of order 2 (iii) The Fano quasi-plane (iv) The linearly 3-punctured projective plane of order 3

b = v + x, with 0 < x I Jv . Finite affine planes are obviously the FLS's for which the equality ( b - v )2 = v holds. In his Ph.D. thesis [27] de Witte proved, conversely, that each FLS satisfying this equality is either an affine plane or a 1-punctured affine plane with a point at infinity. This result suggested to Totten [22],[23] he idea of characterizing the

Theorem 5.3 . For an FLS IL the inequality ( b - v )2 I v holds if and only if IL is one of the following structures . (i) A near-pencil (ii) TheLin cross (iii) (iv) (v) (vi)

Remark 5.4 . From this result, Totten [24] derived a classification of the FLS's for which the equality b = v + 3 holds.

b I v + r - 2. A generalization of Totten's result has been obtained by Metsch [ 171,who characterize3 the FLS's satisfying the inequality b I v + rx - 2 , for some point x . Theorem 5.5 . For an FLS IL the inequality b I v + rx - 2 holds, for some point x, if and only if IL is one of the following structures . (i) Anear-pencil (ii) (iii) (iv) (v)

restricted FLS's, i.e. the FLS's satisfying the inequality ( b - v ) !2 I v .

An affine plane with a point at infinity A projective plane less either all the points af a line or all the points of a line but one An h-punctured projective plane of order n 2 h with n2 + n + 1 lines A simply or projectively inflated affine plane, i.e. obtained from an affine plane by inflating a subset of the line at infinity into a near-pencil or a projective plane

A projective plane or a near-pencil with a point x added to some line A (b-v)-punctured projective plane of order rx-1 An affine plane of order rx-1 with a point at infinity An s-fold inflated projective plane of order n , 1 I s I n, i.e. the FLS obtained from an h-punctured projective plane of order n with n2+n+1 lines by inflating s lines through a point x into subspaces

Remark 5.6. From this result, Metsch derived a positive answer to the following

386 N . Melone

Dowling-Wilson conjecture. If t is the maximum number of lines through a point

missing a common line , then b 2 v + t .

Moreover, in the same paper Metsch characterized also the FLS's for which the equality b=v+t holds (Dowling-Wilson spaces).

Theorem 5.7 . For an FLS IL the equality b = v + t holds if and only if IL is one of the following structures. (i) (ii) (iii) The (3,t + 2)-cross (iv)

b = v + q, q the order of L . Recently, Olanda [19] attacked the characterization problem for FLS's IL satisfying the equality b = v + q , where q denotes the order of IL . Theorem 5.8. Let IL be an FLS of order q satisfying the equality b = v + q and let k denote the maximum line degree. We have (i) If IL has constant point degree q + 1 , then IL is either an affine plane of order q or a

pro'ective plane of order q less q non collinear points (ii) If k has non constant point degree and k = q -t 1 , then IL is a 1-punctured affine

plane of order q with a point at infinity (iii) If IL has non constant point degree and k = q , then b 2 q2 - 2q + 3 and the equality

holds if and only if IL is a 1-punctured affine plane of order q-1 with a triangle at infinity

A projective plane or a near-pencil A linearly t-punctured projective plane of order n 2 t + 1

An affine plane of order n 2 t with a Dowling-Wilson space at infinity

6. RESULTSON PROBLEM (B)

maximum line degree. In the sequel we will call long lines the lines of maximum degree and will denote by k the

The step y = 0 is solved by Theorem 4.2 . i(R) = v + 1 . In this case the answer has been obtained in [15] .

Theorem 6.1 . In an FLS IL the equality i(R) = v + 1 holds for every long line R if and only if IL is either a 1-punctured projective plane of order k - 1 or an affine plane of order k or one of the following structures. (i) The 2-punctured Fano plane (ii) The Fano quasi-plane

1 (1 = v + 2. These FLS's have been characterized by De Vito and Lo Re [7] . Theorem 6.2 . In an FLS IL the equality i (1 1 = v + 2 holds for every long line R if and only if IL is either a 2-punctured projective plane of order k - 1 or a I-punctured affine plane of order k or one of the following structures. (i) The Lin's cross (ii) (iii) (iv)

The Fano quasi-plane with a long line inflated into a triangle The Fano quasi-plane with at most two points of degree three deleted The affine plane of order 3 either with a point or with a triangle or with a K4 at infinity

(v) K5


(vi) The Fano plane with an extra point connected to all the others by a line of degree 2 (vii) The S i n e plane of order 5 with a 1-punctured Fano plane at infinity

(viii) The closed complement of a Baer subplane B of a projective plane of order 4 , i.e. the complement of B with an extra point added to all the lines of minimum degree

1 (1) = v + 3 De Vito , Lo Re and Metsch [9] solved this case.

Theorem 6.3 . In an FLS IL satisfying k 2 7 , the equality i (1 1 = v + 3 holds if and only if I. is either a 3-punctured projective plane of order k - 1 or a 2-punctured affine plane of order k or one of the following . (i) The affine plane of order 7 with the Fano plane at infinity (ii) The closed complement of a Baer subplane of a projective plane of order 9 (iii) 8 I k 5 13 and IL is a projective plane of order k - 1 with a line inflated into an FLS

with constant point degree 4 (embeddable into PG(2,3) )

i(4) = v + l ,v + 2 .The answer has been obtained by De Vito and Lo Re [6] . Theorem 6.4 . In an FLS IL both the equalities i(1) = v + 1 and i(1) = v + 2 hold for the long lines if and only if IL is the affine plane of order 3 either with a line inflated into a K3 or with a near-pencil at infinity or the affine plane of order 4 with the line at infinity inflated into the 2-punctured Fano plane.

Remark 6.5. Theorems 4.1 , 6.1 , 6.2 and 6.3 suggest the following

CONJECTURE ([9]). Apart from a finite number of sporadic cases, every FLS satisfying the inequality i(1) I v + y , for a suitable fixed y, is a y-punctured projective plane or a (y-1)-punctured affine plane.

Recently, Metsch 1181 gave a positive answer to this conjecture.

Theorem 6.6 . If IL is an FLS different from a near-pencil satisfying the inequalities k>7y7 + 40y6 - y and i(1) I v + y for all long lines 1 , then IL is either a y-punctured projective plane or a (y-1)-punctured affine plane.

7. RESULTS ON PROBLEM (C) . The f i s t step in Problem (C) , i.e. the case z = 0 is solved by Proposition 4.3 . Up to

now, Problem (C) has k e n investigated under the following special assumption on the family

(7.1)

30.

The family 3, of painvise intersecting lines consists of all the long lines

Henceforth, k shall denote the degree of the long lines and b, their number.

b, = v - 1 . The solution to this case has been obtained in [16] ,

Theorem 7.1 . If IL is an FLS satisfying (7.1) and the equality b, = v - 1 , then IL is a projective plane of order k-1 either with an inflated line or with an extra point.

b, = v - 2 . This case has been handled in [8] . Theorem 7.2 . If IL is an FLS satisfying (7.1) and the equality b,, = v - 2 , then IL is a

388 N. Melone

projective plane of order k - 1 with t inflated lines , 0 I t S 2 , and with 2 - t extra points or one of the following . (i) The 1-punctured Fano plane

(ii) The Fano quasi-plane with an inflated line of degree 3

bo = v - 3 . This case has been solved by De Vito and Lo Re [5] . Theorem 7.3 . If IL is an FLS satisfying (7.1) and the equality bo = v - 3 , then IL is a projective plane of order k - 1 with t inflated lines , 0 S t 1 3 , and with 3 - t extra points or one of the following . (i) The FLS obtained from a projective plane of order k-1 by inflating the sides of a

mangle in such a way that the vertices form a subspace, and deflating this subspace (ii) The near-pencil on 4 points (iii) The 2-punctured Fano plane (iv) The Fano quasi-plane less a point of degree 3 (v) The 1-punctured projective plane of order 3 (vi) The 1-punctured projective plane of order 3 with at least one inflated line of degree 3

Remark 7.4 . Other results on Problem (C) with some more general arithmetical conditions on b, have been recently obtained by De Vito [4] and Lo Re [14] . Remark 7.5. The results mentioned above suggest the following two problems.

Problem I. Give a general result for the FLS's satisfying (7.1) and the inequality b, 2 v - z for a suitable fixed non negative integer z . Problem I1 . Extend the previous results to general maximal families d o of pairwise intersecting lines, i.e. without the assumption (7.1) .

Concerning Problem I1 , an extension of Theorem 7.1 is the following.

Theorem 7.6. Let IL be an FLS and do a maximal family of painvise intersectin lines with at least three non concurrent lines, and let b, the size of do . If b, = v - 1 , then is a projective plane either with an inflated line or with an extra point or one of the following structures. (i) The (3,~-2)-cross (ii) The double near-pencil on v points, i.e. the FLS on v points with a line of degree v-2 (iii) The 1-punctured Fano plane

Proof. From Proposition 4.3 , the dual IL* of IL is an FLS with b, = v - 1 points and b* I v lines. Thus, the following two cases may occur:

b * = v - l , b*=v (7.2)

Suppose b* = v - 1 . From Theorey 4.1 we have that IL * is either a near-pencil or a projective lane on v-1 points . If IL is a near-pencil, then b,-1 = v-2 lines 41, 1 2 ,..., Rv-i;f 1, have a common point. Since the lines of d., meet each other, the further line R of has at least v-2 points. Hence, either all the lines R e have exactly two points in which case IL is a double near-pencil , or one of the lines Ri kas three p o i p and the other two points in which case IL is a (3,~-2)-cross. Let us suppose now that IL is a projective plane of order q . Thus, b, = v - 1 = q2 + q + 1 , every line of d has at least q + 1 points and every point of IL belongs to 0 or q + 1 lines of 3, . ?t follows that all the


one line of &, . Hence, IL \ (-1 is a projective plane of order q .*suppose now b* = v . We have b,, = v - 1 , b* = v and, from Theorem 5.1 it f$lows that IL is either the 2-punctured Fano plane or a 1-punctured projective plane. If IL is the 2-puncfiured Fano plane, then it is easy to check that IL is the 1-punctured Fano plane. Finally, let IL be a 1-punctured projective plane of order q . Then b, = v - 1 = q2 + q , i.e. v = q2 + q + 1 and, by a duality argument, it is easy to see that there exist q + 1 points xi , x2, ..., x in IL such that every xi belongs to q lines of 3, and all the lines of 3, have degree 8 + 1 . Moreover, every line joining two points xi contains only points xi , i.e. D = ( xi , x2 , ..., xq+l ) is a subspace of IL and IL \ D is an affine plane of order q . Hence, IL is a projective plane with an inflated line and the proof is complete.

REFERENCES

W.G.Bridges, Near 1-designs, J.Combin.Th.,(A) ,(1972),116-126.

N.G.de Bruijn and P.Erdos, On a combinatorial problem, NederLAkad. Wetensch.a, (1948),1277-1279 and Indag.Math.B,(1948), 421-423.

P.De Vito, Su UM classe di spazi lineari con le rette lunghe a due a due incidenti, Pubbl.Dip.Mat. e Appl. Univ.Napoli,B,( 1990).

P.De Vito and P.M.Lo Re, Spazi lineari su v punti con v-3 rette lunghe a due a due incidenti, Ricerche di Matematica,vol.XXXIX,( 1990).

P.De Vito and P.M.Lo Re, Spazi lineari su v punti in cui ogni retta di lunghezza massima t intersecata da v o v+l altre rette, Rendiconti di MatemSerie

P.De Vito and P.M.Lo Re, On a class of linear spaces, Research and Lecture Notes in Math., Proceedings of the Conference "Combinatorics '88",~0l I, (1990),

P.De Vito and P.M.Lo Re and N.Melone, Linear spaces on v points with v-2 mutually intersecting long lines, J. of Geometry,Z,( 1990), 87-94.

P.De Vito and P.M.Lo Re and K.Metsch, Linear spaces in which every long line intersects v+2 other lines, Arch. Math.,(1990), to appear.

J.C.Fowler, A short proof of Totten's classification of restricted linear spaces, Geom.Ded,, U,(1984),413-422.

M.Hal1, Projective planes, Trans.Amer.Math.Soc.,s,( 1943),229-277. H.Hanani, On the number of straight lines determined by n points, Riveon

LematematikaJ, (1951), 10-11. H.Hanani, On ther number of lines and planes determined by d points, Scientific

Publications,Technion (Israel Institute of Technology,Haifa),fi,(1954-55), 58-63. P.M.Lo Re, Sugli spazi lineari fmiti con p2 + p + 1 rette lunghe a due a due incidenti,

Pubbl.Dip.Mat. e Appl. Univ.Napoli,U,( 1990). N.Melone, A structure theorem for finite linear spaces, Research and Lecture Notes in

Math., Proceedings of the Conference "Combinatorics '88",~0l II,(1990),

N.Melone, Sugli spazi lineari finiti le cue rette di massima lunghezza sono a due a due incidenti, Rend.Mat.,VII,U( 1990),349-355.

K.Metsch, Proof of the Dowling-Wilson conjecture, to appear. K.Metsch, Linear spaces in which every line of maximal degree meets only few lines,

A.Bruen, The number of lines determined by n P points, J.Combin. Th ,(A), (1973), 225-24 1.

VII,8,( 1988), 455-466.

322-339.

231-241.

Submitted. D.Olanda, Spazi lineari di tipo affine, Ricerche di Matematica, to appear.

390 N. Melone

[20] R.G.Stanton and J.G.Kalbfleisch, The X-v problem, h = 1

[21] G.Tallini, Spazi di rette finiti e k-insiemi di PG(r,q), ConfSem. Mat.Univ. Bari.192,

[22] J.Totten, Basic properties of restricted linear spaces, Discrete Math.a,( 1975),67-74. [23] J.Totten, Classification of restricted linear spaces, Canad J. Math.,B,( 1976), 321-333. [24] J.Totten, Finite linear spaces with three more lines than points, Simon Stevin ,a, [25] L.E.Varga, A note on the structure of painvise balanced designs, J.Comb.Th.,(A),O,

[26] 0.Veblen and J.W.Young, Projective geometry, Ginn,Boston,(l918). [27] P.de Witte, Combinatorial properties of finite plans, Doctoral Dissertation,

[28] P.de Witte, A new proof opf a theorem of Bridges, Simon Stevin,Q, (1973), 33-38. [29] P.de Witte, Combinatorial properties of finite linear spaces 11. Bull.Soc.Math.Belg.,a,

[30] P.de Witte, Finite linear spaces with two more lines than points, J.Reine

= 3 ,Proc.Second Chapel Hill conf. on Combinatorics.Chape1 Hill (1972), 45 1-462.

(1984). 1-24.

(1977), 35-47.

(1985), 435-438.

Univ.Brussels, (1965). cf.Zbl M,(1967), 13-14.

(1975). 115-155.

Ang.Math.,(Crelle), a,( 1976), 66-73.

Combinatorics '90 A. Barlotti et al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. 39 1

Linear spaces in which every line of maximal degree meets only few lines Klaus Metsch Mathematisches Institut, Arndtstr.2, D-6300 Giessen

Abstract It is shown that a non-degenerate linear space L with v points in which v + c is the

maximum number of lines which meet a line of maximal degree can be obtained from an affine plane by removing at most c - 1 points or from a projective plane by removing at most c points provided that v > ( T c ' + 4 0 ~ ~ ) ~ .

1. INTRODUCTION

In 1948, N.G. Bruijn and P. Erdos [4] proved that every finite linear space has at least as many lines than points with equality if and only if the linear space is a generalized projective plane. This result was proved independently by H. Hanani [8]. Later on he was even able to improve this result as follows [9]: If L is a line of maximal degree of a finite linear space L, then L meets at least v lines with equality if and only i f L is a generalized projective plane. The first result was improved a couple of times in the sense that linear spaces satisfying b = 2) + k ( b is the number of lines, v the number of points) have been classified for small values of k [2,14,17] until J. Totten [13] classified all linear spaces satisfying ( b - I I ) ~ 5 v. In 1975, P. de Witte [16] posed the following problem: Is it possible to generalize the second result in the same way? N. Melone [lo] and P. De Vito and P.M. Lo Re [5,6] determined all linear spaces in which a line of maximal degree meets at most v + 2 lines. In 171, it was conjectured that for a fixed value of c, all except a finite number of non-degenerate linear spaces in which v + c is the maximal number of lines meeting a line of maximal degree is an affine planes with c - 1 points deleted or a projective planes with c points deleted. It is the purpose of this paper, to prove this conjecture (Corollary 3.2).

2. DEFINITIONS AND PRELIMINARIES

A partial linear space consists of a set of v points and a set of b lines such that any two points are contained in at most one line, any line has at least two points, and there are at least two lines.

any two points are contained in a unique line. A linear space is a partial linear space in which

We consider only (partial) linear spaces with a finite number of points and lines. The

392 K. Metsch

number rp of lines through a point p is called the degree of p and the number kL of points on a line L is called the degree of L .

A partial linear space L consisting of some of the points and lines of a linear space L’ is called embeddable in L’. A near-pencil is a linear space with a line containing all but one of the points. A linear space is called non-degenerate if it is not a near-pencil. A generalized projective plane is a projective plane or a near-pencil.

The first result we present is due to Stanton and Kalbfleisch [12]. In order to state it, we define a function f ( k , w ) := k2(v - k)/(w - 1) for all integers w and k satisfying 2 5 k 5 w - 1 . Results 2.2 and 2.3 are slight generalizations of theorems, which appeared first in [l] . For a proof of the generalizations see [ l l , Corollary 14.3 and Theorem 14.41.

2.1 Result. Suppose that L is a linear space with w points, and that k is the degree of a line L of L. Then L meets at least f(k, w) other lines.0

2.2 Result. Let H be a line of an incidence structure I and denote by M the set of lines which are disjoint from H . Suppose that there are integers e 2 0, d , f, g , n, and z satisfying the following properties. (1) H is disjoint from d n f I > 0 lines. (2) If L1 and Lz are intersecting lines of M , then there are at most e lines which miss

(3) If L E M and i f m is the number of lines disjoint to L and H , then n - 1 + f 5 rn 5

(4) 2n > ( d + l ) ( d e - 2f) + e + 21. ( 5 ) n > (2d - l)g + e - 2x. Then there are exactly d sets C of mutually disjoint lines satisfying H E C and ICI 2 n - ( d - l )g t I + 1. Furthermore every line which is disjoint to H occurs in exactly one of these d sets.0

L1, L z , and H .

n - 1 + g .

2.3 Result. Suppose that L is a partial linear space and that there exists an integer n 2 2 such that the following three conditions hold. (1) b 2 n2. (2) For every line H there exists an integer t ( H ) with the following properties: There

exist exactly n t 1 - kH sets c of pairwise disjoint lines with H E C and ICI 2 t ( H ) . Furthermore, every line which is disjoint to H appears in exactly one of these sets C.

(3) It exists a point of degree n + 1 which is joined to every other point. Then I can be embedded in a projective plane of order 71.0

We need the following lemma characterizing the set of few points (and a line) in a projective plane. We remark that A. Bruen’s result [3] on the minimum number of points of a blocking set is an easy consequence of this lemma.

2.4 Lemma. Let u , s and n be non-negative integers satisfying n > 2u + s - 2 and n > u + s2 - 1 , and suppose it exists a projective plane P of order n and a set M of points of P satisfying the following properties. (1) IMI = u(n t 1) t s.

Linear spaces 393

(2) Every line of P intersects M in at least Then u E (0 , l ) and M contains exactly u lines.

points.

Proof. If L is any line which has a point p outside of M , then

u n + u $ s = (MI =

It follows that IL n MI I: u + s. Hence L Now let q be a point of M and denote by z the number of lines L through q with L Then

I X n M ( 2 ILn MI + ( v P - 1). = ( L n MI f u n . P G

M or u 5 IL n MI 5 u + s for every line L. M .

un t u + - 1 = I M - { q } I = )Jlx n M I - 1) 2 zn + (. + 1 - .)(. - I) , x3q

which implies that z ( n + 1 - u ) 5 n + s. In view of n > 2u + s - 2, it follows that t 5 1. Hence, every point of M lies on at most one line L with L C M . This implies that there is at most one line contained in M .

For every line L set Z L = IL n MI - u. Then count pairs (q , L ) with points q of M and lines L satisfying q E L , and count triples (q , q’, L ) with distinct points q, q’ E M and lines L satisfying q, q’ E L to obtain

x ( u + Z L ) = IMI(n + 1) = (un + u t s ) (n + 1 )

and

x ( u + ~ L ) ( u t Z L - 1 ) = lMl(lM1- 1) = (un +

which implies that

LEC

+ s ) ( ~ n + u + s - 1 ) L€C

ZL = un + ( n t 1)s and Z; = un2 + (u + s - uz )n + sz LEI: LEC

We know that 0 I: Z L 5 s or Z L = n + 1 - u for every line L. First assume that M contains no line. Then 0 I: ZL 5 s for every line L . It follows that

un2 + (u + s - 2)). + sz = x Z:, 5 x Z L . s = uns + (n + l)SZ, LEL LEC

which implies that un + u + s - u2 5 us + s2. Using n > u + s2 - 1, we conclude that (u - l)(s* - s) < 0. It follows that u = 0, which we had to show.

M . We have to show that u = 1. In view of H C M , we have n +- 1 I: IM I 5 u(n + 1) + s, which implies that u # 0, since n > u + s2 - 1. Since X H = n + 1 - u and Z L 5 s for every other line L , we have

Now suppose that there exists a (unique) line H

u n 2 + ( u f s - u 2 ) n + s 2 =Ex:. I ( n + 1 - u ) ’ + [ ( C Z L ) - ( n + 1 -.)Is L € C LEC

= (n + 1 - u)2 + [un + s(n + 1) - n - 1 + u]s,

394 K. Metsch

which implies that

(u - l ) n 2 5 (u - l ) (un - 2n -+ u t 1 t ns t s) t ns(s - 1).

Assume by way of contradiction that u 2 2. Then it follows that

n2 5 un - 2n + u t 1 + ns + s + ns(s - 1) = n(u - 2 t sz) -t u t s t 1.

In view of u + s + 1 5 2u +- s - 1 5 n, this implies that n 5 s2 + u - 1, a contradiction. Hence, u = 1 and the lemma is proved.0

Remark. Since a Baer-subplane of a projective plane P of order n has u ( n + 1) + s points (with u = 1 and n = s2) and because a Bur-subplane contains no line of P, we see that the bound n > u t s2 - 1 in Lemma 2.4 is best possible in general. Also the bound n > 221 + s - 2 is best possible as the following example shows.

Let P be a projective plane of even order n and suppose that L is a set of n t 2 lines such that every point lies in exactly two or no line of C (such a set exists in the desarguesian projective planes of order 2'). Let M be the set of points which lie on two lines of C, and set s = 0 and u = ( n + 2 ) / 2 . Then [MI = u(n + 1 ) + s and n = 2.u + s - 2. Furthermore every line which is not in C meets M in exactly u points but there are ICI = n t 2 = 2u lines which are contained in M .

Before we start to systematically study linear spaces in which every long line meets only few lines, we study a special case and prove the result of Hanani [9] (see also Varga [15]) mentioned above.

2.5 Lemma. Suppose that L is a linear space with v points, minimal point degree r and maximal line degree k satisfying r < k. Let L be a line of maximal degree and dcnotc by v + c the number of lines which meet L . If k > c + 2, then L is a near pencil.

Proof. The hypotheses imply that v 5 1 +- r ( k - 1) 5 1 +- ( I c - 1)' = kz - k - c - ( k - 2 - c ) 5 k 2 - k - c. Result 2.1 shows that L meets at least 1 + k2(v - k)/(v - 1) other lines. Since L meets v +- c lines, it follows that (v - 1 + c)(v - 1) 2 k2(v - k), which can be transformed into 1 t b3 - c 2 v (b2 t 2 - c - v) =: g(v).

In order to prove the lemma, we have to show that v = k+ 1. Assume to the contrary that k t 2 5 v 5 kz - k - C. Since g is a polynomial of degree two with negative cocficient in v2, it follows that g(v) 2 g ( k + 2 ) = g(kz - k - c ) = ( k + 2 ) ( k 2 - k - c ) . Hence 1 -f k3 - c 2 (k t 2)(k2 - k - c ) , which implies that (k +- 1)" 2 ( k + l ) ( k - 3) -t 2 . It follows that k < c + 3, which is a contradiction.0

2.6 Theorem (Hanani [9], Varga [15]). Let L be a linear space, v its number of points, and let L be a line of maximal degree. Then L meets at least v - 1 other lines and equality implies that L is a generalized projective plane.

Proof. First notice that, if L is a generalized projective plane, then every line of maximal degree meets exactly v - 1 other lines.

Now suppose that the line L of maximal degree meets v - I + c other lines, c 5 0. Set k = k~ and denote the minimal point degree by r. If k > r then k > r 2 2 2 c + 2 and Lemma 2.5 shows that L is a near-pencil. Assume from now on that k 5 T .

Linear spaces 395

We have v 5 1 + r ( k - 1) with equality only if every point of degree r lies only on lines of degree k. The line L meets at least k(r - 1) other lines with equality if and only if every point of L has degree r . Hence k(r - 1) = v - 1 + c 5 r ( k - 1) + c 5 r ( k - 1). In view of T 2 k, we obtain equality. Hence r = k and v = 1 + r ( k - 1)l-t = k(k - 1). Furthermore, every point of degree r lies only on lines of degree k and every point of the line L has degree r . II p is a point outside of L, then the k lines which join p to a point of L have degree k and cover therefore all 1 + k(k - 1) points of L. It follows that every point has degree r which implies that every line has degree k = r so that L is a projective plane of order k (or the near-pencil on three points in the case k = 2).0

3. THE MAIN RESULT

In this section we shall prove the following theorem.

3.1 Theorem. Suppose that L is a non-degenerate linear space, k its maximal line degree, and suppose that c is an integer 2 2 such that every line of maximal degree meets at most v + c - 1 other lines. If k > 7 c 7 + 40c6 - c then L can be obtained from an affine plane by removing at most c - 1 points or from a projective plane by removing at most c points.

Before we prove this theorem we prove an easy corollary.

3.2 Corollary. Suppose that c is an integer 2 2 and that L is a non-degenerate linear space in which every line of maximal degree meets at most v + c - 1 other lines. If v > ( T c ' + ~ O C ~ ) ~ , then L can be obtained from an affine plane by removing at most c - 1 points or from a projective plane by removing at most c points.

Proof. Let r be the minimal point degree and k the maximal line degree. Then a line of degree k meets at least k(r - 1) other lines and the number of points is v 5 1 + r ( k - 1). Since a line of degree Ic meets at most v - 1 + c other lines, it follows that k ( r - 1) 5 v - 1 + c 5 r(k - 1) + c , i.e. r 5 k + c. Hence v - 1 5 r(k - 1) 5 (k + c ) ( k - 1). The hypothesis on v implies therefore that k > Yc' + 4 0 8 - c , and the corollary follows from Theorem 3.1.0

In the rest of this section we shall prove Theorem 3.1. Throughout we assume that L is a linear space satisfying the hypotheses of this theorem. We denote the maximal line degree by k and the minimal point degree by r . A point will be called real if it has degree r , and a realline is a line with a real point. Furthermore, we shall use the following not at ion.

s is defined by v = 1 + r ( k - 1) - s, b, denotes the number of real lines, n := r - 1, z := b, - (n2 + n + l) , w := 3c(c + 1)/2, for every line L we set d L = n + 1 - kL and t L = CpEL(rP - r ) , t :=max{tLlL is a line of degree k}, and 9 := {LIL is a real line with at most 1 + r / w non-real points}.

Finally, I denotes the incidence structure consisting of all points of L and the lines of

396 K. Metsch

8. We shall apply the embedding theorem 2.3 to show that I can be embedded in a projective plane P. Then we shall denote by M the set of points of P-I and show that, M satisfies the assumptions of Lemma 2.4 so that M consists of a few number of points and (eventually) one line. Finally, we shall show that L is the complement of M in P, which will complete the proof of the theorem.

3.3 Lemma. t + s + ( r - k) I c, r 2 k, t 2 0 and s 2 0. Proof. Lemma 2.5 implies that r 2 k. By definition, there is a point of degree r .

Since k the maximal line degree, it follows that v 5 1 + r ( k - 1) . Hence s 2 0. Since r is the minimal point degree, the definition of t implies that t 1 0.

Let L be a line of degree k. By definition, L meets k ~ ( r - 1) + t L other lines. The hypothesis of Theorem3.1 yields therefore that k ~ ( r - l ) + t ~ 5 v - l + c = r ( k - 1 ) - s t c , which implies that t~ -t s t r - k 5 c. Since t =max{tLIL a line of degree k}, it follows that t + s + r - k 5 c.0

3.4 Lemma. Every point of degree r lies on at least r - s 2 r - c lines of degree k, and every line of degree k has at least k - t 1 r - c points of degree r. Furthermore, every real line has degree at least k - s 2 r - c.

Proof. Lemma 3.3 shows that s+t I c. Since s and t are non-negative, it follows that s 5 c and t 5 c. The first two statements in the corollary follow therefore immediately from the definition of s and t . If L is a real line and p a real point of L , then we have v I k~ t (rP - 1)(k - l ) , which implies that k~ 2 k - s . 0

3.5 Lemma. Let H be a real line and denote by n d H + y the number of real lines disjoint from H . Then -c(c - s) 5 y 5 c2 - c.

Proof. By definition, H has a real point. Lemma 3.4 shows therefore that H meets a line L # 13 of degree k. If p is a real point # L (7 H of L , then p lies on d := d H lines which miss H and these lines are real. Since L has at least k - t real points, it follows that H is disjoint from at least (k - 1 - t ) d = ( r - 1)d- ( r - k + t ) d 2 ( r - 1)d - (c - s ) d real lines. Consequently y 2 -(c - s )d 1 - (c - s)c.

Let q be a real point of H and denote by L l , . . . , L , the lines # Ii through q which have degree k. Since every real point lies on at least r - s lines of degree k, we have w = r - 1 - p for an integer p I s. Let Q be the set of points # q which lie on one of the lines L,. Then IQI = w(k - 1). Let M be the set of lines missing H and consider the set X := { ( p , L ) lp E Q , L E M , and p E L } . Counting X in two ways, we obtain

If L is a real line of M , then L meets at least kL - p of the lines L, and therefore IL n QI 2 kt - p 2 k - s - p. Since nd + y is the number of real lines in M , it follows that

n d + y I w(k - 1)d + wt w(s + p - l ) d + wt = w d +

k - s - p k - s - p

Linear spaces 397

(w + s + p - k ) [ ( s + p - l )d + t ] = wd + ( s + / L - 1 )d + t + k - s - p

In view of w + s + p - k = r - 1 + s - k 5 c, s + p 5 2s 5 2c, t 5 c, and k 2 Yc', it follows that y 5 (w - n)d + (s + p - 1)d + t = (s - 1)d + t . Since (s - 1)d 5 (c - 1)d 5 (c - 1)(c - t ) 5 (c - 1)" - t , the lemma is proved.0

We recall that the number of real lines is denoted by b, = n2 + n + 1 + z.

3.6 Lemma. We have -c2 + s 5 z 5 c2.

Proof. Let L be a line of degree k, set d := r - k, and let d ( r - 1) + y be the number of real lines missing 15. By Lemma 3.4, we have y 5 c2 - c. Because L meets k ( r - 1) + t~ 5 k(r - 1) + t other lines, we obtain

b, 5 1 + k ( ~ - 1) + t t d ( r - 1) + y = 1 + r ( r - 1 ) + y + t 5 1 + r ( r - 1) t c2.

In order to prove the lower bound, let L1 and L2 be two lines of degree k which intersect in a real point and let t , be the number of non-real points of L,. Then t, 5 t . A real point of L1 lies on T - 1 real lines # L1. If p is a non-real point of L1, then each line which joins p and one of the k - t 2 - 1 real points # L1 n of L2 is real. Hence L1 meets at least (k - t l ) ( r - 1 ) + t l ( k - t 2 - 1) other real lines. Furthermore, each of the real points # L1 n L2 of L2 lies on r - k real lines which miss L1. It follows that the number of real lines is at least

1 -t (k - t l ) ( T - 1) + t l ( k - t 2 - 1) + (k - 1 - t 2 ) ( T - k) 1 + k ( T - 1 ) + (k - l)(r - k) - t l t 2 - (t l + t 2 ) ( . - k) 1 + T ( T - 1 ) - ( T - k)2 - t l t 2 - ( t l + t 2 ) ( . - k)

=

=

Since t , 5 t , it follows that z 2 - ( T - k ) 2 - t 2 - 2t(r - k) = - ( T - k + t ) 2 . This proves the statement, since ( r - k + t ) 2 5 (c - s ) ~ 5 c2 - s2 5 c2 - s (notice that 0 5 s 5 c).n

3.7 Lemma. Every point lies on at most T + s real lines. Proof. Let m be the number of real lines passing through a point p . Since every real

line has at least k - s points, we have m(k - 1 - s) 5 v - 1 = r ( k - 1) - s. Assume that m 2 r + 1 + s. Then it follows that k 5 1 + s(s + 1 + T - k) 5 1 + c(c + l ) , a contradiction. 0

3.8 Lemma. If I( and L are intersecting real lines, then there are at most 4c2 - 2c lines which miss h' and L.

Proof. Denote the number of real lines which miss I( by n d ~ + yx and the number of real lines which miss L by n d ~ + y ~ . Furthermore let p be the point of intersection of K and L , and denote by a the number of real lines passing through p . The line L intersects b, - n d L - yL real lines. Since there are at most (kx - l)(k~ - 1) + a real lines which meet K and L , it follows that L meets at least

t b, - n d L - y~ - [ ( k ~ - 1 ) ( k ~ - 1) + a]

n2 + n + 1 + z - n d L - y~ - [ (n - d ~ ) ( n - d ~ ) + a ]

:=

= = n + 1 + 2 + d K ( n - d L ) - yL - a

398 K. Metsch

real lines which miss It'. It follows that the number of lines disjoint to It' and L is at most n d K + Y K - x = d K d L + Y K + y~ - z + (Y - n - 1. Since d K d L 5 c2, Y K 5 c2 - c, yr. I c2 - c , z 2 -c2 + s, and (Y I n + 1 + s (Lemma 3.5, 3.6, 3.7), it follows that there are at most 4c2 - 2c lines which miss h' and L.0

3.9 Lemma. Suppose that H and L are disjoint real lines and denote by n - 1 t y the number of real lines missing H and L. Then y 5 4c2. If H and L have at most 1 + r /w non-real points, then y 2 -cz - c - rc/w (we recall that w := 3c(c + 1)/2).

Proof. Denote the number of real lines which miss H resp. L by n d H t y~ resp. n d L + y ~ . Then L meets b, - n d L - y~ real lines. Since there are at most k H k L which meet H and L , it follows that L meets at least b, - ndr, - y~ - k H k L real lines which miss H . Hence the number of lines which miss H and L is at most

n d H + YH - [b , - n d L - Y L - ~ I I ~ L ]

= n d H + y~ - n2 - n - 1 - z + n d L + y~ + ( n + 1 - d H ) ( n + 1 - d L )

I n - 1 + c 2 + c2 + c 2 + ( c - I)(c- 1) 5 n - 1 $ 4 ~ ' .

= n - 1 t Y H t Y L - z + ( d H - l ) ( d L - 1)

This proves the first part. Now we suppose that H has p~ non-real points and that L has p~ non-real points

with p~ 5 1 + r /w and 5 1 + r /w. Let S be the set of points of L which do not lie on a line of degree k . Notice that every point of S is non-real, since real points lie on lines of degree k (Lemma 3.4). By 3.4, it exists a real point po outside of L. Since po lies on at least r - s lines of degree k , we have IS1 5 s.

Let p be a point of L. We want to find an upper bound for the number z of real lines # L through p which miss H . We have x 5 rp - 1 - k H . If p is real, then rP = r and x 5 d H - 1. If p is not real and not in S, then p lies on a line G of degree k so that rp 5 T + tG 5 r t t . In this case we have x 5 d H - 1 + t . Now suppose that p is in S. By Lemma 3.7, the point p lies on at most r + c real lines. Because p q is a real line for every real point q of H and since H has k H - p~ real points, we obtain x 5 T + c - 1 - ( k l f - p H ) . Thus x 5 dir t c - 1 + p~ in this case. It follows that the number of real lines # L which meet L and miss II is at most

( k L - P L ) ( d H - 1) + ( P L - I S l ) ( d H - 1 ti) + I S l ( d H - 1 + c t p ~ ) =

5 k ~ ( d ~ - 1) + ( P L - IsI)t + IS[(. + k L ( d H - 1) + (1 t - ) ( t + 3) + sc I k L ( d H - 1) + (1 + -)c t SC.

I h ( d H - 1) t p L t t ~ ( c + P H ) T T

211 W

Since H misses d H n t x~ real lines, it follows that the number of real lines which miss L and H is at least

r d H n + ZH - 1 - k L ( d H - 1) - (1 + -)c - sc

W

= d H n + Z H - 1 - (n + 1 - d L ) ( d H - 1) - (1 + L ) c - sc

= d H ( d r , - 1 ) + x ~ - l + n t l - d L - ( 1 + - ) c - s ~ W

r W

r = 12 - 1 + ( d H - l ) ( d L - 1) + ZH - (1 + ;)c - SC.

Linear spaces 399

Hence y 2 ( d ~ - l ) ( d ~ - 1) + ZH - (1 + r / w ) c - sc. Since H and L are disjoint and have real points, we have d H 2 1 and d~ 2 1. Therefore y 2 ZH - (1 + r / w ) c - sc. Since ZH 2 -c(c - s) (Lemma 3.5), this completes the proof of the 1emma.O

3.10 Lemma. There are at most c2w2 real lines with more than 1 + r / w non-real points.

Proof. Let p be a real point, let u be the number of lines of degree < k through p , and denote by L 1 , ..., Lr-g the lines through p which have degree k. By Lemma 3.4, we have u 5 s, and by definition of t , the lines L, have at most t non-real points. Hence, if Q is the set of non-real points lying on one of the lines Lj , then (QI 5 ( r - a)t . Let M be the set of lines L with p $! L and with at least 1 + r / w non-real points. Then every line of M has at least 1 + r / w - u points in Q. Counting triples ( q , q’, L ) with distinct points q, q’ E Q and lines L E M satisfying q, q’ E L shows that

Consequently

( M ( ( r - W O ) ~ 5 ( r - u) w t 5 r w t 5 [ ( r - WO)’ + 2raw]w2t2,

which implies that ( [MI - w2t2 ) ( r - wu)’ 5 2urw3t2. Since r - w u 2 r - ws 2 i, we conclude that [MI - w2t2 5 8 a T . Since w 5 2c(c + 1) and t c, we have 8wt2 5 T .

It follows that [MI 5 w2t2 + c w 2 5 w2t2 + sw2 5 w2(c - $ ) 2 + sw2 . Now ( c - s)’ + s = c2 - 2cs + s2 + s 5 c2 - cs + s 5 c2 - s (we have c 2 2 by the hypotheses of 3.1) implies that [MI 5 w2(c2 - s). Consequently ]MI -+ s 5 wZc2. Since lines of degree k have at most t < f non-real points, the point p lies on at most u 5 s lines with at least 1 + r / w non-real points. Hence there are at most [MI + s 5 w2c2 lines with at least 1 + r / w non-real points.0

2 2 2 2 2 2

Now we are almost in position to prove that I can be extended to a projective plane. Recall that I is the incidence structure consisting of all points of L and of the real lines of L which have at most 1 + f non-real points.

3.11 Lemma. The incidence structure I has at least n2 + n + 1 - $ lines and at least one point of degree n + 1.

Proof. L has b, 2 n2 + n + 1 - c2 real lines (Lemma 3.6), so I has at least b, - w2c2 2 n2 + n + 1 - E lines (Lemma 3.10). Let L be a line of degree k of L. Then L has at least k - t 2 r - c real points. Since there are at most w2c2 < r - c real lines which are not lines of I (Lemma 3.10), it follows that L has a real point which lies only in real lines belonging to I. This point has then degree r = n + 1 in 1.0

3.12 Lemma. The incidence structure I can be extended to a projective plane P of order n.

Proof. This will follow from the preceding lemma and the embedding theorem 2.3, if we can show that for each line H of I there is an integer t ( H ) with the following property: There are exactly d H sets C of mutually disjoint lines with ICI 2 t ( H ) , H E C, and such that every line of I which is disjoint to H is contained in exactly one of these sets C.

400 K. Metsch

Let H be a line of I. The existence of t ( H ) will follow from Result 2.2. In order to apply this lemma, we define the integers

e := 4cz - 2c,

Furthermore we set d := d H , we denote by M the set of lines of I which are disjoint to H , and define 5 := [MI - dn. Since L has at most w2cz real lines which are not lines of I (Lemma 3.10), we have

1 ) -(w2 + l)cz 5 L 5 cz (see Lemma 3.5), 2) If It', L are intersecting lines of M , then at most e lines of M miss It' and L (see

3) If L E M and if a is the number of lines in M which miss L , then n - 1 + f 5 a and

f := -(cz + c + rcJw + wzc2), and g := 4 2 .

Lemma 3.8).

a 5 n - 1 + g (see Lemma 3.9). In order to apply Result 2.2, it remains to show that

4) 2n > (d + l)(de - 2f) + 2c2, and 5) n > (2d - 1)g + e + 2c2[w2 + 11.

Since e 2 0, f 5 0, g 2 0, and d = d~ 5 c, it suffices to prove these inequalities in the case d = c. Then 5) holds, since (2c - l )g + e + 2c2(w2 + 1 ) 5 8c3 + 2c2(w2 + 1) < n. Since r = n + 1 , condition 4) is equivalent to

2r > (c + l ) [c . ( 4 2 - 2c) + 2(c2 + c + - + wzcz)] + 2c2 + 2.

or

rc W

r(1 - ~ c(c + l ) ) > (c + 1)w2c2 + 2c4 + 2 2 + 2 2 + c + 1 . W

This inequality is the reason why we defined w to be 3c(c + 1)/2. The inequality can easily been verified using c 2 2 and 4r 2 27c7 + 160c' - 4c.O

Lemma 3.12 says that I can be embedded into a projective plane P of order n = r - 1 . From now on, we denote by M the set of points of P which are not points of I. Since I has v = 1 + r(k - 1) - s points, we have IMI = (n + 1). + s where u := T - k.

3.13 Lemma. Every point p of L has degree at most n + 1 + c in L and it has degree at least n + 1 - $ in I.

Proof. Let p be a point of L. Then p is also a point of I. Since I is embedded in P and because I has at least nz + n + 1 - lines (Lemma 3.10), the point p has degree at least n + 1 - 2 in I (and therefore also in L). If p lies in L on a line L of degree k, then p has in L degree at most n + 1 + t~ 5 n + 1 + t 5 n + 1 + c. It sufices therefore to show that p lies on a line of degree k.

Assume to the contrary that p does not lie on any line of degree k, and consider p as a point of I. Since every line of I through p has in L degree at most k - 1 = n - u, it follows that every line of I through p meets M in at least u + 1 points. Since p has degree at least n t 1 - 2 in I, it follows that ( n + 1 - 2 ) ( u + 1 ) 5 [MI = (n + 1). + s. This implies that 3c(n + 1 - s) 5 n(u + 1) 5 n(c + l), which is a contradiction.0

Linear spaces 40 I

3.14 Lemma. Suppose that LO is a line of P with [LO n MI 5 u + s. Then the set X := Lo - M , which consists of the points of Lo which are in the linear space L, is actually the set of points of a line of L.

Proof. Assume to the contrary that X is not the set of points of a line of L. We consider first the case that the space L has a line L which contains all points of X and a point p which is not in X . Then the point p is a point of P which is not on Lo. For z E X , the line p z of the space L is the line L. But in P, the lines p z for z E X are all different, since X C LO and p 4 Lo. It follows that the lines p z , z E X , of P are not lines of I. Hence p has degree at most n + 1 - 1x1 5 u + s in I. Lemma 3.13 shows that this is not possible.

L. This implies that Lo is not a line of I or L. Let 23 be the set of lines of L which have at least two points in X . Then 1231 2 2. It follows that the lines of B induce a linear space L(X) = (A!,B) on the points of the set X . The line Lo meets M in at most u + s points. Since I has at least n2 + n + 1 - lines, it follows that there are two points p l and p2 in X = Lo - M which have degree n in I (i.e. every line # Lo of P through p j is a line of I). If H is a line of I through p j , then IH n XI = IH n Lo1 = 1 and therefore H is not a line of the L(X). Since p j has degree at most n + 1 + c (Lemma 3.13), it follows that p , has degree at most c + 1 in L(X). Hence, every line # plp2 of L(X) through pl has degree at most c + 1 in L(X). Counting the number 1x1 of points of L(X) using the lines through pl shows therefore that 1x1 5 y + c2 where y is the degree of the line p1p2 in L(X). It follows that y 2 1x1 - c2 = n + 1 - [Lo n MI - c2 2 n + 1 - c - c2. Let p be a point of X which is not on the line plp2 of L(X). Then p has degree at least y in L(X). Since p has degree at least n + 1 - in I and because every line of I through p meets X only in p (use the same argument we used for the points pj), it follows that p has degree at least y + n + 1 - 2 in L. But every point of L has degree at most n + 1 + c, so y + n + 1 - 5 n + 1 - c, which contradicts y 2 n + 1 - c - c2.0

Now we consider the case that L has no line L with X

Now we can easily complete the proof of Theorem 3.1. Suppose L is a line of P with ILnMI 5 u+s. Then the preceding lemma shows that L-M is the set of points of a line of L. Since every line of L has degree at most k = n + 1 - u , it follows that u 5 IL n MI. Hence every line of P meets M in a t least u points. In view of 1M 1 = ( n + 1)" + s, Lemma 2.4 shows now that u = 0, or that u = 1 and that M contains a line. In both cases it follows that every line L of P is either contained in M or satisfies IL n MI 5 u + s.

If L is a line of P which is not contained in M , then the preceding lemma shows that L - M is the set of points of a line of L. Hence, the complement P-M of M in P is a subspace of the linear space L. Since L and P-M have the same set of points, we have actually L=P-M. Hence, if u = 0, then L is the complement of s points in P. If u > 0, then u = 1 and M contains a line L , and L is the complement of s points in the affine plane P-L. Since s 5 c - u (Lemma 3.3), this completes the proof of Theorem 3 . 1 . 0

PROBLEM. What is the smallest real number a such that there is an absolute constant t satisfying the following condition: If L is any non-degenerate linear space with maximal line degree k and such v + c is the maximal number of lines which meet a line of degree k, then k 2 t c a implies that L can be obtained from a projective plane by removing c points, or from an affine plane by removing c - 1 points. Theorem 3.1 shows

402 K. Metsch

that a 5 7. However it seems very likely that this bound is far from being best possible. A lower bound can be obtained from the complement of a Baer-subplane in a projective plane. In this case we have Ic = (c - 1)’ so that a 2 2.

4. REFERENCES Beutelspacher, A. and Metsch, K., ’Embedding finite linear spaces in projective planes’, Ann. Discrete Math. 30 (1986), 39-50. Bridges, W.G., ’Near 1-designs’, J. Comb. Theory (Series A) 13 (1972), 116.126. Bruen, A., ’Blocking sets in finite projective planes’, SIAM, J. Appl. Math. 21 (1971),

Bruijn, N.G. de and Erdos, P., ’On a combinatorial problem’, Nederl. Akad. Wetensch. Indag. Math. 10 (1948), 1277-1279. De Vito, P. and Lo Re, P.M., ’On a class of linear spaces’, Proceedings of Corribinatorics 88 held in Ravello, 23-28 May 1988. Research and Lecture Notes in Combinatorics Med. Press, vol 1 (1988), 322-339. De Vito, P. and Lo Re, P.M., ’Spazi lineari su v punti in cui ogni retta di lunghezza massima d intersecata da a1 pih v + l altre rette’, Rendiconti di Matem. Serie VII, vol.

De Vito, P., Lo Re, P.M. and Metsch, K., ’Linear spaces in which every long line intersects v+2 other lines’, Arch. Math., to appear. Hanani, H., ’On the number of straight lines determined by n points’, Riveon Lema- tematika 5 (1951), 10-11. Hanani, H., ’On the number of lines and planes determined by d points’, Scientific Publications, Technion (Isreal Institute of Technology, Haifa) 6 (1954-1955), 58-63.

380-392.

8, 455-466.

10 Melone, N., ’Un teorema di struttura per gli spazi lineari’, Pubbl. del Dip. di Matem. e appl. ”R. Caccioppoli” Napoli 58, (1987).

11 Metsch, K., ’Linear spaces with few lines’, Lecture Notes in Mathematics, Springer- Verlag: Berlin-Heidelberg-New York-Tokyo, to appear.

12 Stanton, R.G. and Kalbfleisch, J.G., ’The X - p problem: X = 1 and p = 3. Proc. Second Chapel Hill Conf. on Combinatorics’, Chapel1 Hill (1972), 451-462.

13 Totten, J., ’Classification of restricted linear spaces’, Can. J. Math. 28 (1976), 321- 333.

14 Totten, J., ’Finite linear spaces with three more lines than points’, Simon Stevin 47 (1976), 35-47.

15 Varga, L.E., ’A note on the structure of Pairwise Balanced Designs’, J. Comb. Th.

16 Witte, P. de, ’Combinatorial properties of Finite Linear Spaces IT’, Bull. SOC. Math.

17 Witte, P. de, ’Finite linear spaces with two more lines than points’, J. reine angew.

(A) 40 (1985), 435-438.

Belg. 27 (1975), 115-155.

Math. 288 (1976), 66-73.

Combinatorics '90 A. BarlotU et al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. 403

FLAG-TRANSITIVE BUEKENHOUT GEOMETRIES

Antonio Pasini and Satoshi Yoshiara

1 INTRODUCTION AND NOTATION More than one half of sporadic simple groups are known to act flag-transitively on finite geometries belonging t o Buekenhout diagrams obtained from Coxeter diagrams replacing some of strokes for projective planes -

with strokes for circular spaces C -

or for dual circular spaces C.

e----.

We recall that a circular space is a finite linear space with lines of size 2, namely the system of vertices and edges of a complete graph.

Classification theorems exist for some classes of geometries as above, due to a number of people. We will survey some of those theorems in this paper, choosing certain diagrams for which a classification is known or at least substantial progresses have been done in that direction. This choice cuts off many interesting diagrams for which nice geometries exist and some deep theorems have been obtained. Needless to say that this does not mean at all we dislike those diagrams or despise the work that has been done on them.

NOTA BENE. Throughout this paper only residually connected geometries will be considered.

1.1

We will focus on the following diagrams of rank n = k + m

The Diagrams Considered in this Survey

C 8 8 8 8 . 1 ....._ 0 r - C . . . . . - - (t.AIR) I I L I

k nodes m nodes

404 A. Pasini, S. Yoshiara

C 8 8 8 8 t

< J I J

k nodes m nodes

(Lac,) .....-.-r ..".--.

.-- . * . .. - C 8 (L.Dm) - .....-

< J L '-0 8 k nodes m nodes

where 8 and t are finite orders. We write c.A,-l, c.C,-1 and C . D , - ~ for cl.A,,-l, c'.C,_, and cl.Dn-l, for short. The subdiagram (of type Ah) formed by the first k nodes is called the tail of the

diagram. We do not write orders at the k nodes in the tail of the diagram. Indeed the diagram contains the information that the order is always 1 at each of those nodes. Namely, the tail may also be viewed as a sequence of k strokes marked by c

C C C 1 L ....._ .-.... .. .

This explains the symbol c? in our notation. The remaining m nodes of the diagram form the head of the diagram. The head is a

We also allow m = 2 in $.om, writing Dp for the disconnected rank 2 diagram A i @ A i . subdiagram of type A,, C,, or D, (where D3 is a synonym for A3).

The diagram c"-'.Da looks as follows:

4; 1 ( C R - 2 . D2) *- .....- C

In particular, the diagram C.Da may also be represented as follows: C* C

(c*.c) -0 8

The diagram c"-'.AI is aa follows: C

(C"-'.Al) *- .....-. 1

8

It is a specialization of the following diagram: L .-, , . . . . -0.. (An-1.L)

where L

c----.

denotes the class of linear spaces, as in [6]. When n > 1, the diagram A,-1.L characterizes truncations of ddimensional projective

geometries, with d 1 n and where the truncation is obtained dropping all subspaces of dimension i 2 n; this follows from Theorem 8 of [6], recalling that the Intersection

Flag-transitive Buekenhout geometries 405

Property IP of [7] (Axiom (3) of [S]) is implicit in the diagram in this case (see Proposition 1.3 of this paper). Therefore, the diagram ?'.A1 characterizes truncations of n + 8 - 1- dimensional symplecta.

This finishes the discussion of the case of c'-l.AI. Therefore, we will always assume m > 1 when considering c*.A,.

We will also consider the following diagram in this survey, besides the previous ones:

C C' ( C. A,, - 2 .c* ) 0 - c .....-* 0

a 8 8 8

When n = 3 the diagram C.A,,-~.C' looks as follows (and we write c.c* for c.A1.c*, for short):

C C* (c.c*) 0 0

Another diagram will be considered in Section 9, which is in some sense the dual of c.C3. We will discuss it separately because the classification work for that diagram is still at an initial stage, even if some strong results have already been obtained in the cwe of order 8 = 2.

When a = 1 the diagrams 2.A, , c.An-2.c* and 2.0, are nothing but thin cases of the Coxeter diagrams A, and D,, respectively, whereas the diagram c'.C, is the thin-lined case of the Coxeter diagram C,.

We just obtain Coxeter complexes for the thin cases of A, and D, (see propositions 1.4 and 1.6 of this paper). On the contrary, a complete classification of C,-geometries with thin lines h a s not yet been achieved, but we are not going to discuss this topic in this survey.

Therefore we will assume 8 > 1 in any case. If I? is a geometry belonging to ck.Am, ck.Dm or c.A,-~.c', then we call 8 the order of

I'. If I? belongs to 2.C,, then we call 8 and t the first and second order of r, respectively, and we say that I? is thin on top if t = 1.

1.2 More Notation and Conventions

Notation for Types. distinguish them from orders:

We denote types by bold-face figures as follows, in order to

C - ..... 1 1 . . . .. -0-0 1 2 k - 1 k k t l k t 2 n-1 n

C ' . . . . . A . 1 5 . . * , . -0-0

1 2 k - 1 k k + l k + 2 n - 2 n - 1 n

- - - = ..... - C ..... * 1 2 k - 1 k k + l k + 2 n - 3


C C*

1 2 3 n - 2 n - 1 n I ....._ 1-

Points, Lines, Planes and Blocks.. We call the elements of type 1 potnts. Those of type 2 will be called lines, except when the diagram is c .D2; in this latter case, we call the tlags of type {2,3} ltnes. Elements of type n (of type n - 1 and n in the case of ck.D,) will be called blocks. Elements of type 3 are also called planes. Of course, the words ”block” amd ”plane” are synonymous when n = 3.

The point-line system of a geometry I’ belonging to one of the diagrams we are considering is a graph, possibly with multiple edges. The collineurity graph Q(r) of I? is the simple graph obtained from the point-line system of I? identifying edges with the same vertices, if such edges occur; otherwise, G(r) is nothing but the point-line system of I?.

Classical Types. We say that a geometry I? belonging to cn-’.C2 is of classical type if the C, residues of l? are classical generalized quadrangles] being understood that grids are not included among classical generalized quadrangles even if they arise from a classical orthogonal group O:(q). We include the dual of Hl($) among classical generalized quadrangles.

We use the notation of [lo] for classical polar spaces (in particular, for classical generalized quadrangles). This notation is quite standard and it is the same as in [73] in the rank 2 case, except for generalized quadrangles of symplectic type, denoted by W(q) in [73] but by &(q) in [lo].

Notation for Groups. Henceforth Aut(I?) will denote the group of type-preserving automorphisms of a geometry I?. Given a flag-transitive subgroup G 5 Aut(I?) and an element x of I?, G, will be the stabilizer of x in G and K, will be the elementwise stabilizer in G, of the residue I?, of o, called the kernel of G, on I?,. Hence G,/K, is the action of G, on I?,.

We follow [I] for the notation for finite groups.

1.3 According to the conventions sated in 1.1, 8 = 2 is the minimal value we allow for the (first) order 8 . When 8 = 2 the circular and dual-circular strokes

Plain, Exceptional and Extreme Cases

C C* - - coincide with the following ones, denoting affine planes of order 2 and dual affine

planes of order 2, respectively:

Af A f - 1 2 z 1

Therefore the diagrams c.An-l, C.Cn-1, c.Dn-1 and C.An-2.C’ are special cases of the following, when 8 = 2:

Flag-tramitive Buekenhout geometries 407

The diagram A f .Anml characterizes n-dimensional afEne spaces (see Proposition 1.5 of this paper). Classification theorems also exist for geometries belonging to Af .Cn- l with n - 1 > 2 and satisfying the Intersection Property IP of [ I , for finite geometries belonging to A f.Ca and satisfying the Intersection Property IP, for geometries belonging to Af.Dn-1 and for geometries belonging to Af.An-a.Af* and satiafying the Intersection Property (see sections 3.4, 4.2 and 8 of this paper; no group theoretic hypotheses are needed for this).

Classifications for c.An-l, c.Cn-1, c.Dn-l and c.A,-Z.c* are obtained as a by-product of those theorems when B t 2, apart from one exceptional example for c.C3 with t = 2(= 8 )

(the sc-called Neumaier geometry, Section 3.3). Thus, the cases of c.An-l, c.C,-I, C . D , - ~ and C.A,-p.c* with B = 2 may be considered

as the 'plain' ones. The reader will see that only 3 flag-transitive examples exist for the case of c'l.A, with

B > 2 or k > 1; only 4 for the case of ck.C,,, with na > 2 and B > 2 or k > 1. Four infinite families and 6 isolated examples exist for cnda.C2 with t > 1, and 8 > 2 or n > 3. Only one example exists for c.An-Z.c* with B > 2 and n > 3. Most of those examples arise from sporadic simple groups or in connection with exceptional isomorphisms between small simple groups. Thus, we may regard them as 'exceptional' examples.

We will also see that no flag-transitive examples exist for 2.0, with m > 2 and k > 1 or 8 > 2.

On the other hand, the cases of C"-2.C2 thin on top with B > 2 or n > 3, of F a . D a and of C.C. (= c.Al.c*), look wilder: a lot of flag-transitive examples exist for them obtained from truncations of (often infinite) Coxeter complexes, or in other ways. We may regard these latter cases as the 'extreme' ones.

1.4 Unfolding

The 'extreme' case of c"-~.CZ thin on top is strictly related with the 'extreme' case of

Let I? be a geometry belonging to the diagram ct.D,, and let I" be the shadow geometry of I' with respect to the initial node of the diagram (with respect to the central node of c*.c, when k = 1 and m = 2; the reader may see [79] or [61] for the definition of shadow geometries; they are called "canonical linearizations" in [Sl]).

C"-'.Da.


The geometry I” is thin on top and belongs to $.C,,,. In particular, starting from C ” - ~ . D ~ we obtain F a . C , thin on top.

Conversely, let I?’ be a geometry thin on top belonging to ch.Cm. If I?’ can be obtained as shadow geometry as above from a geometry I’ belonging to >.Om, then we say that I?’ can be unfolded and that I? is the unfolding of I”. It follows from [74] that I” can be unfolded if it is 2-simply connected and satisfies the Intersection Property IP of [7]. Thus, any classification of cL.Cm-geometries thin on top and satisfying the Intersection Property entails a classification for ck .Dm. In particular, a classification of dl-’.D1 must be implicit in any classification of c”-a.Ca thin on top.

Of course, the above definitions can also be stated for Af.D, -1 and Af.Cn-1 thin on top.

1.5 The Properties LL and IP Trivially, no multiple edges occur in the point-line system of a geometry belonging to one of the diagrams we are considering iff the following property holds in it:

(LL) any two distinct points are incident with at most one line.

Needless to say that LL also makes sense for geometries belonging to diagrams obtained from thoae we are considering substituting circular or dual circular strokes

C C* 1 L

with the following strokes, denoting linear and dual linear spaces, respectively: L L*

c-----. c-----.

In particular, we may replace circular or dual circular strokes with affine and dual affine strokes:

A f - A f - The property LL is a consequence of the Intersection Property IP of [7]. The property

LL is weaker than IP in general. However, in most of the cases we are considering the property LL, or some property a bit stronger than it, is sufficent to obtain IP. For instance, by Proposition 2 of [64] and Proposition 1.3 below, we have the following:

Proposition 1.1 The pmpertres LL and IP are equivalent in geometries belonging to the following diagram:

L L* (L.A,-a.L*) o r - - . .... - L

In particular, LL and IP are equivalent in C.An-s.c* and in AZ.A,-a.A f’ By Proposition 2 of [64] we also have the following:


Propmition 1.2 Let I? be a geometry belonging to the following diagram of rank n = m tk:

Then the Intersection Property IP holds in I' if and only if LL holds i n r and IP holds in the residues of t h e points oj I'.

In particular, LL and IP are equivalent in geometries belonging to L.C2 (as usual, we write L-Cn-1 for L1.Cn-1).

By Proposition 1 of [64] we have the following:

Propmition 1.3 The Intersectson Property IP holds m every geometry belonging to the following dragram:

Hence IP holds in every geometry belonging to the following diagram:

In particular, IP holds in every geometry belonging to ck.Am. The next two propositions are trivial consequences of Proposition 1.3 and of well known results on matroids with projective or d n e planes ([6], Section 9).

Proposition 1.4 The Coxeter diagram A,, characterizes n-dimensional projective geometries.

Propmition 1.5 The diagram A f ,An-l characterizes n-dimensional afine geometries.

Proposition 1.4 will be used quite frequently in this paper, often without saying it. We will see in Section 4.2 that LL holds in every geometry belonging to the following

diagram:

including the case of n = 3:

( L J a ) < L 0

Therefore, the property LL holds in every geometry belonging to c.Dn-1 or to A f.Dn-1 (or to Dm). As a consequence of this, IP holds in any geometry of type c.C,,-1 or A f.Cn-1 (or Cm) thin on top and admitting unfolding.


1.6 Some Background on Lie Diagrams

We have already mentioned a result on Lie diagrams (Proposition 1.4). We will also use the following results in this paper.

Proposition 1.6 All geometries belonging to the Coxeter diagram D,, are buildings.

(see [85], Lemma 3.2; see also [64], Theorem 3).

Proposition 1.7 All geometries belongrng to the Coxeter diagram C,, and satzsfying the Intersection Property IP are polar spaces.

(see [MI, Proposition 9; also [64], Lemma 3).

Proposit ion 1.8 Let I' belong to the Coxeter daagram C,, wtth n 2 4, with jinite orders 3 , t and a > 1:

0- ....._ - 3 3 a 3 t

(cn)

Let A u t ( r ) be flag-transitive. Then I' i s a classical polar space.

(see [64], Theorem 9). The assumption n 2 4 is essential in Proposition 1.8. Indeed a finite thick non-building

flag-transitive C3 geometry is well known, discovered by Neumaier [57]. It has uniform order 8 = 2, its full automorphism group is the alternating group A T , it has 7 points, 35 lines and 15 planes and it is flat (we recall that a C3 geometry I' is sad to be flat if all points of I' are incident with all planes of r). The reader may see [64] (section 2.3) for more details on this exceptional example. It is called the AT-geometry.

Proposition l .B Let r be a Jag-transative C, geometry with finite orders a , t and a > 1:

Then I? i s one of the following: (I) a classical polar space; (it) the AT-geometry; (tit) an unknown C3 geometry where (resadues of) planes are sharply flag-transitive

non-classical projectcve planes of order 8 > lo3.

(see [46], Theorem 5).

We can now come to the diagrams t .A , , L.C,,,, c'.D, and c . A ~ - ~ . c * .

Flag-tranritive Buekenhout geometries 41 I

2 THE DIAGRAM @ . A m

The following theorem by Hughes [32] is the starting point for the theory we are going to survey:

Theorem 2.1 Let I? be a geometry belonging to the diagmm 2 . A , (with s 2 2 and m 2 2). Then r i s either an a@ne geometry of order 2 (hence 8 = 2 and k = 1) or one of the Steiner systems S(3,6,22), 5(4 ,7 ,23) or S(5 ,8 ,24) for the Mathieu groups Mz!, MB and Mzr respectively (in these cases we have s = 4, m = 2 and k = 1, 2 and 3 respectively) .

Actually, one more case was left open in [32], with k = 1 and 5 = 10. However, the non-existence of projective planes of order 10 (see [43]) rules out this possibility.

2.1 Links with Other Classification Problems The problem answered in Theorem 2.1 is a special case of the more general problem of claasifying geometries belonging to the diagram L'.A, of rank n = k + m, with finite orders r l , r2, ..., rk, 8 and na 2 2

L b- +....--a b d ..... - L

T1 r2 rk 8 8 5 8 ( L'.A,)

m nodes

As usual, we write L.A.-1 for L1.An-I. Of course, for every h = 0,1, ..., 12 - 1, the diagram L*.A,+k-h is a specid case of L'.A,. In particular, A, (= Lo.A,) is a specid case of L ~ . A , .

Doyen and Hubaut [25] have proved that projective and affine geometries are the only examples for the diagram L.A,-1 when n 2 4.

Counting point-block flags and checking divisibility conditions obtained in this way, we can see that we necessarily have r = 1 in the following diagram:

L A f -b

c 8 - 1 5

Namely, L.A f is in fact c.A f:

Using this, Remark 2 of [40] (page 190) and the result by Doyen and Hubaut [25] we obtain that projective and affine geometries are the only possibilities for the diagram L'.A, when m 2 3, even if k > 1.

The case of m = 2 is still open. The three Steiner systems mentioned in Theorem 2.1 are the only examples known for this case other than projective and f i n e geometries; hence it is likely that nothing else can exist here. However, no proof is known for this conjecture.

412 A, Pasini, S. Yoshiara

A partial result has been obtained by Doyen and Hubaut in [25] for L.Aa. They have proved that, if r is a geometry belonging to L.A2 other than a projective or affine geometry, then the orders r (= rl) and a of I' satisfy one of the following relations:

( r + I ) ~ = a

Actually, the first relation holds in the Steiner system S(3,6,22), with r = 1 and 8 = 4. The second one would give us finite projective planes of non prime power order (for instance, planes of order 10 when r = 1, which do not exist by [43]).

Putting the above result together with the classification of finite flag-transitive projective planes by Kantor [42] and with the classification of finite flag-transitive linear spaces [12], [13], it is not hard to check that, if the flag-transitivity is assumed in the case of L"-'.Aa, then projective and d n e geometries and the three Steiner systems mentioned in Theorem 2.1 are the only surviving possibilities.

or (r + q3 + (r + 1) = a .

3 THE DIAGRAM c.Cn-l, THE PLAIN CASE We consider the diagram C.C,-~ in this section, assuming a = 2 (whence we have a special case of A f .C,,-l). As a = 2, we have t = 1, 2 or 4, see [73].

We begin with a description of some examples belonging to Af.Cn-1.

3.1

Let P be a non-degenerate polar space with thick lines and rank n 2 3. Let H be a geometric hyperplane of P , namely a proper subspace of P meeting every line of P ([19], [7q). Let I' = P - H be the complement of H in P, namely the system of points, lines and singular subspaces of P not contained in H. Then I' is a geometry belonging to the diagram Af.C,,-,. It is called the @ne polar space defined by H in P. If P is defined over a division ring F, then we say that I' is defined Over F.

The points of I' have the same residues in I' as in 'P. It turns out from [19] that Aut(I') is the setwise stabilizer Aut(P)ar of H in Aut(P).

Let K 5 Aut(P) , act semi-regularly on the set of points of I'. Then K defines a quotient I ' /K of I?, with point-residues as in I' (needless to say that r / K = I' if K = 1). If, moreover, K fixes H pointwise, then the quotient I'/K is said to be standard (of course, this definition also includes d n e polar spaces as improper quotients, when h' = 1).

Minimal standard quotients are obtained when K is the pointwise stabilizer of H. They are flag-transitive.

It is proved in [69] that a quotient of an afIine polar space I'/K is standard iff the property LL holds in it (iff IP holds in r / K , by Proposition 1.2 and because point-residues of I ' /K are polar spaces as in I').

AfRne Polar Spaces and Their Quotients

3.2 A Non-Standard Quotient

Non-standard quotients of affine polar spaces also exist (see [69]). We only mention the following example here.

Flag-transitive Buekenhout geometries 41 3

Let P = 9:(2), H = p' for some point p of P, r = F - H and K = Aut(P)H. The quotient r / K has 4 points, 18 lines and 6 planes. It is flat, in the same meaning stated in Section 1.6 for C3 geometries: all points are incident with all planes. The property LL does not hold in this example: any two distinct points are on 3 lines. The automorphism group is S, x S3, flag-transitive.

3.3 The Neumaier Geometry

An exceptional geometry belonging to c.As with 8 = t = 2 (hence belonging to Af.C3) has been discovered by Neumaier [57]. It can b e constructed as follows.

Take a set S of 8 points and the elements, the 2-subsets and the 4-subsets of S as points, lines and planes, respectively. Choose an affine geometry A = AG(3,2) having S as set of points and take as blocks the images of A under the action of the alternating group AS on S. Define the incidence relation between points, lines and planes in the natural way, as symmetrized containment, and state that a point, a line or a plane are incident with a block X if they are a point, a line or a plane of the affine geometry X.

Let r be the geometry defined in this way. The property LL holds in l?, but IP does not hold in it. Indeed I' is flat: all points and all lines of l? are incident with all blocks of r. The residues of the points of I? are isomorphic with the A7 geometry. We have Aut(I?) = Aa, flag-transitive.

We call this geometry the Neumaier geometry.

3.4

The following proposition has been proved in [22]:

Proposition 3.1 All geometnes belonging to A f .C,,-, with n 2 4 and satisfying the Intersection Property IP are standard quotcents of aBne polar spaces.

Some Results on Af.C,_I Geometries

Cuypers [20] has obtained the following result for the case of n = 3:

Proposition 3.2 All locally finite geometries belonging to A f .C3 and satisfying LL are standard quotients of G n e polar spaces.

Proposition 3.3 Let I' be a gag-transctive locally finite geometry belongang to Af.C,-, with n 2 4. Then I' 1s either a standard quotient of an aBne polar space or the Neumaier geornetsy.

Sketch of the Proof. The residues of the blocks (elements of type n) of r are n - 1- dimensional projective geometries (Proposition 1.5), and they are desarguesian because n 2 4. Hence the residues of the points of I' are either polar spaces or the A7 geometry, by propositions 1.8 and 1.9.

Let the residues of the points of I' be polar spaces (classical by [ S I ) . By Seitz's theorem [80] the action on the residue l?. of a point a of the stabilizer G, of a in a flag- transitive subgroup G 5 Aut(r) contains the appropriate Chevalley group for the polar space I?.. Therefore we can imitate an argument used in Case 1 of the proof of Theorem


4 of (691 to obtain a contradiction from the hypothesis that LL fails to hold in r; the reader may see [69] for the details of that argument.

Therefore, LL holds in r. As LL holds in I? and the residues of the points of I' are polar spaces, the Intersection Property IP also holds in I? (Proposition 1.2). Therefore I? is a standard quotient of an d n e polar space, by Proposition 3.1.

If the residues of the points of r are isomorphic to the A7 geometry, then it is straightforward to prove that r is the Nuemaier geometry. 0

Proposition 3.4 Let I? be a flag-tmnsrttve locally finite geometry belonging to Af.C2 and l e t the residues of t h e points o f r be classical generalrzed quadrangles other than S3(3) , Q4(3), Q;(3), H 3 ( 3 2 ) . Then r i s a standard quotient OJ an afine polar space.

Sketch of the Pmf. As for Proposition 3.3, we can use Case 1 and Case 2 of the proof of Theorem 4 of [69] to prove that LL holds in I? (the argument used in Case 2 of that proof is the same as in Lemma 6 of [24]; see also [95], Remark after Lemma 2, and [97], Lemma 6). The conclusion follows from Proposition 3.2.

The cases of S3(3), Q4(3), Q;(3) , H3(32) , excluded in the above statement, and S3(2) are the exceptional C, cases in Seitz's theorem (801 (Theorem C.7.1 of [41]). 0

Proposition 3.6 Let I' be a geometry belonging to Af.C2, thin on top and with lines of size 8 < 00. Assume furthermore that there i s an integer c > 1 such that any two collinear points o f r are rncident with precrsely c common lines. Then I? i s flat (all points are inctdent with all planes) and c = 8 + 1.

(see [69], Lemma 11).

3.5

By propositions 3.1 and 3.2 we immediately obtain the following theorem, originally due to Buekenhout and Hubaut [lo] and Buekenhout [9]:

Theorem 3.6 All geometries belonging to c.C,-, with 8 = 2 and satisfying the Intersec- tion Property IP are standard quotients of aBne polar spaces defined over GF(2).

Classification Theorems for c.C,+, with s = 2

The next theorem easily follows from propositions 3.3, 3.4 and 3.5 (see also [24]):

Theorem 3.7 Let r be a flag-transitive geometry belonging to c.C,,-, with 8 = 2. Then I' is one of the followcng:

(I) a standard quottent of an afine polar space defined over GF(2); (t i ) the pat example of Sectwn 3.2; (II?) t h e Neumaier geometry.

It isalso worth remarking that an afEne polar space defined over GF(2) admits at most one standard proper quotient (the minimal one, when this is aproper quotient). Therefore, all standard quotients of affine polar spaces defined over GF(2) are flag-transitive.


4 THE DIAGRAM c.Dn-l. THE PLAIN CASE

We now consider the diagram c.D,,-I with a = 2 (whence we have a special case of A f We first describe a class of examples belonging to A f .D,-l .

4.1 Afflne D,-buildings Given a building 2) of type D, over a field K, let P = Qt,-,(K) be the polar space associated with 2) (it is a shadow geometry of 2); see 1.4). We can drop a geometric hyperplane H of P, thus forming an affine polar space P - H . This affine polar space can be unfolded (1.4). Let r be its unfoding. The geometry I? belongs to the diagram A f . D n - l . We call it an afine D , building (over the field K).

It is worth remarking that every proper quotient of the affine polar space P-H gathers the two families of blocks of l? (elements of type n - 1 and n) in one family. Hence it destroys the Af.Dn- l diagram and cannot give us any quotient of I?.

4.2 A Result on Af.D,-I We begin with a lemma on the diagram L.D,-l (see Section 1.5):

Lemma 4.1 The property LL holds rn every geometry belongrng to L .Dn- l .

Sketch of the Proof. The proof is just the same as for Lemma 3.2 of [85] (quoted aa Proposition 1.6 in this paper) but for using Proposition 1.3 of this paper to state IP in residues of blocks. The reader is referred to [&I for the details. 0

Corollary 4.2 Let I' be the shadow geometry of a geometry belonging to L.D,-l , with shadows taken wtth respect to the initial node of the L.Dn-l diagram. Then the Intersec- tion Property IP holds in l?.

Prooj. Easy, by Lemma 4.1 and propositions 1.6 and 1.2, recalling that the Intersection Property holds in all buildings and is preserved under taking shadow geometries. 0

Proposition 4.3 All geometries belongrng to the drogram Af .Dn- l are afine D , buildings.

Prooj. This follows by Corollary 4.2 and [74], recalling that a n e D. buildings do not admit proper quotients, as we have remarked above (when n 2 4, or when n = 3 but r is locally finite, then we could also use Proposition 3.1 and Proposition 3.2 instead of [74]). 0

4.3

By Proposition 4.3 we immediately have the following:

Theorem 4.4 All geofizetries belongmg to C . D , - ~ with 3 = 2 are affine D , bvtldtngs over GF(2) .

The Classification Theorem for c.D,-, with s = 2

416 A. Pasini, S . Yoshiara

5 THE DIAGRAMS ck.Cm AND ck.Dm WITH rn 2 3

5.1

Besides afEne polar spaces of order 2, their quotients and the Neumaier geometry (Section 3)) 4 more examples are known belonging to ck.C,,, with m 2 3. They admit the groups Fiza, Fia3, Filar and 3 - Filar (central non split extension) as minimal flag-transitive automorphism groups and have m = 3, k = 1 ,2 ,3 ,3 respectively and orders 6 = 4 and t = 2 (see [6], [a], [76], [48]):

Geometries for the Fischer Groups

C LL-

4 4 2

C --a-

4 4 2

for Fiaz

for Fia3

(the fourth geometry is a triple cover of the third one). The full automorphism groups are Fill . 2, Fia3, Fia4 and 3 ' F i l l , respectively. The residues of the blocks are the Steiner systems for Mas, Mas, Mar, M24 respectively, mentioned in Theorem 2.1. The first geometry is the isomorphism type of the point-residues of the second one and this is in turn the isomorphism type of the point-residues of the third and fourth ones.

5.2

The following theorem has been proved by Meixner [48] (also Van Bon and Weiss [92]), but part of it had already been obtained by Buekenhout and Hubaut [lo]:

The Classification Theorem for ck.Cm with m 2 3

Theorem 6.1 Let I' be a flag-transrtive geometry belonging to c'.C,,, with m 2 3. Then I? i s one of the follozurng:

(i) a standard qvotrent of an afine polar space defined over GF(2); (:i) the Neumaier geometry; ($88) one of the 4 geometnes for Fischer groups of 5.1.

Remarks on the Proof. Since rn 2 3, we have either a = 2 and k = 1 or a = 4 and m = 3, by Theorem 2.1. The case of a = 2 and k = 1 is dealt with in Theorem 3.7.

Let a = 4 and m = 3. It is easily seen using Proposition 1.9 that C3 residues are polar spaces.

Using this information and the dag-transitivity it easy to prove that LL holds when k = 1 (see [24], Lemma 6). Therefore IP holds when k = 1, by Proposition 1.2. Furthermore, the collinearity graph uniquely determines the geometry (Lemma 7 of [24]). Therefore


the hypotheses of [lo] hold when k = 1 and the geometry for Fizz is the only surviving possibility in this case [lo] (see also Proposition 6.1 of [4d]).

Turning to the case of k = 2, we now know that the point residues are isomorphic to the geometry for Fizz. Using this information, Meixner has proved that LL holds (hence IP holds by Proposition 1.2), that the collinearity graph uniquely determines the geometry and, finally, that the example for Fi23 is the only surviving possibility in this case (Proposition 6.2 of [48]).

The case of k = 3 is dealt with in a similar way (Meixner [48], Proposition 6.3). 0

5.3

Taking shadow geometries and using theorems 5.1 and 4.4 we immediately obtain the following:

Theorem 6.2 All flag-tnansitive geometries belonging to $.D, with rn 2 2 are a f i n e Dn buildings over GF(2) (hence k = 1, rn = n - 1 and s = 2).

The ClassiRcation Theorem for c'B, with m 2 3

5.4

Let us consider the diagram L.Cn-l

Links with Other Classiflcation Problems

L (L.Cn-1) 0 - e ....._ -0

r 8 S 8 8 t

where r , 8 , t are finite orders and s > 1. By [25], when n 2 5 this diagram is either Cn or Af.Cn-l (see also Section 2.1 of this paper). We have diecussed these diagrams in sections 1.6 and 3.4, respectively. Therefore, we will assume n 5 4.

= 4 and assume the flag-transitivity. By what we have said at the end of Section 2.1 we have one of the following: r = 1, r = 8 - 1 or r = 8 . Heace we have C,, Af.C3 or c.C3, respectively, and we are driven back to sections 1.6, 3.4 and 5.2.

Let n = 3 and assume the flag-transitivity and C2 residues of classical type (hence t > 1). Putting the classification of flag-transitive finite linear spaces (121, [13] together with Seitz's theorem on flag-transitive subgroups of Chevalley groups [80], it can be proved [B] that C3, Af.C2 and c.C2 are still the only surviving possibilities for L.C2. The cases of C3 and Af.Cz have been examined in sections 1.6 and 3.4; the diagram c.Cz will be discussed in the next section.

What about the case with non-classical C, residues (in particular, with t = 1) ? We remark that at least one flag-transitive example exists with 1 = 1 and n = 3 where r # 1 , s - 1, s (see [36], Example 7; take the shadow geometry of that example with respect to the central node of the diagram).

If we start from the diagram L.Dn-l and take shadow geometries with respect to the initial node of the diagram (with respect to the central node when n = 3), then we are led back to the case thin on top of L.C,,-l. Thus, the case of n = 3 is the very difficult one when we deal with L.Dn-l, too.

Let

41 8 A. Pasini. S . Yoshiara

6 THE DIAGRAM c"-~.C~. EXAMPLES We first deacribe a standard construction to produce geometries belonging to diagrams of rank n of the following form T 2 . X , where X is a suitable class of partial planes, starting from certain geometries of rank 2 or from certain graphs.

C X ( F - 2 . X ) ..... -* a - a

Let A be a geometry of rank 2, the elements of A being called points and blocks, aa usual, and assume the following on A: there is a positive integer k such that every k + 1-subset of a block belongs to a t least 2 blocks, every k + 2-subset of a block does not belong to any other block, all blocks have size 2 k + 2 and, given a k-subset X of a block, the blocks containing X and their k + 1-subsets containing X form a connected graph if we take the symmetrized inclusion as adjacency relation. Then we can insert all :-subsets of blocks of A (i = 2,3, .,., k + 1) as new elements and we obtain a new geometry I? = E(A) of rank k + 2, which we call the enrichment of A. This geometry belongs to a diagram of rank n = k + 2 of the form c k . X for a suitable class X of partial planes depending on A and it satisfies the Intersection Property IP.

Clearly, the converse also holds: every geometry satisfying the Intersection Property IP and belonging to a diagram of the form 2.X can be viewed as the enrichment of its point-block system.

We have already met examples of geometries obtained in this way in Theorem 2.1: namely, the ck.A2 geometries obtained from the Steiner systems for the Mathieu groups Map, M23, Ma4 (we have k = 1 ,2 ,3 respectively).

We will be especially interested in the case where A is the system of vertices and maximal cliques of a suitable graph 9 , provided that the vertices and the maximal cliques of Q form a geometry of rank 2 satisfying the above hypotheses. In this case we say that E(A) is the enrichmentof the graph Q, and we write E ( 9 ) for E(A).

We can turn to the examples now.

6.1 Examples of Classical Type The geometries we will describe in this section are of classical type (in the meaning stated in Section 1.2), they admit a unique minimal flag-transitive automorphism group and in all cases but one either the action induced by the minimal flag-transitive automorphism group on a C2 residue Q contains the appropriate Chevalley group acting on the classical generalized quadrangle Q, or Q = 5 4 2 ) and that action is As = (542))'.

The last example of this section (for A u t ( H 5 ) ) is an exception to the above 'rule'. The C2 residues of that geometry are of type H3(9) (hence the geometry is of classical type) but the action induced on them by the automorphism group of the geometry is ~ 5 4 4 ) . 2im. Thus, in this case we have one of the exceptional flag-transitive actions in classical finite generalized quadrangles allowed by Seitz's theorem ([41], Theorem C.7.1).

(1) The Family Pin+2 Rank n I: 3; orders a = t = 2.


Let I be a set of size 2n + 2 (n 2 3). Define a graph 0 taking the %subsets of I as vertices and stating that two vertices A , B of 0 are adjacent precisely when A n B = 0. Let Pints = E(G) be the enrichment of 9.

The blocks of are the partitions of the 2n +%set I in Zsets (the notation Pint2 should just remind us of this).

The geometry Pint2 has diagram c " - ~ . C ~ , orders 8 = t = 2 and full automorphism group Aut(Pantn) = Sant2 (see [48], [54] and [d] (72)). The minimal flag-transitive automorphism group is Aan+a.

The C2 residues of Pant2 are isomorphic to &(2). For every n = 3,4,5, ..., Pant2 appears as point-residue in Pantl.

The geometry P&tz is simply connected for every n = 3,4,5, .... The first member P j of this family is one of the 2 affine polar spaces of rank 3 and

orders 8 = t = 2. The other affine polar space with this rank and these orders is the first member of the family described in the next paragraph (2).

If we allowed n = 2, then we would obtain &(a) aa 'Pi. This shows that the starting point of this construction is the exceptional isomorphism A6 "= S'3(2).

(2) The Family O(-)n+2

Rank n 2 3; orders a = t = 2. Let f be a non-degenerate quadratic form in PG(n + 1 ,3 ) (n 2 3) of discriminant -1.

We define a graph 0 taking as vertices the points of PG(n + 1,3) of norm 1 with respect t o f and stating that two of those points are adjacent in 0 precisely when the line joining them in PG(n + 1,3) is tangent to the quadric defined by f. By enrichment, we obtain a geometry with diagram C"-~.CZ and orders 8 = t = 2 where the blocks are antipolar frames of PG(n -+ 1,3) (see [48] and [8] (66)). We denote this geometry by O(-)n+z.

We have chosen points of norm 1 to start our construction of O(-),,2. However, when n is even we could choose points of norm -1 instead of norm 1 and we would obtain nothing but another model for O(-)nt2.

The minimal flag-transitive automorphism groups of the geometries of this family are 05(3), O,$(6), O7(3), O,(3), 09(3), Ot0(3) ,... for n = 3 ,4 ,5 ,6 ,7 ,8 , ... respectively.

The C2 residues are isomorphic to S3(2). For every n = 3,4,5, ..., O(-)n+Z appears as point-residue in O( -),,+3.

All these geometries are simply connected, but when n = 5. In fact O(-)7 has a (simply connected) triple cover, which we denote by 3.0(-)7, where the central non-split extension 3 . 07(3) acts flag-transitively [MI.

The first member 0(-)5 of this series is one of the 2 d n e polar spaces of rank 3 and orders a = t = 2. The other affine polar space with this rank and these orders is the first member Pt of the family described in paragraph (1). The construction of 0(-)6 shows that os(3) acts flag-transitively on O(-)5. However, if we regard O(-)5 as an affine polar space, we see that its minimal automorphism group is U42).

Thus, the exceptional isomorphism os(3) p! Ud(2) is the starting point for this family of geometries.

Finally, we remark that if we allowed n = 2, then we would obtain S3(2) 89 O(-),.


(3) T h e Family O ( + ) n + 3

Rank n 2 3; orders a = 4, t = 2. We now start from a non-degenerate quadratic form f in PG(n + 2,3) of discriminant

1 (R 2 3). The points of norm 1 are again taken as vertices and the rest is as in paragraph (2). The geometry obtained in this way will be denoted by O(+),t3. It belongs to C''-~.C~ and has orders a = 4 and t = 2 (see [4d] and [d] (42)).

The minimal flag-transitive automorphism groups are the following ones, for n = 3,4,5,6,7,8, ..., respectively: 0,(3), 0,(3), O$(3), 09(3), O,(3), Ou(3) ,...

The C2 residues are isomorphic to H3(2'). For every n = 3,4,5, ..., the geometry O( i- )n+3 appears as point-residue in O( +)nt4.

All these geometries are simply connected, but when n = 3. In fact, 0 ( + ) 6 admits a (simply connected) triple cover, which we denote by 3 . Q(+)6, where a central non-split extension 3 . 0 , ( 3 ) acts flag-transitively (the center 3 of this extension is inverted by the involution 22 of Aut(U,(3)), notation as in [l]).

If n = 2 were allowed, then O(i-)5 = H3(22): the exceptional isomorphism U4(2) Z O5(3) is involved here, once again.

As in the case of O(-)n+a with n even, we could choose points of norm -1 instead of points of norm 1 to start our construction of O(+),t3 when n is odd, and we would obtain nothing but another model for O(+),+3, However, it may happen that the same subgroup of PG(n + 2,3) acts in two different ways on the two models of O(+),+3 (the same might happen for O(-)nt2, of course). This is in fact the case for two of the subgroups O,(3).2 of A ~ t ( o ( + ) ~ ) , namely those recognizable as U + ( 3 ) . 22 in [l].

Remark . If we choose a point a of PG(n + 2,3) of norm -1, then the points of a' of norm 1 form a model for 0(-),,+2. Therefore, 0 ( - ) n + 2 can be realized aa a subgeometry

Similarly, if we start from a form of discriminant -1 in PG(n + 3,3) and choose a point Q of norm -1, then the points of norm 1 in a' form a model of 0 ( + ) , + 3 . Thus, we obtain a realization of 0(+),+3 as a subgeometry of O(-)nt3. In particular, if we allow n = 2, then we obtain the well known embedding of 9 4 2 ) aa a hyperplane in Qi (2 ) (dual of H3(22) = O(+)5) and an embedding of H3(2') in 0 ( - ) ~ , respectively.

of O(+)n+3.

(4) A 2-cover of O( +)7

The triple cover 3.O(+), of O(+)S also appears as a point residue in a z.C2 geometry where the subgroup 37. O7(3) of Fi'24 acts flag-transitively [4d]. We denote this geometry by 3' 0(+)7, since it is a 37-fold 2-cover of O(+), (but it is not a Scover of O(+),). Thus, O(+)7 is not 2-simply connected (however, it simply connected, namely 3-simply connected).

(5) The Family Un+2 Rank n 2 3; orders a = t = 3. We now start from the Hermitian variety H,+1(22) embedded in PG(n + 1 , 4 ) (n 2 3)

and define a graph taking the non-isotropic points of P G ( n + l , 4) as vertices and stating that two vertices are adjacent when they are orthogonal with respect to H , + I ( ~ ~ ) . By enrichment of 0 , we obtain a geometry belonging to c"-'.C2 with orders a = t = 3 (see [46] and [8] (20)). We denote this geometry by Unta.

Flag-transitive Buekenhout geometries 42 I

The geometry Un+2 has Un+2(2) as minimal flag-transitive automorphism group and C2 residues isomorphic to 543). For every n = 3,4,5, ..., the geometry 'HHRt2 appears as point residue in 3tn+3.

The geometry Un+2 is simply connected for every n, but when n = 4. In fact, 246

admits a 2-fold and a 4-fold cover (the latter is simply connected), where the groups 2 . Cr,(Z) and 22 . US(2) respectively act flag-transitively. We denote these covers by 2 . 'H6 and 2'. 316, respectively.

If n = 2 were allowed, then we would obtain S3(3) as 314. This shows that the exceptional isomorphism U 4 2 ) S 5'4(3) = 05(3) is (once again) the starting point for this family of geometries.

Remark. Constructions similar to those considered in the above paragraphs (2), (3) and (5) are used by Cuypers [21] (and by Fischer [26] before him) to produce generalized Fischer spaces.

Furthermore, the description of minimal quotients of affine polar spaces as tangent geometries [22] is almost the same thing ria the construction used in the above paragraphs (2) and (3).

These analogies would deserve a closer investigation.

(6) Geometries for McL, CQ and 2 x C s Ranks 3, 4 and 4, respectively; orders 8 = 3, t = 9. Let P be the McLaughlin graph on 275 vertices ([47], [29], [5] 11.H). By enrichment of

G we obtain a c.C2 geometry with orders 8 = 3 and t = 9 and admitting the simple group McL as flag-transitive automorphism group. We denote this geometry by I'(McL).

The point-residues of I'(McL) are isomorphic with Q i ( 3 ) and I'(McL) is simply connected [761.

The geometry I'(McL) appears as a point residue in a c2.C2 geometry admitting CQ as flag-transitive automorphism group ([d] (23); also [29], [50] and [94]). We denote this latter geometry by I'(Cq).

The collinearity graph of I'(CQ) is trivial, with 276 points. Thus, r(CQ) cannot be recovered from its collinearity graph. However, it can be recovered from the (unique) two-graph on 276 vertices [29].

The geometry I'(Cs) is not simply connected. Indeed it admits a (simply connected) double cover where 2 x Cg acts flag-transitively ([76], [SO], [94]). We denote this double cover by 2 . I?( Cq) .

(7) Geometries for Suz and Cq Ranks 3 and 4, respectively; orders 8 = 9, t = 3. A c.C2 geometry admitting Suz as minimal flag-transitive automorphism group and

with orders 8 = 9 and t = 3 arises by enrichment from the Patterson graph for the simple group Suz (see [5] 13.7; also [d] (6)). We denote this geometry by I'(Suz).

The point residues of r (Suz) are isomorphic to H3(32). The geometry F(Suz) is simply connected [76].

The geometry r( Suz) appears as a point residue in a simply connected c2.C2 geometry for the simple group (2% (see [50] and [d] (7)). We denote this latter geometry by r (Cs ) .


(8) A Geomet ry for A u t ( H S ) Rank 3; orders d = 9, t = 3. Let 3.1 be the Higman-Sims graph and let Y be the graph obtained from 3t taking the

edges of H as vertices and stating that two edges A, B of 3.1 are adjacent as vertices of Y precisely when A n B = 0 and A, B are the only edges of 3.1 contained in the 4-set A U B. The graph obtained in this way has two families of maximal cliques, of size 10 and 11, respectively [96]. A rank 2 geometry ll can be defined taking the vertices of Y as points and the cliques of Y of size 11 as blocks. By enrichment of ll, we obtain a c.C, geometry with orders = 9 and t = 3 (see [96]), just as for I ' (Suz) . This c.Ca geometry is called the Yoshiara geometry. We denote it by I ' (HS) .

The geometry r ( H S ) admits Aut(HS) as unique flag-transitive automorphism group. The point-residues of I ' (HS) are isomorphic to H3(3'), as in I'(Suz). However, the stabilizer in A u t ( H S ) of a point o acts in the residue of o as L3(4). 2;33. Thus, we have one of the exceptional flag-transitive groups for classical finite generalized quadrangles allowed by Seitz's theorem ([41], Theorem C.7.1).

The geometry r ( H S ) is simply connected ([66], [95]).

6.2 Examples Thin on Top

The examples of this section are thin on top (hence they are not of classical type). In all of them the stabilizer of a block u in a flag-transitive automorphism group acts on the 8 + n - 1 points of u 88 Attn-1 or S a t n - 1 .

(1) The Family Sntl Rank n 2 3. First order d = 2. Let I be a set of size n+ 1 2 4 and let Sntl be the set of all permutations of I, viewed

as subsets of I x I. Let Sntl be the enrichment of the rank 2 geometry having I x I as set of points and Sntl as set of blocks. Sntl is a well known permutation geometry [16]. It belongs to the diagram cn-'.C2, it is thin on top and has first order 8 = 2.

We have Aut(S,,,) = Snt1J2 (wreath product) and Sntl admits two minimal flag- transitive automorphism groups, both isomorphic to Anti x s,,,.

For every n = 3,4,5, ..., the geometry Snt1 appears as point-residue in SntZ. The geometry Sn+l is simply connected for every n = 3,4,5, .... Since, furthermore, IP holds in it, Sntl admits unfolding; the odd and the even permutations form the two families of blocks of the unfolding of Sn+l.

The first member S4 of this family is an affine polar space. The non-standard quotient described in 3.2 is in fact a quotient of Sd. We denote it by S4/P.

(2) The Family Coz(n,s) Rank R 2 3; first order 8 2 2. Let C be a Coxeter complex belonging to the following diagram:


e . . . . . - <

0 - c ... . . - L O

I

a nodes

n - 3 nodes

where n 2 3 and a 2 2. We remark that, if n = 3 then we have a Coxeter diagram of type Aa,+l. If n = 4 but s = 2, then we have Eg. We have a non-spherical Coxeter diagram in all remaining cases (whence C is infinite in those case). In particular, if n = 5 and a = 2 then we have the afEne diagram &. If n = 4 and 8 = 3 then we have the afEne diagram &.

If we truncate the elements of C corresponding to the a - 1 nodes in each of the two right-hand horns of the diagram, then we obtain a c"-'.D3 geometry. We call it Tr(C). Let C m ( n , 8) be the shadow geometry of Tr(C). Then Coz(n, a ) is a c " - ~ . C ~ geometry thin on top and a is its first order. Clearly, C m ( n , a ) is flag transitive and admits many quotients (actually, infinitely many flag-transitive quotient when the Coxeter diagram which we start from is not of spherical type).

For every n = 3,4,5, ... and a = 2,3,4, ... the geometry Coz(n, a ) appears as point- residue in Coz(n + 1 , d ) and is a subgeometry of Cm(n, a + 1).

It follows from [63] (Theorem 1) that Tr(C) is Zsimply connected. Hence, Coz(n, a ) is Zsimply connected, by [74].

Coz(3, a ) is cal!ed Johnson geometry in [18]. We have Aut(Coz(3, a ) ) = 2 x &,+a and 2 x A2,+2 is the minimal flag-transitive automorphism group of coz(3,3). The geometry Coz(3, a ) admits just one flag-transitive proper quotient (over the center 2 2 of Aut(Cm(3, d))), called the halved Johnson geometry. We denote it by Cm(3, a)/2. The full and minimal flag-transitive automorphism groups of coz(3, a)/2 are S2,+2 and A2,+2,

respectively. The geometry coz(3,2) is one of the two affine polar spaces of rank 3 thin on top

defined over GF(2) (the other one is &). The Johnson geometries (and their quotients) and coz(4,2) are the only finite ex-

amples among the geometries Cos(n,a). This makes it clear that no analogue can be found of Theorem 3.4 of [18] to bound the diameter of the collinearity graph of a L.C, geometry when k 2 2.

Remark. A c?.C, geometry thin on top with a = 3 is mentioned in [8] (30'), admitting Ss(2) as flag-transitive automorphism group. There are strong evidences that this geometry is a quotient of Cm(4,3) [59].

(3) Geometries for M22 . 2, L3(4). 22 and 25 : L44) Ranks 5, 4 and 3, respectively; first order a = 3. Choose two distinct points u, b in the Steiner system S(5,8,24) and form a rank 2

geometry II taking as points the points of S(5,8,24) other than a or b and as blocks the


octads of S(5,d, 24) containing one but not both of a and b. By enrichment of ll we obtain a c3.C2 geometry with 8 = 3 and t = 1 ([la], 9.12). We denote it by I'(Mn), because the (unique) flag-transitive automorphism group of this geometry is the stabilizer A u t ( M 2 ~ ) in Ma4 of the unordered pair {a, b}.

The geometry r ( M z 2 ) admits unfolding; the two families of blocks of its unfolding consist of the octads containing a but not b and containing b but not a, respectively. The stabilizers of the blocks of I'(Ma2) form the two conjugacy classes of A6 in Mm, fused by the outer automorphism of Mn.

The point-residues of r( Mz2) admit L3(4).22 aa (unique) flag-transitive automorphism group. We denote the isomorphism type of the point-residues of T'(M22) by r ( L 4 4 ) ) . Trivially, r( L3( 4)) also admits unfolding.

The point-residues of I'(LB(4)) are examples of Cameron-Fisher extensions of odd type (see [55], [Id], [17]; also paragraph (1) of the next section of this paper). We denote their isomorphism type by C F i . Trivially, CF; admits unfolding. However, it is not simply connected. Indeed, it admits a double cover [15]. We denote this cover by 2 . CF;. It follows from [4] (also [27]) that 2 . CF; is simply connected.

The minimal flag-transitive automorphism group of CF; is 2& : L 4 4 ) . The minimal flag-transitive automorphism group of 2 . CF; is 2b : L2(4).

6.3

Many flag-transitive c.C2 geometries thin on top are known besides S4, S4/2a, cog(3 , a), C m ( 3 , s ) / 2 , CF; and 2 . CF;. We will mention them now.

We use [55] as standard source of references for this matter. We warn the reader that some of the examples we will consider in this section can be obtained by different constructions. In these cases we have chosen only one of those constructions, referring the reader to [55] for more information.

In the examples of this section the action induced on a block u by the stabilizer of u in a flag-transitive automorphism group is either a 1-dimensional projective linear group or an &ne group. The geometries of S4 and C 0 ~ ( 3 , 2 ) and their quotients s4/2' and c o s ( 3 , 2 ) / 2 again arise in the present context (note that A4 = L Q ( ~ ) ) . We will also meet CF; and 2 . CF; again (note that A5 Z L 4 4 ) ) .

All geometries described in this section admit unfoldings, but for the flat ones (para-

Further Rank 3 Examples Thin on Top

graph (4)).

(1) Cameron-Fisher Extensions of Odd t y p e References: [55], 2.5; also [la] and [17]. First order 8 = q - 1, where q 2 4 is a power of 2. Construction. Start from a quadratic cone C in PG(3, q ) , where q 2 4 is a power of

2. Let v be the vertex of C, let R be the radical line of C and u, b two distinct points in R - { v } . Take C - { v } as set of points and the planes of PG(3 ,q ) on one but not both of a and b as blocks. The enrichment of this rank 2 geometry is a c.C2 geometry thin on top with 8 = q - 1. We denote it by CF;.

Covers. It is proved in [15] that, for every proper divisor d of q, CF; admits a q-fold cover, call it d . CF;. It is not clear if q/2. CF; is the universal cover of CF; in general


(however, this is true when q = 4, as we have remarked in 6.2(3)).

automorphism group q' : LQ(q) : 2. Flag-Transitivity. The geometry CF; is flag-transitive, with minimal flag-transitive

The action of the stabilizer in q2 : L2(q) : 2 of a block on the q + 1 points of the block is L2(q).

(2) Cameron-Fisher Extensions of Even Type References: [55], 2.6; also [Id] and [17]. First order .I = q - 2, where q 2 4 is a power of 2. Construction. We only must slightly modify the construction of the previous para-

graph (1). We chose a line L of the cone C, a point D € R - { v } and a point b € L - (.}. We take C - L as set of points and as blocks the planes of PG(3, q ) not on II and containing one but not both of a, b. The enrichment of this rank 2 geometry is a c.Ca geometry thin on top with .I = q - 2. We denote it by CF:.

Flag-Transitivity, The geometry CF' is flag-transitive, with minimal flag-transitive automorphism group G = ( (AGL(1,q) x AGL(1 ,q ) ) : 2 ) . 2.

The action of the stabilizer in G of a block on the q points of the block is AGL(1, q). Remark. We have CF: = Sd.

( 3 ) Geometries from Quadrics in PG(3, q ) ( q odd) References: [55], 2.4. First order a = q - 1, where q is an odd prime power. Construction. Let Q be a non-degenerate quadric, embedded in PG(3 ,q ) , q odd.

Let Pt (respectively, P - ) be the set of non-singular points of PG(3,q) of square norm (respectively, non-square norm). If L is a line of PG(3, q) tangent to Q, then L - ( L fl Q) is contained either in P+ or in P- . Therefore the lines of PG(3,q) tangent to Q are partitioned in two families, call them Lt and C-, where a tangent line L belongs to Lt (respectively, to C-) if L - ( L n Q) Pt (respectively, if L - (L n Q) C P-) . The tangent lines contained in a secant plane belong to the 6ame family. Therefore, the secant planes are also partitioned in two families, call them IIt and ll-, where a plane a is in IIf (respectively in ll-) if the tangent lines on a belong to C+ (respectively, to C-) . In fact, if Q = Q i ( q ) and q 3 (rn0d.4)~ then llS consists of the planes a such that a* E P - ; on the other hand, when Q = Q;(q) with q E 3 (rnod.4) or Q = Q$(q) with q I 1 (mod.4), then lI+ consists of the planes a with ( Y ~ E Pt.

Let us take C+ as set of points and P+ U II+ as set of blocks, defining the incidence relation between a point and a block in the natural way, by (symmetrized) containment. By enrichment of thiPi rank 2 geometry we obtain a c.C2 geometry thin on top with s = q - 1. We call it Tg( Q).

1 (rnod.4) or Q = Q,f(q) and q

Flag-Transitivity. The geometry Tg( Q) is flag-transitive. The minimal flag-transitive automorphism group G and the action induced by G , on

(a) Q = Q i ( q ) ; G = O;(q). 2 (= O;(q) x 2 when q f 3 (rnod.4)); action induced on

(b) Q = Q,f(q); G = (Lz(q) x L ( q ) ) . 2 (= L2(q) x L2(q) x 2 when q 1 (mod.4));

the q + 2 points of a block I are as follows:

blocks: PGL( 2 , q).

action induced on blocks: Lg(q).


When Q = Q i ( q ) with q 9 3 (rnod.4) or Q = Q$(q) with q I 1 (mod.4), then Tg(Q) admits a flag-transitive proper quotient (over the center 22 of A u t ( T g ( Q ) ) ) . We denote this quotient by Tg(Q)/2.

Remark . We have T g ( Q i ( 3 ) ) = cos(3 ,2) and T g ( Q i ( 3 ) ) = S,. The geometry Tg(Qi(5)) can also be obtained as the shadow geometry of the geometry of Example 7 of [36] when q = 4 (recall that L2(4) Z L2(5)). The quotient Tg(Q$(5))/2 is the geometry described in [Id], 9.12(iii) (see also [17]).

(4) F l a t Examples. The geometry S4/21 is flat: all points are incident with all blocks. A general construction for flag-transitive flat c.C2 geometries thin on top is given in [55], 2.7. They have first order q - 2, where q 2 4 is a power of 2 and automorhism group G = (AGL( 1, q ) x 2. The action on the q points of a block is AGL( 1, q).

Needless to say that S4/P is included in that construction.

6.4 A Few Remarks on Non-Flag-Transitive Examples

Many non-flag-transitive C " - ~ . C ~ geometries thin on top can be obtained taking quotients of geometries of type &+I or Cos(n, a) . Another non-flag transitive c.C2 geometry thin on top with first order a = 3 is constructed in [4].

A family of (non-flag-transitive) c.C2 geometries with orders a = 4-1, t = q + l ( q 2 5, odd prime power) is constructed by Thas [84]. The point-residues of the geometries of this family are isomorphic to non-classical generalized quadrangles of type AS(q). The construction by Thas also works when q = 3 . In that case it gives us a standard quotient of an afEne polar space over GF(2).

7 THE DIAGRAM c"-~.C~. THEOREMS 7.1 The Classical Case The next theorem is the complete classification of flag-transitive c.C2 geometries of classical type. It is obtained glueing together contributions by a number of people: Buekenhout and Hubaut [lo] (this is in fact the most conspicuous contribution to this classification), Buekenhout [9], Weiss and Yoshiara [95], Del Fra, Ghinelli, Meixner and Pasini [24]. Also Goethals and Seidel [29], Blokhuis and Brouwer [2], [3].

Theorem 7.1 Let r be a flag-transrttve c.C2 geometry of classical type. Then I? is one of the following:

( 8 ) a standard quotient of an afine polar space of mnk 3 over GF(2); (at) one of the geometries 0 ( + ) 6 , 3. O(+), or&, descrtbed an (3) and (5) of Section

( t i t ) one of the geometrresJor McL, Suz or Aut(HS), mentaoned in (6)-(i?) of Section 6.1;

6 .1 .

The geometries Pi and O(-)s ((1) and (2) of Section 6.1) are affine polar spaces, whence they are included in (i).

Flag-tramitive Buekenhout geometries 421

A partial classification for the general case has been obtained by Meixner [48], [50] (also Weiss [94], in the case of a = 3 and t = 9).

Theorem 7.2 Let Then I? 1s one of the following:

be a flag-tmnsataue C"-'.Cz geometry of classical type with n 2 4.

( 8 ) one of the geometries for Cq, 2 x Cq and C q mentroned an (6) and (7) of 6.1; (88) one of P;);I+p, O(-),,t2, O(+)n+3 orunt, (Section 6.1, (1)-(3) and (5)); (iii) one of 3 . O( -)7, 2 ' 246 or 2' . 246 (Sectron 6. I, (2) and (5)); (rv) a C"-'.C, geometry with n 2 6 and c3,C2 residues isomorphic to 3 . O(-)7; (v) a c"-'.C' geometry with n 2 4 and c.Ca residues womorphic to 3 . 0 ( + ) 6 ;

(vi) a c"-'.Ca geometry wrth n 2 5 and 2.C' residues isomorphac to 2 .US or 2' ,248.

The geometry s7 . 0 ( + ) 7 mentioned in (4) of Section 6.1 is the only example known for (v). No examples are known for (iv) and (vi).

Problem. Can we rule out (iv) and (vi) from the above theorem ? Can we prove that 3'. O(+), the only example for (v) ?

7.2 A Few Comments on the Proofs of the Above Theorems Various methods have been used by the authors quoted in Section 7.1 for the proof of theorems 7.1 and 7.2. However, the initial main steps of the proof are quite standard. We now briefly explain them, focusing on the rank 3 case.

Let I? be aa in the hypotheses of Theorem 7.1 and let G 5 Aut(r) be flag-transitive. By Theorem 3.7 we may assume that r has first order a > 2.

A well known theorem by Seitz [do] (Theorem C.7.1 in [41]) gives us a detailed information on flag-transitive automorphism groups of finite classical generalized quadrangles. On the other hand, the stabilizer G, of a block u acts 2-transitively on the a + 2 points of u. Comparing the information we have by Seitz's theorem with the classification of 2-transitive permutation groups [14] it is not difficult to check that the group induced on a point-residue must be classical with the only exception of L3(4) . 2:33 acting in H3(32)

In any case, the stabilizer G,, of a point-block flag (a, u) acts on the 8 + 1 points of u other than a as a group between L'(a) and Aut(L'(a)). By a theorem of Suzuki [83] (also Tits [86]) we have a = 3,4 or 9 and G, acts as MI1 on the 11 points of u when a = 9.

Furthermore, the stabilizer G, of a point a acts primitively on the set of lines on a (points of the generalized quadrangle I?,, residue of a). Using this information it is easy to prove that LL holds in I' (see [24], Lemma 6; also [95], Remark after Lemma 2; and [97], Lemma 6(2); compare with the proofs sketched for propositions 3.3 and 3.4).

At this stage we know that point-residues are isomorphic to H3(3') when a = 9, but the range of possibilities left for point-residues is still too large when 5 = 3 or 4. We must prove that only S3(3), 0;(3) and 4 4 2 ' ) can occur.

There are two ways to prove this. We can exploit a group theoretic analysis to get a contradiction in the cases to rule out, as in [97]. The property LL is the only geometric information on r which we use if we work in this way (besides geometric informations on point-residues, of course).

(see ~ 4 1 ) .

42 8 A. Pasini, S . Yoshiara

Otherwise, we cam collect information on the collinearity-graph G(r) of r and on stabilizers in G of certain configurations in G(r), using the information we are getting on G(r) to improve our knowledge of G and conversely; a graph- or group-theoretic contradiction is eventually obtained in the cases to rule out. This is the method chosen by Buekenhout and Hubaut in [lo] (see also [68]). We call it the geometric method, just to have a name for it, even if it is not at all entirely geometric.

A preliminary step to make if we want t o use the geometric method is showing that r can be recovered from its collinearity graph G(r) and that the local structure of G(r) can be recovered from the local information we have on I' (on point-residues and point- stabilizers). In practise, this amounts to know that (LL holds, as we have said above, and that) the blocks of are the maximal cliques of (?(I?). This latter property can be proved to hold true whenever the action on I?, of the stabilizer G, of a point a contains the appropriate classical group (see [24], Lemma 7); but that property does not hold in the Yoshiara geometry for Aut (HS) (Section 6.1 (d)), where the action of G, on I?, is exceptional. However, we are only considering the cases of s = 3 and 4 at this stage; in these cases, we can recover r from o(r) in the above sense. It is worth remarking that the possibility to recover I' from G(r) was actually assumed as an hypothesis in [lo], together with LL.

Anyhow, we can prove that the isomorphism types of the point-residues are just those we want. The last step is proving that the examples listed in (ii) and (iii) of Theorem 1 are the only possible ones. Again, we can work in two different ways to obtain this goal.

We can use a geometric method to get control over G(r) and over certain subgroups of G, eventually reaching the conclusion with the help of some graph-theoretic result (as [29] for I'(McL) or [2] and [3] for O(-)s and 3 . 0 ( - ) 6 ) or some group-theoretic result (as [82] for r(Suz)). This is the method used by Buekenhout and Hubaut in [lo] for cases with s = 3 and 4 and by Meixner in [49] to characterize I' (Suz) . One of the first steps in this method is getting control over configurations of points adjacent with two points at distance 2 in G(r) (see also [68]); then we must get control over configurations of points adjacent with a point a and having distance 2 from another point b a t distance 3 from a, if we have not already proved that G(r) has diameter 2; and so on, till when the distribution diagram of G(r) is completely determined.

The case leading to the Yoshiara geometry r ( H S ) is the only one that looks (almost) impossible to deal with in this way, as we have remarked above. Of course, we do not claim that nobody will ever be able to do that; we only remark that, in order to work in a geometric style in this case, we need some trick to recognize the &cliques of the adjacency graph not contained in any block, using the local information we have on point-residues and stabilizers; this seems rather difficult to do.

On the other hand, we can also work in a more algebraic style, as in I971 (also [95]; and [65], [66], [673). We can exploit the information we have on the action of G, and G, in the residues I?, and I'. of a and 2~ to reconstruct the (possible) structure(s) of the parabolic8 G, and G,, where ( c I , ~ ) is a point-block flag. When we have done that, we have also determined presentations for G. and G, by means of generators and relations, or the information we have got on G, and G. implicitly entails such presentations. We can now go on in two slightly different ways.

We can reconstruct all possibilities for a complete set of generators and relations


presenting G, involving presentations for G, too (where T is a line on Q in u) . Then, if we want to save time we can ask a computer to check which of those possibilities survive (hoping to obtain some answer in any case, of course) and eventually we identify the surviving groups; otherwise, we may do this job 'by hand' (it is not so terrible as we might believe before trying it), ruling out the sets of relations that afterwards turn out to be inconsistent and recognizing precisely those groups we wished to obtain in the surviving sets of relations.

This is the style chosen in [97] for a uniform proof of Theorem 7.1 (and [95], for the cases with a = 9; also [94] for the case of 8 = 3 and t = 9). We call this method the method by generators and relatrons.

However, we can try to give some more indirect argument to compute the number of possibilities for the structure of G, (with r a line on a in u ) and for the amalgamated product of Go, G, and G,. For instance, in most of the C.Ca cases we are considering this can be done computing the dimension of certain linear subspaces of the kernel K,, of G, (called subspaces of drscriminating vectors in [65] and [67]). Then we check if the distinct possibilities for G, and for those amalgams are as many as the known simply connected examples with those parabolics Go and G,, that we are considering. Needless to say that we must preliminary know which of the known examples are simply connected and, if some of them is not simply connected (as O( +)6), we should know its universal cover. For instance, the most part of [66] is devoted to prove the simple connectedness ofI'(HS) (however, checking simple connectedness is not difficult at all in the other cases and can be done exploiting upper bounds for diameters of collinearity graphs [67])

If the possibilities we have computed for amalgams are as many as the simply connected examples that we know (in particular, ifjust one possibility existsfor that amalgam and just one simply connected example is known), then we can use the well known correspondence between amalgamated products and universal covers ([89], [go]; also [62], pages 334-336) to conclude that we already know all simply connected examples. Proper quotients (when they exist) are easy to study in the cases we are considering. We are done.

This is the method used in [65], [66], [671 for a uniform proof of Theorem 7.1. We call it the amalgam method, even if the main idea of it, namely the correspondence between universal covers and amalgamated products of parabolics, is also implicit in the method by generators and relations.

Turning to Theorem 7.2, Meixner [4(3], [50] has proved that theorem by the geometric method, combining it with the amalgam method in [50] for the case leading to I'(Cs). It is worth remarking that the collinearity graph of I'(CQ) is trivial, whence a geometric approach to this case might seem to be hopeless. However, things are not so bad at all; indeed the two-graph of %subsets of blocks can be used in this case instead of the collinearity graph, and the conclusion can be obtained by [29]. The uniqueness proof for r (Cq) and 2 . q Cq) has also been obtained by Weiss [94] using generators and relations.

7.3 The Case Thin on Top

In this section I' is aflag-transitive F a . C , geometry thin on top and G is a flag-transitive subgroup of Aut(I'). As usual, 8 2 2 is the first order of r.

430 A. Pasini. S. Yoshiara

If u is a block of I?, the stabilizer G , of u in G acts n - 1-transitively on the a -t n - 1 points of u. Therefore, the action G,/K,, of G, on the points of u contains AItn-l when n 2 7.

Theorem 7.3 Let n = 3 and G,/hrU 2 A,+2 (wrth u a block). Then I? 1s one of the followrng: cox(3 , a ) , Cox(s,a)/2, &, &/2', 2 . CF; or CF; .

The reader may see [70] for the main part of the proof. A case waa left open in [TO], corresponding to possible flat examples, but it is proved in [56] that S4/2' is the only flat example satisfying the hypotheses of this theorem.

It is likely that much more could be proved:

Conjec ture 7.4 the flag-transstrue cn-'.C2 geometries thrn on top w:th A,,+,-l or Sn+,+l rnduced on blocks are precisely those considered tn Section 6.2 (including proper quotients, when they ezist).

An almost complete proof of this conjecture is given in [56]. We say "almost complete" because for the moment a few gaps still remain to be filled in that proof in the case of a = 3. Of course, this would only give us a classification of the 2-simply connected examples satisfying the above hypot heses, whereas infinitely many proper quotients exist of Cos(n, a ) when n 2 5 and when n = 4 and a 2 3, which still satisfy those hypotheses. However, a classification of 2-simply connected examples is often the only goal we can hope to reach in cases like this, where infinite examples exist admitting infinitely many quotients.

A proof of Conjecture 7.4 would also give us a proof of the following, as a by-product:

Conjec ture 7.6 The geometrresSn+l and Cos(n, 8 ) are the only 2-simply connectedflag- transatwe cn.C2 geometrtes thm on top of rank n 2 7 .

Something can also be proved on flat flag-transitive c.C2 geometries thin on top. For instance, if Aut(I') is assumed to be solvable, then the only examples that can arise are as in (4) of Section 6.3 (see [56]). The flat geometry S4/2' is the only flat c.C2 geometry thin on top that can appear as a residue in higher rank flag-transitive C-2.C2 geometries [561.

7.4 Problems

Let I? be a flag-transitive c.C1 geometry, G a flag-transitive subgroup of I? and u a block of I?. The stabilizer G, of u acts 2-transitively on the a + 2 points of u . The list of 2-transitive finite permutation groups contains many cases. However, only 4 of them are realized in the known examples, namely the following ones:

(i) alternating or symmetric groups on a t 2 points; (ii) the Mathieu group Ml1 on 11 points (only in l"(Suz) and I '(HS)); (iii) one dimensional projective linear or semilinear groups over GF( a + 1) (when a = 2

(iv) one dimensional affine linear or semilinear groups over GF(a + 2) (when a = 2 or 3 these ace the same as A,+a or S,+p)

these are the same as A, or S4).

Flag-transitive Buekenhout geometries 43 I

Is this list so poor because it must be so, or only because our imagination is poor ? Of course, this question concerns the case thin on top and it is in some sense the

same as asking whether the examples of sections 6.2 and 6.3 already form a more or less complete catalogue, or not.

In the classical case, Theorem 7.1 cuts the matter short and the way in which that theorem is proved (by Seitz's theorem [80] and Suzuki's theorem [33]) also explains why the above list must be so poor in this case.

As for non-classical cases with t > 1, only 2 non-classical flag-transitive thick finite generalized quadrangles are presently known (with ( a , t ) = (3,5) and (15,17); see [41], pages 97-93). They are members of an infinite family of non-classical generalized quadrangles usually denoted by T;(O) (see 1731).

Three flag-transitive c.Cz geometries have recently been found by Yoshiara [loo], having the generalized quadrangle Ta+(O) of order (3,5) or its dual as isomorphism type of point-residues. In these geometries the stabilizer G, of a block u acts on the 5 (or 7) points of u as S, (as 5'7 or ,542)) if residues of points have orders (3 , s ) (respectively, (523)).

What about the case with orders (15,17) or (17,15) ?

7.5

By Section 1.4, classifying C"-'.D2 geometries is essentially the same thing as classifying F 2 . C 2 geometries thin on top. However, something more can be said on C , - ~ . D Z geometries, concerning the Intersection Property IP.

By a theorem of Sprague [81] concerning the diagram L*.L, we have the following:

A Few Remarks on C " - ~ . D ~

Theorem 7.6 Let r be Q c*.c geometry satisfying IP. Then I? IS obtained truncating a Coxeter complex of type Az,+l 0s an (2) of Sectron 6.2 (where s IS the order of I?).

It is worth remarking that, even if IP holds in the shadow geometry of every C*.C

geometry r (Lemma 4.1 and Proposition 1.2), nevertheless IP need not hold in I?, in general.

Let now I? be a flag-transitive C " - ~ . D ~ geometry of order s, satisfying IP. Then the c.D2 residues of I? are obtained truncating Coxeter complexes of type &,+I, by the above theorem. Hence the stabilizer G. of a block u of J? in a flag-transitive subgroup G of Aut(r) acts as A,+,-1 or S,+,-l on the 8 + n - 1 points of u. If we can prove Conjecture 7.4, then we could also prove that I' is obtained by truncation and possibly taking quotients from a Coxeter complex, as in Section 6.2 (2).

8 THE DIAGRAM c.A,+~.c*

In this section r is a geometry belonging to c.AR-2.c* and s 2 2 is the order of I?. We recall that c.A,-2.c* is a special case of Af.An-2.Af' when 8 = 2.

We recall that LL and IP are equivalent in this context (Proposition 1.1).


8.1

Let p and H be a point and a hyperplane of P = P G ( n , K ) , n 2 3. If we drop from P the point p together with its star and the hyperplane H together with everything it contains, then we obtain a geometry r = P - { p , H } belonging to Af.A,-z.Af* and satisfying the Intersection Property IP. We call a n afine-dual-afine geometry (defined over the division ring K).

Standard quotients of I? can also be defined, factorizing over subgroups N of the elementwise stabilizer of the star of p and of the hyperplane H in P G L ( n + 1,K) . The geometry r / N still belongs to Af.A, - , .Af* .

We allow N = 1, so that to include r among its standard quotients, as the improper quotient.

It is easily seen that LL (whence IP) fails to hold in all proper standard quotients of r. In particular, if N is the elementwise stabilizer of H and of the star of p and either p E H or K # GF(2) , then r / N is flat: all points are incident with all blocks. Trivially, this standard quotient is the minimal one and it is the only flat quotient of I?.

When K = GF(2) and p E HI then the geometry I' admits just one proper standard quotient, the flat one. When K = GF(2) and p g! H, then does not admit any proper standard quotient.

A Preliminary Result on Af.An-z.Af*

The following has been proved by Lefevre and Van Nypelseer [44], [93] (also [45]):

Propmition 8.1 A11 geometries belonging to A f .A,-2.A f and satisfying LL are afine- dual-afine geometries.

8.2 An Example for H S

A geometry for the simple group HS is well known, with diagram c.As.c* and order a = 4 ( [8 ] (49); also [34]; [66], page 150). It satisfies the Intersection Property IP and HS is its minimal flag-transitive automorphism group. We denote it by 'HS.

The following has been proved by Hughes [34], [35]:

Theorem 8.2 The geometry 315 i s the unrque geometry belongmg to c.A2.c* wrth .I = 4 and satisfying LL.

It is remarkable that no group theoretic assumptions are necessary for this theorem. On the other hand, the property LL is essential, at least in the proof given in [34], [35] (the only one known for this theorem).

8.3 Assembling Proposition 8.1 and Theorem 8.2 with Theorem 2.1 we obtain the following:

Theorem 8.3 Let belong to C.A,,-~.C' with n 3 4 and let LL hold rn I?. Then r ;s erther an afine-dual-afine geometry defined over GF(2) or the geometry 'HS for the simple group H S .

The Diagram C.An-Z.C* with rz 2 4

Assuming the flag-transitivity instead of LL we have the following:


Theorem 8.4 Let I' be a flag-tmnsitiue geometry belonging to c.A,-2.c+ with n 2 4. Then 1s either the geometry '?is for H S OT a standard quotient of an afine-dual-afine geometry defined over GF(2).

Proof. We have s = 2 or 4, by Theorem 2.1. The residue of a block of I' is AG(n - 1,2) or 5(3,6,22), by Proposition 1.5 or Theorem 2.1. Dually for residues of points.

If LL holds, then we can apply Theorem 0.3 and we have finished. Therefore, we assume that LL does not hold.

As LL does not hold in r, there are distinct collinear points of I' incident with at least two distinct common lines. By the flag-transitivity of G = Aut ( r ) , the same situation occurs for any pair of distinct collinear points. Therefore, the stabilizer G, in G of a point a of r acts imprimitively on the set of hyperplanes (if 8 = 2) or blocks (if s = 4) of the dual of the residue I?, of a. An imprimitivity class for this action is the set of lines on a incident with another given point. Lines forming the same imprimitivity class appear as parallel hyperplanes or non-intersecting blocks in the dual of I',. However, G, acts as M22

on I?,, = S(3,6,22) if s = 4 and M2p acts primitively on the set of blocks of 5(3,6,22). Therefore we have s = 2 and the imprimitivity classes of G, are the parallelism classes of hyperplanes of AG(n - 1,2) (dual of I?,). Hence I? is flat and it is now easy to check that r is in fact the flat quotient of the affine-dual-dne geometry obtained from PG(n ,2 ) dropping the star of a point p and a hyperplane H with p E H.

(Remark. A special case of the above with, n = 4 and s = 2, is also considered in [301.) 0

Problem. Can we prove that every locally finite flag-transitive Af.An-2 .Af* geometry is a standard quotient of an affine-dual-affine geometry, by some modification of the above proof ?

8.4 The Diagram c.c*

Many examples exist for c.P (= c.Al.c+). The (point-block systems of those) examples that satisfy LL are called semtbiplanes [33]. Trivially, assuming LL is the same as assuming that the geometry is the enrichment of its point-block system.

The semibiplanes where the point-block system is a symmetric %design (equivalently, the collinearity graph ia trivial) are called biplanes [39], [33].

A classification program for C.C* geometries looks rather hard to accomplish, even if the flag-transitivity is assumed. However, some results can be obtained if some hypotheses are assumed on the action of the stabilizer of a block on the a + 2 points of that block, at least when the order s is small (see [30]).

We only mention some flag-transitive examples here. Starting from the Coxeter complex of type D,+a, we truncate the elements correspond-

ing to the first s - 1 nodes of the D,+2 diagram

< . ....._ 0-

s - 1 nodes

434 A . Pasini, S . Yoshiara

then we obtain a flag-transitive semibiplane with order 8 ( a + 2 points in a block and 8 + 2 blocks on a point). The (type-preserving) automorphism group of this semibiplane is 2'+l. S,+p.

Another example is the biplane with 11 points and 11 blocks ( a = 3 ) for Lz(l1) ( [8] , (26)). It arises from the exceptional 2-transitive action of Lz(l1) on 11 points. We denote this C.C* geometry by I'(La(l1)).

A semibiplane with a = 10 is known admitting A412 as flag-transitive automorphism group ([el (32) ) . The group L2(11) is also involved here, now as point-stabilizer and as block-stabilizer (in its natural Ztransitive action on the 12 points of a block, or on the 12 blocks on a point).

An infinite family of semibiplanes of order 8 = q - 2 (q odd prime power) can be constructed as follows. In PG(3, q) , choose a non-incident point-plane pair (p, A ) and a non-degenerate conic C on A . Take as points the points not on A , other than p and belonging to lines projecting C from p. As blocks we take the planes of PG(3, q ) not on p, other than A and meeting A in a line tangent to C. The point-block incidence relation is the natural one, inherited from PG(3 ,q ) . A c.c* geometry can be defined by enrichment of this semibiplane. It has order 8 = q - 2 and q . L2(4) as minimal flag-transitive automorphism group. We denote this geometry by r(q. Lz(4)).

The semibiplane F ( q . L 4 q ) ) admits flag-transitive quotients, the minimal one being obtained factorizing over the group Z,-1 of all homologies of PG(3, q ) of center p and axis A . We denote this quotient by I'(q. L 2 ( q ) ) / ( q - 1).

A construction slightly similar to the above can be made in PG(3 ,4 ) . Given a non- incident point-plane pair (p, A ) in PG(3 ,4 ) , let '?i be a hyperoval in A. We take as points the points other than p, not on A and lying on lines projecting 7i from p . As blocks we take the planes of P G ( 3 , 4 ) not on u, other than r and meeting A in lines external to 7i. By enrichment, we obtain a C.C* geometry of order s = 4 , which we denote by r ( A 6 ) . This geometry corresponds to case (i) of Result 2 in [30]. It admits 3.56 (respectively 3.A6) as full (respectively, minimal) type-preserving flag-transitive automorphism group. It also admits a flag-transitive proper quotient r ( A 6 ) / 3 obtained factorizing over the group of homologies of PG(3 ,4 ) with center p and axis A . This quotient is flat (all points are incident with all blocks) and admits S, (respectively, A6) as full type preseving (minimal flag-transitive, respectively) automorphism group.

The above example is evidently related with the exceptional isomorphisms ,544) Z

The diagram c.c' can also be viewed as the rank 3 case of the following diagram of &(5) 2 As.

rank k +- h (with 1 5 h 5 k and 8 2 2):

c a c* .-c , . . . . -* L -r.....-,

I c > ( 4 2 ) ' ) L

k nodes h nodes

A standard example for this diagram can be constructed as follows, generalizing the construction of C.C* geometries from D,+p Coxeter complexes (see the beginning of this section). Let C be the Coxeter complex in the following diagram of rank a t k t h, where s 2 2:


k nodes

I' I I \.-L ..... -L*

8 - 1 nodes c /

h nodes

If we truncate the elements of C corresponding to the first s - 1 nodes of this diagram, then we obtain a geometry belonging to $. (ch)* . We call it COZk,h(8 ) . The geometry Coxk,h(s) is >simply connected (by Theorem 1 of [63]), flag-transitive and it is infinite in all cases but the following ones (we recall that we have assumed h 5 k):

h = k = 1 (the Coxeter diagram is D,+a); s = 2 and h = 1 (the Coxeter diagram is Dk+3);

s = h = 2 and k 5 4 (the Coxeter diagram is D5 or El+(); s = 3, h = 1 and k 5 4 (the Coxeter diagram is D5 or Ek+*); a 5 5, h = 1 and k = 2 (the Coxeter diagram is D5 or E,+3).

When Coxk,h( 8 ) is infinite, it admits infinitely many flag-transitive quotients. Trivially, if h 5 h' and k 5 k', then Cmk,h(8) is a residue in Coq, ,h , ( s ) ; if 8 5 8' then Coxk,h(a) is a subgeometry of cOxh ,h( 8 ' ) .

An example of different kind is known, belonging to c a d with s = 3 and MIl as flag-transitive automorphism group [8] (27). It has 12 points and c.c* residues isomorphic to r(L411)). It arises from the exceptional 3-transitive action of M11 on 12 points, as follows: fix one of those 12 points, make a model of r(La(l1)) with the remaining 11 points, then move this model around, letting MI1 act on those 12 points. We denote this geometry by rl(M11)

A 2.c. geometry with 8 = 7 also exists [52], admitting r ( 9 . L2(9))/8 as point-residue and Mll as flag-transitive automorphism group. It arises from the natural 4-transitive action of Mll on 11 points, similarly as the previous c a d geometry for Mll arose from the exceptional action of Mll on 12 points. We denote this geometry by ra(M11).

The geometry ra(Mll) also admits a >simply connected double >cover [52] with point residues isomorphic to r(9. L49)) /4 and 2 x M11 as flag-transitive automorphism group.

The geometries I'l(M11), I'4M11) and its double cover also appear as block-residues in suitable C?.(C*)~ geometries [52].

The stabilizer of a block in a flag-transitive c a d geometry is evidently 3-transitive on the s + 3 points of the block. If this action is not contained in A'Ut(Lz(8 + 2)), then C O S ~ , ~ ( ~ ) , its flag-transitive quotients and rl(MI1) are the only possibilities that can arise [52] (one more example is mentioned in [52] where S, x 2 acts flag-transitively, but it is only a chamber system, not a geometry). This is a first important step towards the classification of flag-transitive ck.(ch)* geometries with k + h > 2.

Problem. If we only consider Theorem 2.1, diagrams of the form ck.A2.(cA)* with s = 4 and 1 5 k, h 5 3 5 k + h seem to be consistent. However, no examples are known for them. Can we prove that those diagrams are inconsistent, at least if the flag-transitivity is assumed ?

436 A . Pasini, S. Yoshiara

9 THE DIAGRAM c.C:

In this section we consider the following diagram: c

0 L L O 8 t t

with s 2 2 (otherwise we have thin-lined cases of the Coxeter diagram F4, which we are not going to discuss here). The elements are called points, lines, planes and symps (or blocks), from left t o right.

9.1 The Case of s = 2 When s = 2 the diagram c.C; is a special case of the following:

A f ( A f. c; ) r - 0

Examples for this diagrams can be obtained from buildings of type F4 dropping hyperplanes, as for affine polar spaces. A hyperplane of an Fa building M is a proper subspace H of the point-line system of M such that every line of M meets H. For instance, if p is a point of M , the set of points at distance < 3 from p in the collinearity graph of M is a hyperplane, which we denote by p'. Dropping H from M is dropping all points, lines, planes and symps contained in H .

A geometry obtained in this way will be called an ufine F4 building (defined over the division ring K if its planes are d n e planes over K).

Affine F4 buildings defined over GF(2) are the 'plain' examples for c.C;. However, plain examples do not seem now to have the relevant role that they analogues have in the cases of c.Cn-lr c,D,-1 and Af.A,-a.Af*. For instance, affine F4 buildings over GF(2) need not be simply connected [99]. Furthermore, only three d n e F4 buildings are known over GF(2) ([99]; two of them arise from F4(2) and one arises from Ei(2); only one of them is simply connected). On the other hand, a number of examples exist with first order 8 = 2 that do not seem to have anything to do with affine F4 buildings. We list them here, only giving the orders, the minimal flag-transitive automorphism group G, sometimes a few additional information, and references (when available).

We do not include universal covers of non-simply connected affine F4 buildings in this list (the reader is referred to [99] for more information on them). We omit the case of t = 1 (we will discuss it later).

(1) s = t = 2. G = 24 , A,. The residues of the points are isomorphic to the flat A,-geometry. The geometry itself

We have got knowledge of this geometry by Meixner [53]. is flat: all points and all lines are incident with all symps.


It can be constructed as follows. Start from the realization of the symplectic generalized quadrangle Q = Ss(2) in PG(3,2) , viewed as system of linear subspaces of Vect(4,2), and let P be the system of all cosets of linear subspaces forming Q and all vectors of Vect(4,2) (we remark that P is a model for a standard quotient of an affine polar space of rank 3 over GF(2); see [22]). We take as symps the images of P under 2' . A7 (5 2' . As 5 AGL(4,2)). As points, lines and planes we take the points, the lines and the planes of AG(4,2). The incidence relation is the natural one, defined by containment. Residues of symps are isomorphic to P.

This geometry is the unique flag-transitive c.C* geometry with the Neumaier geometry as point-residue [loo].

(2) a = 2, t = 4. G = 2 , CQ. This geometry is simply connected and it admits a (unique) proper quotient where

CQ acts flag-transitively. References: [8] (73), [76], [51], [99].

(3) s = t = 2. G = Fi22.

This geometry appears as a subgeometry in the next example. References: [loll.

(4) s = 2, t = 4. G = Fitar. References: [loll.

A remarkable amount of work has already been done for the case of s = 2 (I t ) ([51], [99], (1011; also [loo]) and a complete classification seems to be in reach for this case, at least if C2 residues are assumed to be classical.

9.2 The Case of s 2 3 and t 2 2

We continue the above list, turning to cases with s > 2 (and t > 1).

(5) a = t = 3. G = Fi'24. This geometry appears as a subgeometry in the next example. References: [ll], [8]

(24)) [7r31.

(6) a = 9, t = 3.

References: [ l l] , [8] ( 8 ) , [78]. G = M (= 4).

9.3 The Case of t = 1 If we start from the following diagram


( c*.c. 0 2 )

and take shadow geometries with respect to the central node of the diagram, then we obtain c.C,' thin on top. Unfoldings of geometries thin on top and belonging to c.C; are defined as for geometries thin on top and belonging to c.Cn-, or C " - ~ . C ~ , replacing (the central node of) c*.c.D2 for (the initial node of) c.D,-1 or c " - ~ . D ~ . It follows from [72] that every 2-simply connected geometry belonging to c.C,', thin on top and satisfying the Intersection Property IP can be unfolded. Thus, any classification of c.C,' geometries thin on top and satisfying the Intersection Property entails a classification for the diagram

Needless to say that the unfolding construction is more general than stated above. For instance, if M is the F4 building obtained as shadow geometry from a DI building, with respect to the central node of the diagram, and if l? is the affine F4 building obtained dropping a hyperplane from M , then I' can be unfolded and its unfolding belongs to the following diagram

c*.c.D~.

At least one c*.c.Dz geometry with 8 = 2 arises by an 'affine' construction as above. On the other hand, infinitely many C*.C.Da geometries can be constructed as follows.

We start from a Coxeter complex in the following diagram of rank 3 8 + 1 (s 2 2): 8 nodes

and truncate the elements corresponding to the 8-1 external types in each of the three horns of the diagram, only saving the central node and the 3 nodes adjacent to it. The geometries obtained in this way are simply connected (by Theorem 1 of [63]). Of course, all these geometries are infinite and admit infinitely many flag-transitive quotients.

The example (78) of [a] (with diagram c.C,', orders 8 = 3, t = 1, flag-transitive automorphism group 5 4 3 ) ) seems to arise as a quotient of a shadow geometry of a truncated Coxeter complex as above [60].

Needless to say that we could also consider Coxeter diagrams with n - 1 > 3 horns. Truncating them as above and taking shadows with respect to the central node we would obtain the rank n analogue c.Cz-, of c.C;.


10 TWO PROBLEMS We finish with some informal comments on two theoretical problems suggested by the results and the examples we have reported about in this paper.

10.1 Spherical and Non-Sperical

The examples obtained truncating Coxeter complexes show that a Buekenhout diagram might look like a spherical Coxeter diagram without being really ’spherical’ at all, if we say that a diagram is of ’spherical type’ when some finiteness condition on diameters of chamber systems is implicit in it, at least if finite orders are assumed. For instance, the following diagrams of rank 3 are ’spherical’: c.C, (Section 3 of [HI), L.pG (where pG means partial geometries; see [31], [37], [23], [38]), C2.L (see [58]). The diagram L.C,-1 is also spherical (see [71]). On the other hand, the case thin on top of c2.C2 with a > 2 is not spherical at all.

Is it possible to recognize whether a diagram is ’spherical’ or not, by some general criterion, only looking at labels on its strokes and at orders at its nodes ?

What we presently know on this matter is not sufficent a t all to hazard an answer to this question. We have a necessary condition for a diagram to be spherical in our sense, due to Ronan [75] (see also [63]) and a sufficient condition due to Tsaranov (911, but the gap between these two conditions is very large, and most of interesting diagrams other than Coxeter diagrams fall in the no-man’s-land between them.

This question also has some practical relevance: if we want to determine all examples in some class of flag-transitive geometries using coset enumeration, it is preferable if we know in advance if the examples we have to cope with will be finite and, if so, how large they will be at most.

10.2 Afflne Constructions

The constructions we have used for d n e polar spaces, affine D, and Fd buildings (also the well known construction of affine geometries from projective geometries) are instances of the following general construction.

Given a building B of spherical type and a type i at which B is thick, let H be a hyperplane of the shadow geometry Shi(B) of B with respect to i , namely a proper subspace of the point-line system of Sh;(B) meeting every line of Sh;(B). Drop H from Sh;(B) and unfold what is left.

For instance, if i is one of the two nodes at the horns of a D, diagram, then this construction gives us the following:

If i is the central node of the diagram D4, then we obtain the diagram Af* .Af .Dz of Section 9.3.

We can do more. If B admits an involutory diagram automorphism 6 such that 6(i) # i, then we can drop both H and 6 ( H ) from Sh;(B) and S h q ) ( B ) respectively. For instance,

440 A . Pasini, S. Yoshiara

affine-dual-affine geometries (Section 8.1) arise in this node at a horn of D,, then we obtain the following:

way. If we do the above with a

If i is one of the two extremal nodes of F,, then the following arises:

If B is of type D1 and i is one of the three extremal nodes of the diagram, then we can also work with a triality instead of duality, thus obtaining the following:

(4) 4: Af

Is it possible to develope a bit of geometric theory on this matter ?

Replacing c for Af. As usual, if we do these constructions on buildings defined over GF(2), then we obtain diagram with c and c* replaced for A f and A f *. For instance, from (I), (2), (3) and (4) we obtain the special cases with a = 2 or a = t = 2 of the following diagrams:

By Theorem 2.1, we necessarily have a = 2 or 4 in (l'), and a = 4 only in the rank 4 case (which is nothing but c.&). Similarly, a = 2 in (?)in the case of rank n 2 5 and a = 2 or 4 in the case of rank 4. The rank 3 case of (2') is nothing but c.c*, hence we have no restrictions on a in this case. Similarly, no restrictions on 8 can be given in case (4'). Geometries for this diagram can be obtained by truncation from Coxeter complexes belonging to Coxeter diagrams with a central node to which 3 projective strokes and a string of a - 1 projective strokes are attached (the string is the part to truncate). All

Flag-transitive Buekenhout geometries 44 1

geometries obtained in this way are infinite and admit infinitely many quotients. An example for (1’) is mentioned in [8] (71), with order B = 2 and rank 5. It is in fact

obtained from a DS building over GF(2) by the above construction. An example belonging to (3’ ) (with orders B = t = 2) is also given in [8] (77). It

admits Ma2 as flag-transitive automorphism group, whence it has nothing do do with the building of type F42) .

An example for (4’) (with a = 2) is desribed in [8] (63). It admits U 4 3 ) as flag- transitive automorphism group. It seems KO] that we can obtain it as a quotient of a truncated Coxeter complex of affine type D4. We do not know if this means that we cannot obtain it also from a D, building over GF(2) .

References

[l] J . Conway, R. Curtis, S. Norton, R. Parker and R. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford 1985.

[2] A. Blokhuis and A. Brouwer, Uniqueness of a Zara graph on 126 points and non- existence of a completely regular two-graph on 288 points, in Papers Dedicated to J. J. Seidel, (P. Doelder, J. de Graaf and J. Van Lint eds.), Eindhoven Univ. Techn. Report 84WSK-03, August 1984, 6-19.

[3] A. Blokhuis and A. Brouwer, Graphs that are locally generalized quadrangle, with complete bipartite p-graphs, unpublished, 1983.

[4] A. Blokhuis and A. Brouwer, Locally 4-by4 grid graphs, J . Graph Th. 13 ( l889) , 229-244.

[5] A. Brouwer, A. Cohen and A. Neumaier, Distance-Regular Graphs, Springer 1989.

[6] F. Buekenhout, D~agrams for geometries and groups, J . Comb. Th. A 27 (1979)) 121-151.

[4 F. Buekenhout, The basrc daagram of a geometry, in Geometries and Groups, L. N. 893, Springer (1981), 1-29.

of Age ( J . Mc Kay ed.), Contemp. Math. 46, AMS (1985), 1-32. [8] F. Buekenhout, Diagram geometries for sporadic groups, in Finite Groups Coming

[9] F. Buekenhout, Eztensrons of polar spaces and the doubly transitive symplectic groups, Geom. Dedicata 0 (1977), 13-21.

[lo] F. Buekenhout and X. Hubaut, Locally polar spaces and related rank 3 groups, J. Algebra 45 (1977), 391-434.

[ll] F. Buekenhout and P. Fischer, A locally dual polar space for the Monster, unpublished.

[12] F. Buekenhout, A. Delandtsheer and J. Doyen, Finite linear spaces with Jag- tmnsitiue groups, J . Comb. Th. A 49 (1988), 268-293.


(131 F. Buekenhout, A. Delandtsheer, J. Doyen, P. Kleidman, M. Liebeck and J. Saxl, Linear spaces with flag-transitive automorphism groups, Geom. Dedicata 36 (1990), 89-94.

[14] P. Cameron, Finite permutation groups andfinite simple groups, Bull. London Math. SOC. 1 3 (1981), 1-22.

[15] P. Cameron, Covers of graphs and EGQs, to appear in Discr. Math.

[16] P. Cameron and M.Deaa, On permutation geometries, J. London Math. SOC. 20 (1979), 373386.

[17] P. Cameron and P. Fisher, Small extended generalized quadrangles, European J. Comb. 11 (1990), 403-413.

[18] P. Cameron, D. Hughes and A. Pasini, Extended generalized quadrangles, Georn. Dedicata 36 (1990), 193-228.

[19] A. Cohen and E. Shult, Afine polar spaces, Geom. Dedicata 36 (1990), 43-76.

[20] H. Cuypers, Locally generalized quadrangles with afine planes, to appear.

[21] H. Cuypers, Geometries and permutation groups of low Tank, Part 11 (Generalized Fischer spaces), Ph . D. Thesis, Rijksuniveristeit Utrecht, 20 March 1989.

[22] H. Cuypers and A. Pasini, Locally polar geometries with afine planes, t o appear.

[23] A. Del F’ra and D. Ghinelli, Diameters bounds for locally partial geometries, Euro- pean J. Comb. 12 (1991), 293-307.

[24] A. Del Fra, D. Ghinelli, T. Meixner and A. Pasini, Flag-transitive extensions of C,-geometries, Geom. Dedicata 37 (1991), 255273.

[25] J. Doyen and X. Hubaut, Finite regular locally projective spaces, Math. Z. 110 (1971), 83-88.

[26] B. Fischer, Finrte groups genernted by 3-transpositions, Inventiones Math. 13 (1971), 232-246; and University of Warwick Lecture Notes (unpublished).

[27] P. Fisher, Edended 4 x 4 grtds, European J. Comb. 12 (1991), 383-388.

[28] D. Ghinelli, Flag-transitive L.C2 geometries, in preparation.

[29] J. Goethals and J. Seidel, The regular two-graph on 276 vertices, Discr. Math. 12 (1975), 143158.

[30] G. Grams and T . Meixner, Some results about flag transitive diagram geometries using coset enumeration, to appear.

[31] S. Hobart and D. Hughes, Extended partial geometries: nets and dual nets, European J. Comb. 11 (1990), 357-372.


[32] D. Hughes, Extensions of designs and groups: projective, symplectic and certain afine groups, Math. Z. 89 (1965), 199205.

[33] D. Hughes, On designs, in Geometries and Groups, L. N. 893, Springer (1981), 4367.

[34] D. Hughes, Semi-symmetnc 3-desgns, in Finite Geometries (N. Johnson, M. Kalla- her and C. Long eds.), L.N. in Pure and Applied Math. 82, Dekker N.Y. (1982), 223-235.

[35] D. Hughes, On the non-existence of 0 semr-symmetric 3-design with 78 points, Ann. Discr. Math. 18 (1983), 473-4dO.

[36] D. Hughes, On some rank 3 partial geometries, in Advances in Finite Geometries and Designs (J. Hirschfeld, D. Hughes and J. Thas eds.), Oxford U. P. (1991), 195- 225.

[37j D. Hughes, &tended partial geometries: dual 2-designs, European J. Comb. 11 (1990), 459471.

[38] D. Hughes, Partial geometries of rank n, these Proceedings.

[39] D. Hughes and F. Piper, Design Theory, Cambridge Univ. Press, 1985.

[40] W. Kantor, DimenJion and embedding theorems for geometric lattices, J. Comb. Th. A 17 (1974), 175195.

[41] W. Kantor, Genemiited polygons, GABS and SCABS, in Buildings and the Geometry of Diagrams, L. N. 1181, Springer (1986), 79-158.

[42] W. Kantor, Primittve permutation groups of odd degree and an application to finite projective planes, J. Algebra 106 (1987), 15-45.

[43] C. Lam, L. Thiel and S. Swiercz, The non-esistence offinate projective planes of order 10, to appear in Canadian J. Math.

[44] C. Lefevre-Percsy, Infinite ( A f , AP)-geometries, J. Comb. Th. A 55 (1990), 133- 139.

[45] C. Lefevre-Percsy and L. Van Nypelseer, Finite rank 3 geometries with a#ine planes and dual afine point-residues, to appear in Discrete Math.

[46] G. Lunardon and A. Pasini, Finite C,-geometries: 0 survey, to appear in Note di Matematica.

[47] J. Mc Laughlin, A simple group of order 898.128.000, in Theory of Finite Groups (Braner and Sah eds.), Benjamin N. Y. (1969), 109-111.

[48] T. Meixner, Some polar towers, European J. Comb. 12 (1991), 397-415.

[49] T. Meixner, A computer-free proof of a theorem of Weiss and Yoshiara, to appear.

444 A. Pasini. S. Yoshiara

(501 T. Meixner, Two geometries related to the sporadic groups C s and Cs, t o appear.

[51] T. Meixner, A geometric characterization of the simple group C s , J. Algebra (to appear).

[52] T. Meixner, Two diagram geometries related to Mil, preprint, Summer 1991.

[53] T. Meixner, private communication.

[54] T. Meixner and K. Metsch, An infinite family of non-regular near polygons, to appear.

[55] T. Meixner and A. Pasini, A census of known flag-tmnsitiue extended grids, t o appear.

[56] T. Meixner and A. Pasini, Some classification theorems for flag-transitive extended grids, in preparation.

[57l A. Neumaier, Some sporadic geometries related to PG(3,2), Arch. Math. 42 (1984), 89-96.

[58] D. Pasechnik , Dual linear extensions of generalized quadrangles, to appear in Eu- ropean J. Comb.

[59] D. Pasechnik, private communication.

[60] D. Pasechnik, private communication.

[61] A. Pasini, Canonical linearizations of pure geometries, J. Comb. Th. A 35 (1983), 10-32.

[62] A. Pasini, Some remarks on covers and apartments, in Finite Geometries (C. Baker and L. Batten eds.), Dekker (19135), 223-250.

[63] A. Pasini, Covers of finite geometries with non-spherical minimal circuit diagram, in Buildings and the Geometry of Diagrams, L. N. 1181, Springer (1986), 218-241.

[64] A. Pasini, Geometric and algebraic methods tn t h e classification of geometries belonging to Lie diagrams, Ann. Discr. Math. 37 (19813)~ 315-356.

[65] A. Pasini, A survey of finite flag-transitrue locally classical c.Ca-geometries (groups for geometries in given diagrams I), to appear in the Proceedings of the Conference in honour of A. I. Maltsev, Novosibirsk August 1989.

[66] A. Pasini, Groups for geometries cn given diagrams 11: on a characterization of the group Aut (HS) , European J. Comb. 12 (1991), 147-158.

[67l A. Pasini, A classification of a class of Buekenhout geometries exploiting amalgams and simple connectedness (groups for geometries in given diagrams 111), t o appear in Atti Sem. Mat. Fis. Univ. Modena.


[68] A. Pasini, Remarks on double ovoids i n finite classrcal generalized quadrangles wrth an applrcataon to extended generalrzed quadrangles, to appear in Simon Stevin.

[69] A. Pasini, Quotrents of afine polar spaces, Bull. SOC. Math. Belgique 42 (1990), 643-658.

[70] A. Pasini, On extended grids with large automorphism groups, Ars Comb. 29 B (1990), 65-83.

[71] A. Pasini, A bound for the collinearity graph of certain locally polar geometries, J. Comb. Th. A 58 (1991), 127-130.

[72] A. Pasini, Shadow geometries and simple connectedness, to appear.

[73] S. Payne and J. Thas, Finite Generalized Quadrangles, Pitman, Boston 1984.

I741 S. Rinauro, On some extensions of generalized quadrangles of grid type, J. Geometry 38 (1990), 158-164.

[75] M. Ronan, On the second homotopy group of certain simplicia1 complexes and some combinatorial appltcations, in Quart. J. Math. Oxford 32 (1981), 225-233.

I761 M. Ronan, Coverings of Certain Finite Geometries, in Finite Geometries and De- signs (P. Cameron, J. Hirschfeld and D. Hughes eds.), Cambridge U.P. (1981), 316-331.

[77] M. Ronan, Embeddings of hyperplanes of discrete geometries, European J. Comb. 8 (1987), 179-185.

[78] M. Ronan and G. Stroth, Minimal pambolic geometries for the sporadic groups, European J. Comb. 5 (1984), 59-91.

(791 R. Scharlau, Geometrical realizations of shadow geometries, Proc. London Math. SOC. 61 (1990), 615-656.

[80] G. Seitz, Flag-transitive subgroups of Chevalley groups, Ann. Math. 97 (1973), 27- 56.

[81] A. Sprague, Rank 3 incidence structures admitting dual linear, linear diagrams, J. Comb. Th. A 38 (1985), 254-259.

[82] B. Stellmacher, Einfache gruppen, die von einer konjuiertenklassen V O ~ elementen der ordnung 3 erzeugt werden, J. Algebra 30 (1974), 320-354.

[83] M. Suzuki, 7kansitrue extensions of a class of doubly transitive groups, NagoyaMath. J. 27 (1966), 159-169.

[84] J. Thas, Extensions of finite genelalazed qwdrangles, Symposia Math. 28 (1985), 127-143.


[85] F. Timmesfeld, Tits geometries and parabolic systems en finitely generated groups, I, II, Math. Z. 184 (1983), 377-396, 449-487.

dimension, Bull. SOC. Math. Belgique 23 (1971), 481-492. [86] J. Tits, Non-existence de certaines extensions tnansitives I. Groups projectifs a une

[87] J. Tits, Buildings of Spherical Type and Finite BN-pairs, L. N. 386, Springer 1974.

[88] J. Tits, A local approach to buildings, in The Geometric Vein (C. Davis et alt. eds.), Springer (198l), 519-547.

[89] J. Tits, Buildings and groups amalgamatcon, London Math. SOC. L. N. 121 (1986), 110-127.

[go] J. Tits, Ensembles ordonnes, immeubles et sommes amalgamees, Bull. SOC. Math. Belgique 38 (1986), 367-387.

Comb. 11 (1990), 189-204. [91] S. Tsaranov, Representation and classtfication of Cozeter monoids, European J .

[92] J. Van Bon and R. Weisls, A charactenzation of the groups Fi22, Fi23 and Fiar, to appear.

[93] L. Van Nypelseer, Rank n geometries with afine hyperplanes and dual afine point residues, to appear in European J. Comb.

[94] R. Weise, A geometric chanactenzation of the groups McL and Cq, to appear.

[95] R. Weiss and S. Yoshiara, A geometric characterizatron of the groups Suz and HS, J. Algebra 133 (1990), 182-196.

Comb. 11 (1990), 81-93.

generators and relations, European J. Comb. 12 (1991), 159-181.

[96] S. Yoshiara, A locally polar geometry associated with the group HS, European J .

[9fl S. Yoshiara, A classification of flag-transitive classtcal c.C2-geometries by means of

[98] S. Yoshiara, Embeddings of flag-transitive classical ~ocally polar geometries of rank 3, to appear in Geom. Dedicata.

[99] S. Yoshiara, On some eztended dual polar spaces, I, to appear.

[loo] S. Yoshiara, On some non-classtcal extended dual polar spaces, to appear.

[loll S. Yoshiara, On some extended dwl polar spaces, 11, in preparation.

Flag-transitive Buekenhout geometries

Addresses of the authors

Antonio Paaini Department of Mathematics, University of Siena. Via del Capitano 15. Siena 53100 (Italy). and Department of Mathematics, Faculty of Engineering, University of Naples. Via Claudio 21. Naples 80125 (Italy).

Satmhi Yoshiara Department of Information Science, Faculty of Science, Hirosaki University. 3 Bunkyo-Cho, Hirosaki, Aomori 36 (Japan).

441



Collineations of t h e generalized quadrangles associated wi th q-clans

S. E. Payne

Department of Mathematics, Campus Box 170, University of Colorado, P.O. Box 173364, Denver, Colorado 80217-3364

Abstract

with a flock of a quadratic cone), a description of its collineation group is given. For each known finite generalized quadrangle associated with a q-clan (and hcnce

I. INTRODUCTION

Beginning in 1980 with the discovery by W. M. Kantor [6] of a nonclassical generalized quadrangle (GQ) with parameters ( q 2 , q ) for any prime power q with q f 2 (mod 3), there has grown up a considerable literature studying GQ and related structures associated with a set of matrices known as a q-clan. By a result of J. A . Thas [20] these are equivalent to flocks of a quadratic cone, to certain spreads of PG(3,q), etc. In spite of progress made by H. Gevaert and N. L. Johnson [3]; L. Bader, G. Lunardon, and J. A. Thas [2]; S. E. Payne and L. A. Rogers [16]; W. M. Kantor [7]; to mention only four of several recent articles, the connection between GQ and flocks has remained primarily algebraic. But now for q odd a very beautiful geometric construction has been found by N . Knacr [9] (see also [19]).

In the papers cited above, certain information about the collineation groups of the various GQ appears, sometimes as specific information about the projective collineations in PFO(3, q ) leaving a conical flock invariant, sometimes as a complete determination of the collineation group of the GQ. We feel that it is worthwhile to collect in one refercnce a rather complete (from at least one point of view) description of the collineation group of each GQ. We do so here.

The determination of the automorphism groups of the various GQ was cornpletcd largely before the construction by N. Knarr and Section IV of [19] were available. Of course, for q = 2“ that construction is still not available. And W. M. Kantor’s comments in [7] concerning the full automorphism groups of the GQ were only mildly helpfiil in giving a “complete” description of them. So in fact we need the standard construct,ion, starting with q-clans.

450 S.E. Payne

11. q-CLANS AND GQ

For an arbitrary prime power q = p e , p prime, let F = GF(q) and let Ic denote the group consisting of the set

K = { ( a , c , P ) E F2 x F x F 2 : a,P E F 2 , c E F } , (1)

together with the binary operation

( a , c , P ) . ( a ' , c ' , P ' ) = ( a + a ' , c + c' + P. a',P + P'). ( 2 )

(Here a * ,B represents the usual dot product of vectors in F 2 . ) A 2 x 2 matrix A over F is called anisotropic provided aAaT = 0 has only the

trivial solution a = (0,O). For each t E F , let At be a 2 x 2 matrix over F . Then the set

C = {At : t E F } (3)

is called a q-clan provided that whenever s , t E F , s # t , the matrix A, - At is anisotropic.

For the purpose of belonging to a q-clan, a matrix A = ( z :) is completely

determincd by three quantities: z, 2, and y + w. So if B = (z', !:), we write A = B

provided z = d, z = 2 , and w + y = w' + y'. For arbitrary q wc usually assume A is upper triangular, and if q is known to be odd we may require A to be symmetric.

Given a q-clan C, write ICt = At + AT, t E F . We coulcl assume without loss of

generality that A0 = I<" = (: :) (see S. E. Payne arid J. A. Thas [18], p. 213). Wc sometimes do this, but the examples arising from the flocks discovered by J. C. Fisher (Example 5 below) show that this is not always convenient.

Now define subgroups of Ic having order q2.

A(m) = {(a,O, P ) E Ic : P E F ' } , A ( t ) = {(a,aAtaT,aI<t) E Ic : a E F 2 } , t E F. (4)

And also groups of order q3:

A*(m) = { ( o , c , P ) E K : c E F, P E F 2 } , A * ( t ) = {(a,c,aICt) E K: : a E FZ, c E F } , t E F. (5)

So A ( t ) 5 A*( t ) for t E F U {m}. Put J = { A ( t ) : t E F U {m}}, and J'* = {A* : A E J } . Then J is a 4-gonal family for K, i.e., J and J * satisfy the two properties of W. M. Kantor [GI with s = q2, t = q, so that a GQ of order ( q 2 , q ) ma.y be constructed (see especially [8], [12], [13]). This GQ S(C) = (P , B , I ) , with pointset P , lineset B , and incidence I is given as follows.

Collineatiom of the generalized quadrangles 45 1

Points: i) (m)

ii) right cosets A*( t )g , g E K, t E F U {a}. iii) elements g of Ic. Lines:

a) [A(t)l, t E F u {O}. b) right cosets A(t )g , g E K, t E F U {m}.

Then (a) is incident, with each line of type a) and none of type b). The point A*(t )g is incident with [A(t)] and with each line A(t )h contained in A*(t)y. The point g of type iii) is incident with each line A(t)g of type b) containing it.

111. AUTOMORPHISMS OF S(C)

It is our goal to give a more or less explicit description of the automorphisni (= collineationj group of S(C) for each nonlinear q-clan C lcnown at the present time. In this section we consider some general results. A very helpful one is the following.

111.1. (S. E. Payne and J . A. Thas [-I 71). If S(C) is not classical (i.e., if C is not linear (see [20])), the point (m) is fixed by the full collineation group of S(C).

If one automorphism of S(C) can be found which moves the line [A(m)], then usually the complete group can be found with the aid of the following.

111.2. (S. E. Payne and L. A . Rogers [IG]; S. E. Payne and J . A . Thas (191). Let

C = {A t : t E F } be a q-clan

(1 91 this restriction was removed, with a corresponding increase in the complexity of the expression for 6 as in (1) below.) Let 6 be a collineation of S ( C ) which fixes (m), [A(m)], and (8,0,8). Then the following must exist:

(; :). In (In [IS] it was assumed that A0 = I<o =

i) a permutation t H t of the elements of F . ii) X E F , X # 0.

iii) 0 E Aut(F) . (iv) D E GL(2, q) for which Ax - XDT(At - &)"D - A? is skew-symmetric (with zero

diagonal) for all t E F .

Conversely, given cr, D, X, and the permutation 5 H Z satisfying condition iv), the following automorphism of K induces a collineation of S(Cj:

When q is odd and each At E C is symmetric, condition iv) of 111.2 may be repla.ced by iv)' A, = XDT(At - A O ) ~ D + A , for all t E F .

452 S.E. Payne

Suppose 8, = B(al, D,, X a ) , i = 1,2, and f3 = el . 6,2 (O1 followed by 82) . Then

Represent points of PG(3, q ) by homogeneous coordinates written as horizontal vectors = ( x O , X ~ , X ~ , X ~ ) and planes with vertical vectors g = [ y 0 , ~ 1 , ~ 2 , ~ 3 ] ~ . Then x and jj are incident if and only if Cz,y, = 0. The cone K : x: = zozz with vertex V = (0, O,O, 1) has a flock F ( C ) associated with the q-clan C as follows.

The matrix At = (2 z:) (in upper triangular form!) of C corresponds to a plane

xt = [ x t , y t , t t , 1IT which meets I< in a coiiic section Ct = rt n I<. The conics Ct, t E F , partition K - { V } precisely because C is a q-clan (see [20]). So F'(C) = {Ct : t E F } is the flock associated with C, but it is the set F(C) = { ~ t : t E F } with which we will need to work.

For 8 = #(g, DIX), write D = (E i ) . Define a projcctive sendinear

a = O(a,D,X) = B ( ( T ~ . u ~ , D ~ ~ D z , X ~ ~ X ~ ) . (2)

-

collineation To of PG(3, q ) as follows (defined on planes of PG(3,q)): XU2 Xab Xb2 xo x u

(3) ~ X U C X ( d + bc) 2Xbd .) [ :]

Ad2 ZT

0 1 T o : [ i] ( Xc2 Xcd

0 0 Then To fixes the cone I< and leaves invariant F(C) precisely when 6 is a collineation

of S(C). And the map T : 6 H To is a homomorphism from the subgroup of Aut(S(C)) fixing [A(m)], (M), and (g,O,o) onto the subgroup of PrO(4, q ) leaving F(C) invariant. The kernel of T is

N = {e, : (a,c,p) (aa,a2c,ap) : a E F } . (4) Starting with one flock of A' when q is odd, L. Bader, G. Lunardon, and J. A .

Thas [2] have produced a set of q additional flocks. The set S of q + 1 flocks will be called a BLT-set (after W. M. Kantor [7 ] , although he used the term in a difierent (but equivalent) way). In [16] it is shown that each of the q "new" flocks is obtained by recoordinatizing S(C) so as to interchange the line [A(m)] and some other line through (m). It follows that two flocks of a BLT-set are projectively equivalent if and only if the corresponding pair of lines of S(C) are in the same orbit of Aut(S(C)). This was explicitly observed in [7] and in [19].

For the remainder of this section we assume A0 = Iio = (: :) , so lit is invertible

Consider the map B : K: H K: defined by

It is easy to check that 8 is an automorphism of K: interchalging A(m) and A(0) and mapping ( a , a A t ~ x ~ , a l r ' ~ ) to (aKt,-ctAtaT, -a). So if p = aKt and t # 0, this image is (p , /3(-I\Tr1AtK;1),f?Tl P ( - K ; ] ) ) . This shows that

if t # 0.

8 : ( a , c , p ) H (p , c - a . p, -a). ( 5 )

0 replaces At with - K;'AtK;', 0 # t E F. (6) And we may think of 6, as replacing A0 with Ao. Fkom here on it is convenient to

consider separately the cases q even and q odd.

Collineations of the generalized quadrangles 453

111.3. If q = 2" and if there is an integer i for which C =

{ A t = ( t t2 , - l t i ) : t E F } , then there is a collineation B of S(C) for which 8 :

( t i 1 (t")i 1 2 i - 1 ) .

[ A ( m ) ] ts [A(O)], and 6 : [A( t ) ] H [A(t- ' ) ] for 0 # t E F . (In fact, 6 is induced by the automorphism of X: given in (51.)

PROOF: For At as given, it is easy to compute that -I{clAtI<y' 3

The analogue for q odd is slightly more involved.

111.4. Let q be odd, and suppose there is a nonzero a E F and an integer i for which

C = { A t ( a t , 2at2, ' - l ) : t E F } . Then there is a collineation 4 of S(C) which

interchanges [ A ( m ) ] and [A(O)], and also interchanges [ A ( t ) ] and [A(t-')I for 0 # t E F .

t ut'

PROOF: Since At is given in symmetric form, Kt = 2At, I{;' = 2-'AT1, and / 1 - 1 - t - l \ - -

-Kt-lAtKt-l = - t - l ( t - l ) ~ i - ~ ). (SO apparently a # 2!) The map 4 ( a - 2 ) 4 (a-Z)a

(oi,c,p) H (a,dc,dp) replaces At with dAt (cf. [16], 111.3). So put d = 2(a - 2 ) to

Finally, we include one last rather trivial observation. For each g E K, let B(g) denote the permutation obtained by "right multiplication by g." Clearly B(g) induces a collineation of S(C), and { d ( g ) : g E K} acts regularly on P - (m)' and fixes each line through (co).

In the remaining sections of this report we give some sort of description of the collineation group of the known nonclassical S(C). This always includes the order IAut(S(C))l and usually includes a specific description of enough automorphisins to easily (in theory!) describe every possible automorphism of S(C). The ordering of the examples is taken from [lo].

IV. EXAMPLES 1-4

Example 1 (Classical). The classical S(C) arise from the q-clan given by C =

{ A t = (i s:) : t E F } , where x 2 + bx + c is a fixed irreducible polynomial over

F . This S(C) is isomorphic to H ( 3 , q 2 ) arising from a nonsingular hermitian surface in PG(3, q2), and is the point-line dual of the Q(5, q ) arising from the nonsingular elliptic quadric in PG(5,q) . Their collineation groups are well known classical groups, so we choose to pass over this case. (But see [18] for a proof that this q-clan really gives the classical examples.)

454 S.E. Payne

Example 2 (W. M. Kantor [6]). Let q 3 2 (mod 3) and put At =

t E F . By 111.3 and 111.4 there is a collineation interchanging [ A ( m ) ] and [A(O)]. Define e, : K: -, K: by

For each y E F , 0, yields an automorphism of S(C) which fixes [ A ( m ) ] and maps [ A ( t ) ] to [A(t + y ) ] , t E F . Also Or . Bz = y, z E F . When checking these details, keep in mind the paragraph following I1 (3).

So to complete a determination of Aut(S(C)) we must determine all those collineations that leave invariant [ A ( m ) ] , [A(O)], and (n,O,o). From Section I11 we know that all such collineations have the following appearance when interpreted as permutations of the planes that yield the corresponding flock:

Here a, b,c ,d ,X E F with X # 0 # a d - bc, 0 E Aut(F). Using y~ = 3 ( z ~ ) ' = 3(1)2, it is easy to see that b = 0 and a c = 0. Then ad - bc # 0 forces a # 0 = c , and

d = Xu3. So for 0 # aX, 0 E Aut(F), put D = (; to obtain the corresponding

collineation 0 : ( a , c , p ) H (X-'a0D-', X-lc" 7 DUD). (3)

In particular, putting X = a-l gives 6 : ( cy , c ,p ) - ( a (; ."I) I ac,

p (; the following:

which fixes [ A ( m ) ] and [A(O)] and maps [ A ( t ) ] to [A(a t ) ] . This proves

IAut(S(C))l = q6(q + l ) ( q - 1)'e, and Aut(S(C)) is triply transitive on the lines through (m). (4)

Writing e(a, c , /I) for right multiplication by g = ( a , c, p) , we see that

is a group of collineations acting sharply transitively on the lines not meeting [ A ( m ) ] . Unfortunately 7 does not fix the points of [ A ( m ) ] .

Collineationr of the generalized quadrangles 455

Example 3 (Kantor [S]). Let q be odd, 0 E Aut(F), m = E F . Put At =

, I

For each r E F , define 4,. : K: -+ K: by

Then cf, = (4,. : r E F } gives a maximal group of symmetries about the point A*(m) (fixing all points collinear with A*(m)) .

For each r E F, 0 # r , define 6, : K: ---t K: by

Then 6, yields a collineation of S(C) that permutes the lines through (m) as follows:

It follows that (6, : 0 # T E F} U {id} is a maximal group of symmetries about the point A*(O). Thus each point of (a)' is a center of symmetry (and hence regular (see [18] for definitions)), so that S(C) is a translation GQ with respect to each line through

So now consider a collineation 6 of S(C) which fixes [A(m)], [A@)] , (o,O,o), and of

course (m). We want to determine D = ( i i ) with ad - bc # 0, X E F with X # 0,

and 6 E Aut(F), for which

(m).

AT = XDTAfD for all t E F and some permutation t H ?. ( 8 )

a2t6 - b2nx6tbb act6 - bdm6tb6 0 A,= A (act6 - bdm6tb6 C2t6 - d21n6tob = (i

> '

We assume u # id to avoid the classical case, so act' - bdrnbtbb = 0 for all tb E F. Hence

ac = 0 = bd; ad - bc # 0. (9)

From ? = X(a2tb - b2m6tmb) and -mSb = X(c2t6 - d2mbtbb) we find that

XC2t6 + (mXUa2" - Ad2 77-L ' )t - ma6+'Xab20t026 = 0 for all tb E F . (10)

Case 1. u2 # id. Then b = c = 0 = mX'a2" - Xd2mb. So fer 0 # a, X E F , 6 E Aut(F), put d = k A q a " / m ? .

456 S.E. Payne

Case 2. u2 = id # u. Here Xc2 - m"6+1Xab2a = 0 = mX"aZa - Xd2m6, so * e

d = f X F a " / m ? a n d c = & m z z b". (11)

Here b = c = 0 is allowed, but if bc # 0, then by (9) a = d = 0.

In either Case 1 or Case 2, set b = c = 0, 6 = id, a = 1. Then for arbitrary nonzero X E F put d = A+. It follows that j = At, so Aut(S(C)) is triply transitive on the lines through (m). And the order of Aut(S(C)) is as follows:

lAut(S(C))l = q6(q + l ) ( q - 1)'2e, if u2 # id. lAut(S(C))l = q6(q f l ) ( q - 1)'4e, if u2 = id # 0. (12)

It is now easy to determine all whorls about (o,O,o), i.e., those collineations fixing all lines through (a,O,a). So t H ? should be the identity permutation. Then ? = Xa2t6 - Xb2m6tU6 = t for all t E F (with u # id) yields two cases:

Case 1. b = c = 0, 6 = id, X = a-'. Here we have

Remark: If a = X = 1 and the sign "-" is chosen, the collineation ( a , c , p ) ++

( a (i -4) , c, /3 ( ")) appears to be a whorl about (a, 0,a) of a type claimed

in [ll] not to exist. (The error there is in line 12, p. 350; viz., each point 3 of S; must lie on some line of S; n S,+1 .)

Case 2. If a = d = 0, bc # 0, then 6 = u-' = u, -Xb2m6 = 1. Working out the form of the corresponding collineation, we find

It follows that the group of whorls about (8,0,0) is twice as large as claimed in [ll] for the case 0' # id, and is four times as large as might have been predicted for the case u2 = id # u.

Example 4 (W. M. Kaiitor [8] for q odd; S. E. Payne [13] for q even). For

q = f 2 (mod 5), At = (; $ ) , t € F .

By 111.3 and 111.4 we know there is a collineation interchanging [A(m)] and [A(O)]. So let 6 be a collineation fixing (Ti, O , o ) , [A(m)] and of course (m). Then 6 = 6(u, D, A )

for some D = (; fi). And

? = Xa't" + 5Xab3t" + 5Xb2t5" + 0 5j3 = 2Xact" + 5X(ad + bc)t3" + 10Xbdt'" + 503

5t5 = Xc2t" + 5Xcdt3" f 5Xd2t5" + SO5.


Writing out the fact that the second row is five times the cube of the first, we obtain

5Z3 = 5U3 + 2Xact" + 5X(ad + b ~ ) t ~ ~ + 10Xbdt5"

= 5(U + Xa2t" + 5Aabt3" + 5Xb2t5")3 (16)

b = 0. (17)

for all t' E F . Computing the coefficient of ( t " ) I 5 in (16), we have 0 = 54A3b6, implying

This simplifies (16) to

5t3 = SO3 + 2Xact" + 5Aadt3" = 5[a3 + 3U2Xa2t" + 3UA2a4t2" + A3a6t3"]. (18) # 0, then a = 0. But Comparing coefficients on (t")', we see 0 = 150A2a4. So if

a = b = 0 is impossible, since ad - bc # 0. So - 0 = 0. (19)

c = 0. (20)

d = A2a5. (21)

Now compare coefficients on t' : 2Xac = 0. So

This leaves 5Xad = 5X3a6, or

And now it is easily checked that we have exactly the following collineations: For 0 # A,

a E F , CT E Aut(F), D = (; X P , 5 ) 7

Here B : [A(t)] H [A(Z)] where ? = Aa2tu, t E F . So Aut(S(C)) has two orbits on the lines through (m), one of which is { [ A ( o o ) ] , [ A ( O ) ] } . And, finally,

lAut(S(C))l = q5(q - 1)22e. (23) Remark: For q = 2 e , e odd, Example 4 gives two inequivalent flocks, a result not

obtainable from the BLT-theory since q must be odd for that. For example, interchanging [A(m)] and [ A ( 1 ) ] yields the q-clan

) ( t 3 - c = { ( 0 t 5 - 1 : 1 # t E F t - 1 t 3 - 1

For an alternative approach to this remark see N. L. Johnson [4]. Apology: Many of the individual collineations appearing in Examples 2 , 3, and

4 have appeared previously either explicitly or implicitly in any of several publications not cited here. The authors with the strongest grounds for feeling offended at our inattention to their contributions are probably L. Bader, N . L. Johnson, W. M. Kantor, and G. Lunardon (in alphabetical order). However, in no case are we aware of any claim by these authors to having completely determined the full automorphism group of some specific S(C). So we have decided not to attempt to cite all the relevant articles. Instead, we suggest that the interested reader consult the excellent survey article by N. L. Johnson [5] and its list, of references.

458 S.E. Payne

V. EXAMPLE 5 (THAS-FISHER GQ [20])

The q-clans for these GQ are somewhat troublesome to describe, but we need an explicit description in order to give their automorphisrns. The following treatment comes from [14] and [15].

Let F = GF(q) where q = p e , p an odd prime. Let E be a quadratic exteiisioii of F with primitive root C. Put i = ( ( q + ' ) / ' , so 2 q = -i and ( 1 , i ) is an F-linear basis of E. Then w = <*+I is a primitive root (and hence a nonsquare) of F with i 2 = 7u. Put z = C q - l , so z generates the unique multiplicative subgroup of E having order q + 1. Write z = Ti + $2 for some fixed a, 5 E F . Then

For each integer k (modulo q + 1) define the following two elements of F .

Recall a: - w-'bi = 2/(1 + T i ) =@E F. (Eq. 40, [14]) (4)

Finally we can describe the q-clan associated with Fisher's flock

c = { [ t 0 -wt O ] : t 2 -22 / (1+S i )=O } u { [ -;;J : o 5 2 j 5 q - 1 1 . ( 5 )

Unfortunately when q 1 (mod 4) this description of C does not include the zero matrix, and it is not convenient to adjust all the members of C to arrange for the zero matrix to be included. So choose a to for which t i - 2/( 1 + n) = 0. (If q E 3 (mod 4)

of course we could choose to = 0.) Then use A. = At , = (2 O ) in (1) of

Section 111. Since any flock derived from the Fisher flock (by the method of [2]) is projrctively

equivalent to the original ([l], [17]), the collineation group of S(C) must be transitive 011

the lines through (00). So in the present context we content ourselves with an explicit description of those collineations which fix [ A ( ~ o ) ] . These were determined in [19].

For arbitrary 0 E Aut(F), for b ,d E F , ( b , d ) # (O ,O) , and for X =

f(l + E ) y ( d 2 - b2w)-'w('-"), put D = ( i fi) where a = d w q and c = b i i i 7 .

Then the corresponding colliiieation of S(C) (use (1) of Section 111) is

-wto

'7 +I


It follows that IAut(S(C))l = g 5 ( g + I ) ( $ - 1)2e. ( 7 )

This description of Aut(S(C)) is unsatisfactory in two respects: it fails to give an explicit description of any collineation moving [A(co)], to an arbitrary line through (co), and it does not reveal the interesting nature of some of the individual collineations. Here is an example:

0 : (a , c , P) H ("AT, c, PA-'), where A = (8)

This 0 fixes [A(co)] and all lines [A(t)] with t2 - & =

Other examples are given in [19].

(corresponding to planes [t, 0, -wt, I]'), and moves the others in a cycle of length ( g + 1)/2 (cf. [14]) .

VI. EXAMPLES 6 AND 7

Example 6 (H. Gevaert aiid N. L. Johnson [3]; from "likeable" planes of W. M. Kantor) . Here g = 5e > 5 (to avoid the classical case), and k =PG F . Then

, t E F . In [16] it is shown that S(C) is independent, of

the choice of nonsquare k, and Aut(S(C)) is completely determined. The line [A(oo)] is fixed by each automorphism of S(C).

At = ( 3:" k-lt(1 3t2 + kt2)2 )

For an arbitrary but fixed x E F , put B, = (;

yields a collineation of S(C) that maps [A(t)] to [A(t +.)I, t E F. And the collinea.tions that fix [A(oo)], [A(O)], (co), and (0,0,6) are precisely those of the form

where d = h k q , r~ E Aut(F), 0 # a E F . It follows that

lAut(S(C))l = g 6 ( g - 1)2e. (3)

As in the case with Example 2, there is a group 7 of collineations generated by the 0, in (1) and by right multiplication by members of A*(m). Here also 7 acts regularly on the lines of B - [A(co)]'. In both Examples 2 and 6, 7 has other conjugates in Aut(S(C)) which act wgularly on B -[A(..)]'. Even though 7 does not fix each point of [A(oo)], we conjecture that there is some general way to describe the action of 7 on B - [A(..)]* so as to recover S(C).

460 S.E. Payne

Example 7 (H. Gevaert and N. L. Johnson [3]; from some planes of Ganley).

Here q = 3", rn = @ € F, At = t3 ), t E F . In [17] all collineations of

S ( C ) are determined. We assume q 2 27 to obtain k(C) which are neither classical nor isomorphic to any other type already considered (cf., [17]).

As t H At is an additive map,

dr : ( a , C, P ) (a , c + aAraTi P + aKr) (4)

gives a collineation fixing [A(oo)] and mapping [A(t)] to [A(t + r ) ] , for all ~ , t E F . As [A(co)] is fixed by Aut(S(C)), we need only consider those collineations also fixing [A(O)] and (o,O,o). These are precisely of the form

where r-4 = nu-', 0 # a E F , CT E Aut(F). It follows that

lAut(S(C))l = qG(q - 1)2e.

VII. REFERENCES

1. L. Bader, Derivation of Fisher flocks, preprint. 2. L. Bader, G. Lunardon, and J. A. Thas, Derivation of flocks of quadratic cones,

3. H. Gevaert and N. L. Johnson, Flocks of quadratic cones, generalized quadrangles

4. N. L. Johnson, Derivation of partial flocks of quadratic cones, Rend. d i Roma, to

5. N. L. Johnson, Translation planes and related combinatorial structures, Combina-

6. W. M. Kantor, Generalized quadrangles associated with Ga(q), Jour. Combin.

7. W. M. Kantor, Note on generalized quadrangles, flocks and BLT-sets, preprint. 8. W. M. Kantor, Some generalized quadrangles with parameters (q ' , q ) , Math. Zeit.

9. N. Knarr, private coinniunication, 1090.

Forum Math. 2 (1990), 163-174.

and translation planes, Geom. Dedicata 27 (1988), 301-317.

appear.

torics '90, Gaeta, Italy, May 20-27, 1990.

Theory (A), 29 (1980), 212-210.

192 (1986), 45-50.

10. S. E. Payne, A census of finite generalized quadrangles, in Finite Geometries, Buildings, and Related Topics (eds. W. M. I<antor, R. A. Liebler, S. E. Payne, E. E. Shult), Clarendon Press, Oxford, 1990.

11. S. E. Payne, A garden of generalized quadrangles, Algebras, Groups and Geometries

12. S . E. Payne, Generalized quadrangles as group coset geometries, Congressus Nu- 3 (1985), 323-354.

merantiurn 29 (1980), 717-734.

Collineations of the generalized quadrangles 46 I

13. s. E. Payne, A new infinite family of generalized quadrangles, Congressus Numer-

14. S. E. Payne, Spreads, flocks and generalized quadrangles, JOUT. of Geometry 33

15. S. E. Payne, The Thas-Fisher generalized quadrangles, Annals of Discrete Mathe-

16. S . E. Payne and L. A. Rogers, Local group actions on generalized quadrangles,

17. S. E. Payne and J. A. Thas, Conical flocks, partial flocks, derivation and generalized

18. S. E. Payne and J. A . Thas, Finite generalized quadrangles, Pitman, London, 1984. 19. S. E. Payne and J. A. Thas, Generalized quadrangles, BLT-sets, and Fisher flocks,

20. J. A. Thas, Generalized quadrangles and flocks of cones, European JOUT. Combin.

antium 49 (1985), 115-128.

(1988), 113-128.

matics 37 (1988), 357-366.

Simon Stevin, to appear.

quadrangles, Geom. Dedicata, to appear.

to appear.

8 (1987), 441-452.



Projective embedding of fibered groups and the Suzuki groups

S . Pianta Dipartimento di Matematica, Universitb Cattolica, Via Trieste 17,25 12 1 Brescia, Italia

1. INTRODUCTION AND PRELIMINARY NOTIONS

Let (5,.) be a group with identity 1 and $ a family of proper non-trivial subgroups of 5 such that: (i) F o r X , Y E $ : X n Y # ( l ) impliesX=Y; (ii) u $ = G ; hence $ is afibration of 5 and (5$,.) is called afibered group. As it is well known to any fibered group @,$,a) it is possible to associate the geometric structure of an incidence space ( l ) (5,%), by setting % := (Xg I X E $ , g E $), so that the triple (5,%,.) is a so called fibered incidence group (see e.g. [7]): in fact the group T := (zS : 5 -) 5 ; x + xg I g E 5 ) of right multiplications acts on (5,%) as a collineation group and the lines through the point 1 are subgroups of ( 5 , s ) (precisely the subgroups of the fibration 5). Furthermore if, for any Xg, Yh E %, we define Xg II Yh iff X = Y, then II is an equivalence relation on % with the property that each equivalence class is a partition of the set 5, namely il is aparallelism relation. Thus the triple Two parallel structures C = (5,2,, I I ) and c' = (5',%', 11') are said to be isomorphic if there exists a bijection cp : 5 + 5' such that: (i) for any L C 5 : L E % i f and only if L cp E %'; (ii) for any L, M E % : L I I M if and only if L cp 11' M cp. If C = C', then cp is an automorphism. An automorphism z of a parallel structure C is called a translation if either it is the identity or it is fixed-point-free and for any line L E % : L z II L.

:= (5,%,//) is aparallef sfructure, in the sense of Andd [l].

Then a fibered incidence group is nothing else than a translation parallel structure (see [I]), where T acts as a regulrlr group of translations. These structures have been studied by many authors and in some cases it has been proved that they are strictly related to affine spaces. In particular when I 5 I c - and either (Cj,.) is abelian (hence elementary abelian: see [2]) or it is a p-group of exponent p and nilpotence class 2 2 , then there exists a field F and a finite

An incidence or linear space ( &,%) is an incidcnce structure of points and lines such that any two points are incident with exactly one line, any point being incident with at least two lines and any line with at least two points. As usual we shall idcntify each linc L with the set of points incident with L.

464 S. Pianta

dimensional affine space A(F) such that 5 is the set of points and $ is a family of affine subspaces of A(F) pairwise intersecting only in a point u (results of Biliotti [3] and Biliotti- Herzer [4]). Similar results under stronger assumptions have been found for the infinite abelian case by Biliotti ([3]) and Karzel-Maxson ([8]). On the other hand, if we consider the particular class of those fibered groups which are obtained from quadratic associative algebras in the way described by Karzel (the so-called kinematic spaces derived from kinematic algebras: see [7]), we can look at the corresponding geometric structures as "trace spaces" of projective spaces (p, 3) (i.e. such that 5 E 9, 3, = {L := L n 5 I EE 3, I En 5 I 2 2)) . In both these two classes of examples the geometric structures involved can be considered in some sense "embedded" into a suitable affine or projective space, hence into a projective space in any case, in such a way that the translation group T is a subgroup of collineations of the projective space itself and the lines of % are improper or proper subsets of suitable projective subspaces (lines in the latter case). In this paper I try to generalize these two situations by means of a notion of projective embedding introduced in Q 2, which is appropriate to represent all the examples afore mentioned and also a class of finite fibered groups, that are the so called Suzuki groups (sections 3 and 4).

-

2. T H E PROJECTIVE EMBEDDING: DEFINITION AND EXAMPLES

Let (5,$,-) be a fibered group, n a projective space and u a given point of n. Let us denote by A,, a family of projective subspaces of n with the property: (*) V M , N E & L , , : M + N * M n N = { u ) , I&L,,I>l . We define a projective embedding of ($,$,a) into n to be a triple of injective mappings cx : G + n , p : $ + a, , y : T + Aut n such that: i) 1 a = u ; ii) V X E $, X p is the projective closure of the set X a: we shall denote it by E; iii) y is a monomorphism and for any z E T the restriction of z y to 5a is exactly the mapping za := cx-l z a. In order to simplify our notations, we may assume 5 C n, 9' := (x I X E $), T' :=

T y, z' := z y, hence z = z'

Examples a) Let (5,$,.) be a finite fibered group.

'5 *

It is known that if 5 is an abelian group (see [3]) or a p-group of exponent p and nilpotence class S 2 (see [4]), then the points and lines of the corresponding incidence structure C = (5,%, II) can be regarded as resp. the points and a family of subspaces of an affine space A(F) of finite dimension over a field F, on which the group T acts as a collineation group. Obviously X can be embedded in the projective space n ( F ) of the same dimension provided one considers the projective closures of the lines of C. Also T

Projective embedding of fibered groups 46 5

acts on n (F) as a subgroup of collineations. b)Let (&,F) be an associative algebra with identity over the commutative field F. Let (V,.) be the group of all units of &. Assume that (&F) is a kinematic (i.e. quadratic)

algebra, that is for any a E d. : a2 E F + Fa. Denote by 5 := V/F* := (cp(u) I u E V) and $ := [ c p (F + Fu)* n 5 I u E V W ) (where cp is the canonical map cp : &* + &*/F* ; a + F*a). Then (5,$,.) is a fibered group, called the kinematic space derived from the kinematic algebra (&$PI (see e.g. [71). We note that the so constructed fibered group admits an embedding into the projective space n = (&*/F*, 3) (2) which is canonically derived from the algebra (A.,F). In fact 5 C d*/F*, 3, = ( L = En 5 I EE 3, I En 5 I 2 2) i.e. to each line L E 3, we may associate its projective closure c, and finally the translation group T acts as a collineation group on n.

c) If @,$,a) is a fibered group which admits a projective embedding into a projective space n, for any subgroup 3 c 5 with .% $ we may consider the induced fibration $% := (X n 3 I X E 5, I X n 3 I f ( 1 ) ) so that (.%,$,,.) is still a fibered group which can be embedded into the same space n in a natural way. So, for instance, the two-dimensional projective special linear group PSL(2,F) over a field F can be embedded in this sense into the projective space n = PG (3,F), being a subgroup of PGL(2.F) which belongs to the class b) of our examples (see e.g. [7]).

The next two sections are devoted to describing another class of finite fibered groups which admit a projective embedding into a suitable finite projective space n.

3. THE DEFINITION O F Sz(q) AND SOME PROPERTIES O F ITS SUBGROUPS LATTICE

Let K := GF(q) be the Galois field of order q = 22m+1 (m 2 1) and n : K + K be the automorphism defined by xn = X ~ ~ + I , so that xn2 = x2 for each x E K. The so called Suzuki groups, denoted by Sz(q), are a class of non abelian simple groups of order q2 (q-1) (q2+1) discovered by M.Suzuki ([lo]). Sz(q) was defined (see [lo], p.133) as the subgroup of GL (4,q) generated by S, M, H, where: S : = ( s (a,b) : = I + ae21 + be31 + (an) e32 + (a2 (an) + ab + bn) e41 + (a (an) + b) e42 + + ae43 I a, b E K = GF (q)) (3) has order q2 and is a 2-Sylow subgroup of Sz (9);

M is a cyclic group of order q-1 generated by diag h2rn, h-2m, x-1-2m), with (h) = (K*,.>;

As usual E := ( cp Fa + Fb)* I a,b E A*, Fa # Fb) . We denote by I the idcntity and by ei, the elements of the canonical basis in the algebra Mat (4,q) of

4 x 4 matrices over h e field K .

466 S . Pianra

H : = eI4 + e23 + e32 + e41 (3) is a permutation matrix of order 2. It is well known (see [ 5 ] , XI, Theorem 3.10, for example) that Sz(q) admits a fibration 8 consisting of the conjugates of S, M, U and V where U and V are cyclic of orders q+2m+l+l and q-2m+l+l respectively. Any subgroup of Sz(q) either is isomorphic to Sz(q') for some subfield GF (4') 2 GF (q), or it is a subgroup of the normalizer N(X) of a group X of 8 ( [ 5 ] , p.194).

(3 .1 ) If X = S , N(S) = SM (cf. 191, (21.5) and (24.2 c)), so that the number of conjugates of S in Sz(q) is q2+1. If X is not conjugate to S , it has index 2 or 4 in N(X) (cf, [ S ] , X I , Th. 3.10): in particular it is the only subgroup of 5 contained in N(X).

Finally it will be to observe that no subgroup X in 8 is contained in a group isomorphic to Sz(q'), q' < q. This is obvious when IXI = q2 since a 2-Sylow subgroup of Sz(q') has order (q')2 < 92. For the remaining subgroups in 8, it is an easy consequence of the fact that they are cyclic of odd order. As a matter of fact such an X would have to be contained in a subgroup of odd order, hence a cyclic subgroup, of the corresponding fibration 8' of Sz(q'). But this is impossible since the smallest cyclic subgroup in 8 has order q-2m+1+1 > q'+2m'+1 +1 which is the order of the largest cyclic subgroup in 8'. (For the last inequality let q' = 22m'+1, q = (q')2h+1 so that m = 2hm' + h + m', that means m-m' 2 3. Hence 2m-1 2 2m'+3 - 1 > 2m + 1. Finally 2m+l > 2m'+l implies 2m+l (2m- 1) > > 2m'+l (2m'+ I), hence 22m+l - 2m+l > 22m'+l + 2m'+l i.e. q-2m+' + 1 > q'+ 2""+1 + 1).

4. THE EMBEDDING OF Sz(q) INTO A PROJECTIVE SPACE OF DIMENSION 15

Let R be the algebraic closure of K = GF(q) and F any intermediate field, K S F I R. The full matrix algebra Mat (4,F) may be considered as a vector space of dimension 16 over F and, for any subset X of Mat (4,F), let us denote by FX the vector subspace of the vector space (Mat (4,F), F) generated by X: this subspace is actually a subalgebra whenever X is a subgroup of GL (4,F). The Suzuki groups are given in section 3 in a linear representation of degree 4 over F, which is absolutely irreducible, that means FG = Mat (4,F) where G := Sz(q) (see [6 ] , (9.2)): an elementary proof of this known fact may also be found in our corollary to theorem 2. Let (p : Mat (4,F)* + PG( 15,P) be the canonical projection of the set of non-zero vectors of Mat(4,F) onto the corresponding projective space PG(15,F). We show that the images under (p of the subalgebras generated by any two subgroups of 8 have exactly one projective point in common, via the following:

Theorem 1. In the above notation FX n FY = FI where X , Y are distinct subgroups of the fibration 8 of G, I is the identity matrix so that FI is the centre of Mat (4,F). Proof. We consider three cases separately. Case 1) 1x1 = IYI = 92.

Projective embedding of fibered groups 467

Up to conjugation we may assume X=S. The involution H defined in section 3 conjugates S to its transpose St so that [ S : Ns (H-1 S H)] = [S : S n M(H-1 S H)] = [S : S n MSt] =

Since the conjugates of S are q2+1, S acts by conjugation transitively on the q2 2-Sylow subgroups distinct from it. So we may assume Y = s-1 H S H s for some s E S. This implies FX n FY = FS n F (s-1 H S H s) = s-l (FS n FSt) s = s-l (FI) s = FI. Case 2) 1x1 = 92, IYI + 92. In this case Y is abelian of order prime to the characteristic of F, hence Y is diagonalizable over a finite extension Pof F (this is a consequence of Maschke Theorem and of the fact that every irreducible representation of an abelian group G of order n over a field F which contains a primitive n-root of 1 is linear: see e.g. [6], (1.9) and (2.6)). Assume z E FX n FY. FY G FY implies that z is conjugate (under GL(4,F)) to a diagonal matrix and z E FX implies that its characteristic polynomial is (x -P )~ for some p E F (we are still assuming X = S). It follows z E FI. Case 3) 1x1 # q* and IYI @ q2. X and Y are both cyclic so that FX and FY are abelian subalgebras of Mat(4,F): in particular,if (X,Y) denotes the group generated by X and Y, F X n F Y is contained in the center of F (X,Y). So our theorem is proved when (X,Y) = G since FG = Mat (4,F) as observed at the beginning of this paragraph. Otherwise by (3.1) (X,Y) can only be contained in a conjugate of the normalizer SM of S . Since I S . M I = q2(q-1) is not divisible by q2 f 2m+l + 1 we have I X I = I Y I = 9-1, i.e. X and Y are both isomorphic to M. Thus we may assume X=M=(x) and Y=(sxk) for some SE S\( I ] , k E 2. Let z E FX n FY. Since FX = ( ( a p ) I a,P,y,G E F ) one has F X n FY 3 z = (a p ) Y f j Y t i and z S X ~ = sxh z. Since xh z = z xh it follows zs = sz ;

[ S : I] = 92.

2+n

1 +n u = a + a b + b n

v = a + b

zs - sz = = o

a a(P-a) = a(6-y) = a"(yP) = 0 and b(y-a) = ~ ( 6 - a ) = ~ ( 6 - p ) = 0. a = P = y = b and Z E FI

a=O* b # O ( o r s = I ) and u = b R , v = b 3 a = P = y = 6 and Z E FI.

468 S. Pianta

So it follows from Theorem 1 that a Suzuki group G has a projective embedding into n := PG(15,F) given by the triple of functions (9, , ' ) where cp is the canonical map, x := (FX) cp for each X in & and, always denoting by T the group of right multiplications of G, T' := {g (pr : ll + Il; a cp + a g cp I g E GI. Remark. The more general problem of finding an embedding a : G + ll for the Suzu- ki group G=Sz(q) such that " V X , Y E % := [Fg I F E &, g E G ) with X f Y , I Xa n Yar I I; 1" remains open.

-

Furthermore it is possible to calculate the dimensions of the subalgebras of Mat (4,F) generated by the subgroups of &, hence the dimensions of the projective subspaces of 9'. In fact we have:

Theorem 2. Let X E &. Then: dimF (FX) = 6 if 1x1 = q2 dimF (FX) = 4 otherwise.

Proof. i) If IXI = q2 we may assume X = S and FX = FS C [ plI + p2 (ezl + e44 + p3 e31 + p4 e32 + pLs eql + p6 e42 I pi E F , 1 I i I 6 ) has dimension at most 6. On the other hand consider the following matrices of S = {s (a,b) (as in 8 3)) : s1 := s (0,O); s2 := s (1,O); s3 := s (v,O); s4 := s (0,l); ss := s (0,v); s6 := s ({,O), where v + V K is a fixed element of K and 5 is an indeterminate. Let A be the 6 x 6 matrix whose i-th row consists of the entries of the mamx si (1 I i I 6) of positions resp. ( l , l) , (2,1), (3,1), (3,2), (4,1), (4,2) (taken always in this order). It is easy to chek that the first 5 rows of A are linearly independent: since the 6" row of A is 1,c. 0, c2mt1, c2+2mt1, c1+2m+1, the equation det A = 0 is non-trivial of degree 5 2 + 2m+l c 22m+l < IFI. It follows that there exists E F which is not a root of the above equation and this proves that the vector space FS has dimension 6. ii) Assume IXI f q2. Then X is abelian of order not divisible by 2 = char (F) hence X is diagonalizable (over a finite extension Fof F): this implies that FX has dimension at most 4. In order to prove that FX has dimension 4, it is enough to show that the generator x of X has minimum polynomial of degree 4 because, i n this case, I, x, x2, x3 are linearly independent. If IXI = 9-1, then X is conjugate to M which is generated by x = diag (Xlt2", h2m, h-2m, h-1-2m). So x has 4 distinct eigenvalues which are roots of its minimum polynomial and we are done. So assume IXI = q f 2m+1 +l. We have q2+ 1 = (q + 2m+l + 1) (q - 2m+l + 1) so that (1x1, q2 - 1) = 1 and (1x1, q2 + q + 1) = 1 from which it follows (1x1, q - 1) = (1x1, q2 - 1) =

Let (x) = X: since x must have at least one non-trivial eigenvalue in a finite extension of F whose order is a divisor of 1x1, the minimum polynomial of x (over F) cannot have degree 1,2 or 3 and the theorem is proved.

(IXI,q3 - 1) = 1.

Projective embedding of fibered groups

Corollary. FG = Mat (4,F).

469

Proof. It is easy to see that (FS + FM + F (H-1SH)) = (FS + FM + FSt) is the set of matrices (%,) E Mat (4,F) having azl = a43 and a12 = a34 so that it spans a vector space of dimension 14. Considering that FG is a subalgebra of Mat (4,F), i.e. FG is closed under the product, one gets dimF FG = 16.

A(3KNowLEDGEMENT. The author is grateful to Prof. M.C.Tamburini for her precious comments and suggestions.

5. REFERENCES

[l] AndrC J.: Uber Parallelstrukturen: Teil I, Teil 11, Marh.2. 76 (1961), 85-102, 155- 163.

[2] Baer R.: Partitionen abelscher Gruppen. Arch Math. (Basel) 14 (1963), 73-83. [3] Biliotti M.: S-spazi ed 0-partizioni. Bofl. U.M.I. ( 5 ) 14-A (1977), 333-342. [4] Biliotti M., Herzer A.: Zur Geometrie der Translationsstrukturen mit eigentlichen

Dilatationen. Abh. Math. Sem. Hamburg 53 (1982), 1-27. [5] Huppen B., Blackburn N.: Finite groups III. Springer Verlag - Berlin, Heidelberg,

New York 1982. [6] Isaacs I.M.: Character theory of finite groups. Academic Press. New York, San

Francisco, London 1976. [7] Karzel H., Kist G.P.: Kinematic algebras and their geometries. Rings and Geometry

(Kaya et al. eds.) NATO AS1 series. C160 (1985), 437-509. [8] Karzel H., Maxson C.J.: Kinematic spaces with dilatations. J . of Geometry 22

(1984), 196-201. [9] Luneburg H.: Translation planes. Springer Verlag - Berlin, Heidelberg, New York,

1980. [lo] Suzuki M.: On a class of doubly transitive groups. Ann. Math. 75 (1962), 105-145.


Combinatorics '90 A. Barlotti et al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. 47 I

Codes, block designs, Frobenius groups, and near-rings

G. F. Pilz

Institut f~ Mathematik, Johannes Kepler Universitat Linz, A-4040 Linz, Austria

Abstract Starting from a finite Frobenius group Qs (or, equivalently, from a finite planar near-

ring), one can construct tactical configurations (in particular, block designs) via several constructions. If one takes the rows or the columns of the corresponding incidence mamx, one obtains various (non-linear) equal-weight-codes, some of them being also equal- distance-codes. The usual parameters of these codes can be described (mainly by l@l) and some of these codes turn out to be "maximal" in the sense that they reach the bounds A(n,d,w) for the sizes of equal-weight(=w)-codes with given length n and minimal distance d. Some of these constructions yield new formulas for A(n,d,w).

Let us start with a finite Frobenius group G with (additively written) kernel (N,+) and Frobenius complement Qs. See e.g. [ l l ] for the basic facts on Frobenius groups. In particular, (N,+) is then nilpotent and @ can (and will) be considered as a fixed-point-free automorphism group of (N,+). We will study the orbits Q(n), n#O. of Q on N, as well as the "enlarged orbits" Qo(n) := Q(n) u (0). These orbits give rise to tactical configurations, which are "often" block designs. This was discovered by Ferrero in [8] and neatly described in [2]. Cf. also [13].

More precisely, let G,N,Qs be as above, and let B, , := Qs,(n) + m for n,m E N, nN. If B = (B,, I n,m E N, n#O) then (N,@ is always a tactical configuration (see [2]) with

parameters (v,b,r,k), given by v = INI, k = lQl+l, b = + tv, r = s+tk, where s is the

number of those non-zero orbits Qo(n) which are subgroups of (N,+), and t is the number of the remaining non-zero enlarged orbits. Moreover, (N,B is a balanced incomplete block (= BIB) design iff s=O (then h = l@l+l) or t=O (then h=1).

vs

412 G.F. Pilz

A slightly different and "always successful" construction was presented in [7]. Let 9P

be the collection of all Q,(n)+m (n,m E N, n#O). Then (N,B*) is always a BIB-design

with parameters (v,b,r,k,h), such that v = INI, b = *r , r = v-1, k = 101, and

h = lQ,l-l.

Sometimes the collection 5 := (&(n)+m I n,m E N, n#O) with &(n) := Qo(n) u ao(-n)

also gives rise to a BIB-design (N,B)l see [6], but we will concentrate on the two designs (N,!B) and ( N I P ) described above.

The role of Frobenius groups can also, and in some sense more accurately, described by near-rings. A (right) near-ring (N,+,.) consists of a group (N,+), a semigroup (N,.), and the (right) distributive law (a+b)c = ac+bc for all a,b,c E N. If na = nb holds for all nE N, a and b are called equivalent multipliers (denoted by a I b). If each equation xa =

xb+c with a b has precisely one solution and INkI 2 3, N is called a planar near-ring. See e.g. [131. We now describe the connection between Frobenius groups and planar near- rings, following [2].

v(v-1)

-

(A) Let G be a Frobenius group with kernel (N,+) and complement 0. As mentioned above, Q, is a fixed-point-free automorphism group on N. Suppose that for each @E 0.9 # id, id-@ is also bijective. Select orbit representatives ( ni I iE I ) of N w.r.t. Q, and a subset S = { nj I jE J E I ) thereof containing 0. If n,m E N, define a product n-m by n.m := 0 if m E Q,(nj) with jE J; if m E Q,(ni), i E N, there is precisely one @E Q, with m = $(ni); define nqm := @(n) in this case. Then (N,+,.) is a planar nearring. N has zero divisors iff S # (0).

(B) Let (N,+,.) be a planar near-ring. For aE A, let fa : N -) N be the map n + na. Then Q, := (fa I fa # zero map) is a fixed-point-free automorphism group on (N,+), and we anive at a Frobenius group.

In (B), (a I fa = 0) corresponds to u Q,(nj) in (A). Hence to each Frobenius group,

there exists a collection of planar near-rings (depending on the choice of J), which might or might not be isomorphic. And conversely.

The blocks are, however, given by Q,,(n)+m = nN+m and @(n)+m = nN*+m. Hence all planar near-rings obtained from Frobenius groups as in (A) yield the same designs.

To each block design (N,B we can associate its incidence matrix M N , ~ = (mij) in the following way. Let N = (nl ,..., n,] and B = (B1, ..., Bb). Then mij = 1 if ni E Bj, and mij = 0 otherwise. Hence M N , ~ is a vxb-matrix over { 0,l).

jcJ

Codes, block designs, Frobenius groups and near-rings 41 3

Next, we take the rows of M N , ~ . They form a subset CN,B consisting of v vectors of length b. Each of these vectors consists of r "ones" and b-r "zeros", if (N,@ is a BIB- design with parameters (v,bs,k,h). Hence C N , ~ is a binary equal weight (*) code of size v = IN1 and length b, see [12], 191. Also, it can be shown that each two such codewords have the same Hamming distance d = 2(r-h). These parameters were given in the beginning of this paper. CN,g is called the row-code of (N,%). For much more on the interplay between codes and design see e.g. [3].

An excellent way to get suitable Frobenius groups (= suitable planar near-rings) was described in [5] and [9]. Take a finite field IF, of order q = p*, and as the group of automorphisms induced by multiplications with elements a. In the language of near-rings,

we choose a divisor t of q-1 and a primitive element g of (IF;,.). Define a new

multiplication *tin IF: by ga *t gp := ga+p-[pls, where [PI, is the remainder of P modulo

s = q. Then N = (F,,+,*J is a planar near-ring and the corresponding BIB-design (N,@ has parameters

(v,b,r,k,h) = (q. 9s_ t+l , s, t+l, 1) if t = p - 1 for some mln = (q,qs,q+s-l,t+l,t+l) otherwise.

The design ( N , P ) has parameters

regardless of the "shape" oft. See [2]. (q,qs,q-l,t,t- 1)

We are particularly interested in the first of these three cases. Observe that this first case yields a symmetric design (i.e. b=v) iff q = t2+t+l. The codes Cq,t corresponding to the

designs with h=l above have q codewords of length 3, weight s and equal distance 2(s-1) (see [9]).

In the same way, one can start with a BIB-design with parameters (v,b,r,k,h), form the incidence mamx M and take the columns of M as a code CN*g, the column code of (N,@. CN.9 has b codewords of length v and equal weight k (see [9]).

Let A(n,d,w) denote the maximal number of codewords in a binary equal (=w) weight code of length n and minimal distance d. Formulas for A(n,d,w) are much looked for, and give also other combinatorial insights (for instance, in discrete ball packing problems). A given code ICI can be regarded as "good" if ICI is not "much smaller" than A(n,d,w).

Definition: Let C be an (n,d,w)-code. Then C is called maximal if ICI = A(n,d,w).

Theorem 1: a) Column codes CNsB derived from BIB-designs with h=l are maximal. b) All row codes CN,g are also maximal.

414

Proof.

G.F. Pilz

If B is a BIB-design with parameters (v,b,r,k,l) then we have the relevant parameters n=v, d = 2k-2, w=k, ICN,gI = b (cf. [9]). By Cor. 5 of [12], p. 527, A(n,d,w) S

L F L T L ... L T ... 1, where 6 = and LxJ denotes the greatest integer Ix.

Hence A(n,d,w) 5 L, L,!d = L F J = = b = ICNsgl, since v-1 = r(k-1) and vr =

bk. This shows that the upper bound A(n,d,w) = L$ L@ is reached by ICN."I. For any BIB-design we get

n n-1 n-w+6 d

v v-1 vr

v[r2-rb+(r-h)b] = vr2 - hvb = b Pr2 -6 - hv 1 = b(rk-hv) = b(r-h), since r-h = rk - hv by r(k-1) = h(v-1). Hence the row code C N , ~ with v codewords of length n = b, equal weight w = r and minimal distance 2(r-h) reaches the Johnson bound ([12], vol. 2, p. 525):

6n w -wn+6n

v 5 A(n,d,w) I L J = v,

where again 26 = d. Hence CN,g is maximal.

Part b) of Theorem 1 was already mentioned in [8] (Th. 2.6). For row codes CN,g with h = 1 we get by [12], p. 527

IcN,d L, L, The row code derived from no. 45 in the tables of [ 11, p. 616, with v = 41, b = 82, r =

10, h = 1 (hence 6 = 9) shows that we will not get equality in general. For the special case Cq,t, however, the situation is clear:

Theorem 2: For each prime power q = p" and each divisor t of q-1 of the form t = pm - 1 we get

b b-1 p" = ICq,J 5 L 'T L-JU = p2n-2m

(with equality iff n = 2m).

Proof. From [9], table I, we get the parameters

n = b = p"(p"-l) , d = 2( - l), w = r = @ and ICq,J = p". prn(prn-- 1) P -1 P -1

Now by Cor. 5 of [ 121 (cited in a)), we get for these parameters

L y L T J J = L b b-1 p"(p"-l)(p'"-l) rn L( p"(p"-l) - 1)( p1-1 1)11= p (p -l)(p"-l) prn(prn-l) p -1

Codes, block designs, Frobenius groups and near-rings 415

In order to compute X, let u := n-m.

= Pn-rn

Altogether, by Theorem 1, pn = lCq,J = A(n,d,w) I L- L-JJ = ~"-~p"-m. b b-1 r r-1

So we get new formulas for some A(n,d,w)'s:

Corollary: a) If p is a prime and n,m E N with mln then A(pn-m x, 2x-2, x) = pn, where x =

pn-l pm-1

(row codes with h = 1) and A(p",2pm-2,pm) = p-mx (column codes with h =

1). b) If q = pn is a prime power and q-1 = st then A(qs,2(q-t),q-l) = q (row codes from !If'

with parameters q.t) and, if t is not of the form pm-1, A(qs,2(q+s-t-2),q+s-l) = q.

Remark: The column codes CNvS with h>l are not maximal in general: Take, for instance, the

(circular) design !If' with q=13 and t=4. By Table I of [9], we get n=13, d=4, w=4, and 39 codewords. By [4], Table I-A, A(13,4,4) = 65 > 39.

The author thanks Dr. James R. Clay (Tucson, Arizona) for most fruitful discussions.

REFERENCES 1 T. Beth, D. Jungnickel and H. Lenz, Design theory, Bibl. Inst. Mannheim, 1985. 2 G. Betsch and J.R. Clay, Block designs from Frobenius groups and planar near-rings,

Roc. Conf. Finite groups (Park City, Utah), Acad. Press (1976), 473-502. 3 W.G. Bridges, M. Hall and J.L. Hayden, Codes and designs, J. Combin. Theory 31A

4 A.E. Brouwer, J.B. Shearer, N.J.A. Sloane, W.D. Smith, A new table of constant weight codes, IEEE Trans. on Information Theory 36 (1990), 1334-1380.

5 J.R. Clay, Generating balanced incomplete block designs from planar near-rings, J. Algebra 22 (1972), 319-331.

(1981). 155-174.

476 G.F. Pilz

6 J.R. Clay, More balanced incomplete block designs from Frobenius groups, Discrete Math. 59 (1986), 229-234.

7 J.R. Clay, Circular block designs from planar near-rings, Ann. Discr. Math. 37

8 G. Ferrero, Stems planari e BIB-disegni, Riv. Mat. Univ. Parma (2) 11 (1970), 79-96. 9 P. Fuchs, G. Hofer and G. Pilz, Codes from planar near-rings, IEEE Trans. on

Information Theory 36 (1990), 647-651. 10 R.L. Graham and N.J.A. Sloane, Lower bounds for constant weight codes, IEEE

Trans. Inform. Theory 26 (1980). 37-43. 11 B. Huppert, Endliche Gruppen I, Springer, Verlag, Berlin, 1967. 12 F.J. McWilliams and N.J.A. Sloane, The theory of error-correcting codes I, 11, North-

13 G. Pilz, Near-rings, North-Holland/American Elsevier, Amsterdam, Second, revised

(1988), 95-106.

Holland, Amsterdam, 1977.

edition, 1983.


The flag-transitive affine planes of order 27

Alan R. Prince

Department of Mathematics, Heriot-Watt University, Edinburgh, Scotland

1. INTRODUCTION

The purpose of this paper is to describe a classification of all

the flag-transitive affine planes of order 27.

translation planes of order 27 which have a translation complement of

order divisible by 7. This is a more general classification btit, in

fact, all the planes we obtain are flag-transitive. A result of Wagner

[ 6 ] asserts that a flag-transitive affine plane must be a translation

plane. We describe the planes by specifying corresponding spreads in a

6-dimensional vector space V over GF(3). We represent V by the field

GF(3 ) which is obtained by adjoining to GF(3) a root 8 of the

irreducible polynomial x

primitive root of GF(3 ) . There are two important linear transformations

of V : the Frobenius map a defined by xa - x3 and the map corresponding to multiplication by 8, x + x8, which we shall denote simply by 8. (The

context will distinguish whether 8 is used to denote an element of V or a

linear transformation of V.)

important role in our investigation. Our main result is the following.

In fact, we determine all

6

6 - x5 - 1. The element 8 is actually a

6

The subgroup <B,a> of GL(V) plays an

Theorem. There are, up to isomorphism, precisely four translation planes

of order 27 which have a translation complement of order divisible by 7.

All are flag-transitive and, moreover, any flag-transitive affine plane

of order 27 is isomorphic to one of these planes. The four planes

418 A.R. Prince

correspond to the following spreads in the vector space V:

The translates of the subspace < 1 , 8 2 8 , 8 5 6 > by the subgroup <013> of

W V ) ;

The translates of the subspace <1,0, 064> by the subgroup <d3> of

GL(V) ; 3 11 60 6 4

The translates of each of the subspaces <1, 0, 0 >, <B , 0 , 0 >

by the subgroup <026> of GL(V);

The translates of each of the subspaces <1, 0, 0 >, <0 ,0 ,0 >,

<e4,e60,e119>, <e5,

23 2 31 33

by the subgroup <e52> of GL(V).

Remark. The plane described in (a) is the Desarguesian plane while that

described in (d) is the affine plane discovered by Hering [ 2 ] which has a

collineation group acting doubly transitive on the points.

The general theory of spreads and translation planes is described

in [ 3 ] .

2. THE STRATEGY OF THE PROOF

We describe, in this section, the basic idea of the classification

and the context in which we work. As in the previous section, V denotes

6 the 6-dimensional vector space over GF(3) represented by the field GF(3 )

obtained by adjoining to GF(3) an element 0 , which satisfies d 6 - 0 and which generates the multiplicative group GF(3 ) . A spread in V is

a collection of subspaces of dimension 3 with the property that each

nonzero vector of V is contained in a unique member of this collection.

Clearly, a spread in V must consist of (36 - 1 ) / ( 3 - 1) = 28 subspaces,

which are called the components of the spread. The primitive root 0 has

multiplicative order 728 and 0 3 6 4 = -1.

namely (0,Bi, -0 ) for i - 0 , 1, 2 , . . , 3 6 3 .

5 + 1 6 *

3

Thus, V contains 364 1-spaces,

i A 1-space may therefore be

The jlag-transitive affine planes of order 27 419

specified by 0' where the exponent i is taken modulo 364.

contains 13 1-spaces and hence associated to any 3-space are 13 exponents

mod 364.

when taken modulo 52. The special 3-spaces are precisely those 3-spaces,

the seven translates of which under the subgroup <@52> of GL(V) form a

partial spread, that is any two distinct translates have only the zero

vector in common. A flag-transitive affine plane of order 27 must be a

translation plane [ 6 ] and correspond to a spread in V invariant under a

subgroup of GL(V) which acts transitively on the 28 components. This

means that the subgroup leaving the spread invariant must have order

divisible by 7. Since GL(V) = GL(6,3) has Sylow 7-subgroup of order 7,

we may assume that the spread is invariant under the subgroup <@52>.

Hence, the components of the spread are special 3-spaces. Our aim is to

determine all spreads invariant under the subgroup <@52>.

are the union of four <t952torbits of special 3-spaces. To describe

which unions of four orbits are spreads, we introduce a graph r with

vertices corresponding to the <e52t~rbits of special 3-spaces.

Associated to each vertex is a well-defined set of 13 residues mod 52.

Two vertices are joined by an edge in r if and only if the corresponding sets of residues mod 52 are disjoint. The spreads invariant under <052>

correspond to the 4-cliques of r . The group <@,a> induces a group of

automorphisms of r , and hence acts on the set of all 4-cliques of r . If

two 4-cliques are in the same orbit under this action then the

corresponding spreads give isomorphic translation planes. A computer

search shows that there are four orbits of 4-cliques of r under the action of <@,a>. The corresponding translation planes are actually

non-isomorphic (see next section) and yield the four planes described in

A 3-space

We call a 3-space special if these 13 exponents are distinct

Such spreads

-

the Theorem.

480 A.R. Prince

3. THE DETAILS OF THE COMPUTATION

If we consider the <@>-orbit of a special 3-space there is a

representative in the orbit which contains the vector 1 (in fact, in most

cases, there are 13 such representatives), A computer search shows that

there are precisely 196 special 3-spaces containing the vector 1.

list these solutions in Appendix 1, giving the exponents i mod 364

corresponding to the 13 1-spaces <@ > in each 3-space, as well as the

corresponding residues mod 52. (The full set of special 3-spaces

consists of taking the <@>translates of these). In Appendix 2, we list

which of these solutions are in the same <@,a+orbit. The columns in the

tables give solutions in the same <@>orbit, and the action of <a> on the

solutions is also described. Note that a(1) - 1, so that <a> acts on the

set of special 3-spaces containing 1. There are 5 <@,a>orbits of

special 3-spaces, with representatives <1,9,@23>, <1,@,@ >, <1,@,@ >,

<1,@ , @ >, which are solutions #1, #2, #7, #30, #121

respectively. Since we are interested in finding a representative for

each orbit of 4-cliques under the action of <@,a>, our next task is to

find all 4-cliques containing a vertex in r corresponding to one of these 5 representatives. A computer search shows that only the following cases

arise:

We

i

28 64

7 3 1

(i) #1, (#125)s2, (#169)s4, (#47)@ 5 ;

(ii) #2, (#3 )e3 , (#62)e5, (#ioo)e8;

(iii) #7 , (#78)e2, (#38)e3, (#116)@13;

3 4 (iv)

(v)

#7, (#78)e2, (#iso)e , (#8o)e ;

(#121), (#121)e, (#121)e2, (#121)e . 3

(Here, the special 3-spaces are given as a translate of one of the

solutions containing 1. Note that the vertex corresponding to solution

The flag-transitive afJine planes of order 27 48 I

#30 is not contained in any 4-clique.)

containing solution #1 (i.e. the corresponding vertex) and since solution

#62 is in the same <@,asorbit as solution #1, it follows that the

4-cliques corresponding to (i) and (ii) are in the same <B,a>-orbit. It

is not hard to verify that cliques (i), (iii), (iv), (v) are in different

<B,a+orbits

<B,a> on the 4-cliques of I'.

the planes described in the theorem and that these planes are pairwise

non-isomorphic.

Since there is a unique 4-clique

showing that there are precisely 4 orbits in the action of

We now show that these orbits correspond to

We note that solutions #7 and #121 correspond to 3-spaces in which

the 13 exponents mod 364 actually remain distinct when considered as

residues modulo 13.

<B These are cases (a) and (b)

of the Theorem (and cases (iv) and (v) above). Note also that solution

#121 is the 3-space corresponding to subfield GF(3 ) of GF(3 ) and

consists of 0 and the powers of B 2 ' , which is a primitive root of GF(3 ) .

Thus, case (a) of the Theorem corresponds to the Desarguesian plane.

Thus, the translates of each of these 3-spaces by

> give a spread invariant under <@I3>. 13

3 6

3

If we consider spreads invariant under <B26> then the components are

special 3-spaces with the 13 exponents remaining distinct when viewed

modulo 26. We

further spread

Finally,

the additional

do not have any additional 3-spaces but we do obtain the

described in (iii), giving rise to case (c) in the Theorem.

when we consider spreads invariant under < L ? ~ ~ > , we have

orbits, 1, 2, 4 of special 3-spaces but these give rise to

only one additional spread (up to isomorphism), namely (i) above, which

is (d) of the Theorem.

By direct calculation, the subgroup of <@,a> leaving the spread

482 A.R. Prince

invariant can be determined in each case :

52 39 3 (a) (b) <0l3,e7a2>; (c) (d) <e , e a >.

Although these subgroups need not be the full translation complement,

they are the Sylow 7-normalisers in the full translation complement

(1, Lemma 13.51. It follows that the corresponding translation planes

are pairwise non-isomorphic. It is also immediate that planes (a), (b),

(c) are flag-transitive since the subgroups of <#,a> in these cases are

transitive on the components. For (d), this is not the case. However,

the affine plane of order 27 discovered by Hering [2] is a

flag-transitive plane (in fact, its collineation group is doubly

transitive on points) and so must be one of our examples.

plane has translation complement SL(2,13), which has Sylow 7-normaliser

D28, it must be the plane described in (d) of the Theorem.

completes the proof of the Theorem .

-

Since Hering's

This

In conclusion, we remark that the planes described in (b) and (c)

of the Theorem were discovered by Narayana Rao, Kuppuswamy Rao and

Satyanarayana [ 4 , 5 ] .

REFERENCES

1. D. A . Foulser, The flag-transitive collineation groups of the finite

Desarguesian affine plane, Canad. J. Math. 16 (1964), 443-472.

2. C. Hering, Eine nicht-desarguessche zweifach transitive affine Ebene

der Ordnung 27, Abh Math. Sem. Univ. Hamburg 34 (1969), 203-208.

3 . H. Luneburg, "Translation Planes," Springer-Verlag,

Berlin/Heidelberg/New York, 1980.

4. M. L. Narayana Rao and K. Kuppuswamy Rao, A new flag transitive

affine plane of order 27, Proc. Amer. Math. SOC. 5 9 (1976), 337-345.

The Jag-transitive affine planes of order 27 483

5. M. L. Narayana Rao, K. Kuppuswamy Rao and K. Satyanarayana, A third

flag-transitive plane of order 27, Houston J. Math. 10(1984), 127-145.

6. A. Wagner, On finite affine line transitive planes, Math. Z.

87 (1965), 1-11.

APPENDIX 1

In this appendix, the 196 special 3-spaces are listed.

the first line gives the associated 13 exponents mod 364, while the

second line gives the reduction of each exponent mod 52.

For each solution,

SOLUTION 1

0 1 23 0 1 23

SOLUTION 2 0 1 28 0 1 28

SOLUTION 3 0 1 33 0 1 33

SOLUTION 4 0 1 38 0 1 38

SOLUTION 5 0 1 44 0 1 44

SOLUTION 6 0 1 57 0 1 5

SOLUTION 7 0 1 64 0 1 12

SOLUTION 8 0 2 11 0 2 11

128 24

128 24

128 24

128 24

128 24

128 24

128 24

287 27

359 47

359 47

359 47

359 47

359 47

359 47

359 41

123 19

334 22

252 44

70 18

141 37

74 22

272 12

276 16

266 6

290 30

84 32

163 7

148 44

97 45

172 16

83 3 1

297 37

263 3

181 25

146 4 2

131 27

195 39

225 17

188 32

62 10

200 44

230 22

303 43

332 20

119 15

337 25

166 10

321 9

68 16

220 12

275 15

197 41

179 23

89 37

147 43

94 42

79 247 77 27 39 25

277 65 158 17 13 2

316 136 143 4 32 39

270 189 113 10 33 9

206 296 72 50 36 20

202 235 153 46 27 49

137 269 340 33 9 28

75 207 151 23 5 1 47

484 A.R. Prince

SOLUTION 9 0 2 23 0 2 23

SOLUTION 10 0 2 25 0 2 25

SOLUTION 11 0 2 28 0 2 28

SOLUTION 1 2 0 2 34 0 2 34

SOLUTION 1 3 0 2 5 1 0 2 5 1

SOLUTION 14 0 2 59 0 2 7

SOLUTION 1 5 0 3 1 3 0 3 13

SOLUTION 1 6 0 3 28 0 3 28

SOLUTION 1 7 0 3 29 0 3 29

SOLUTION 1 8 0 3 44 0 3 44

SOLUTION 1 9 0 3 47 0 3 47

SOLUTION 20 0 3 88 0 3 36

SOLUTION 2 1 0 3 1 3 2 0 3 28

287 27

287 27

287 27

287 27

287 27

287 27

20 20

20 20

20 20

20 20

20 20

20 20

20 20

1 2 3 1 9

1 2 3 1 9

1 2 3 1 9

1 2 3 1 9

1 2 3 1 9

1 2 3 1 9

349 37

349 37

349 37

349 37

349 37

349 37

349 37

3 3 4 22

2 9 3 33

252 44

2 8 4 2 4

310 5 0

8 0 28

6 1 9

252 44

203 4 7

7 4 22

1 3 4 30

1 5 2 4 8

222 14

290 30

107 3

8 4 32

343 3 1

355 4 3

114 10

2 0 4 48

8 4 32

227 1 9

97 45

292 32

1 7 1 15

2 9 1 3 1

307 47

336 2 4

1 1 3 9

147 4 3

1 8 6 30

2 7 4 14

2 3 1 23

296 36

114 1 0

220 1 2

77 25

3 4 1 29

2 2 1 1 3

57 5

292 32

267 7

304 44

213 5

1 7 4 1 8

1 4 2 38

1 1 0 6

268 8

1 2 5 2 1

100 4 8

242 34

357 45

1 1 5 11

56 4

270 10

1 8 8 32

282 22

6 9 1 7

237 29

1 9 5 39

59 7

1 8 1 25

7 9 27

267 7

1 6 0 4

2 7 2 1 2

1 3 4 30

3 1 1 51

313 1

288 28

335 23

6 9 1 7

1 0 3 5 1

8 2 30

99 47

2 4 9 41

95 4 3

2 5 4 46

215 7

2 2 4 1 6

332 2 0

269 9

257 4 9

14 2 38

236 28

1 7 9 23

8 0 28

6 5 1 3

200 4 4

3 1 1 5 1

1 7 3

172 1 6

47 47

341 29

259 5 1

1 7 0 14

1 9 8 4 2

274 14

326 14

339 27

210 2

192 36

283 23

216 1 7 8

SOLUTION 22 0 5 28 0 5 28

SOLUTION 23 0 5 33 0 5 33

SOLUTION 2 4 0 5 38 0 5 38

SOLUTION 25 0 5 4 3 0 5 4 3

0 5 4 9 0 5 4 9

SOLUTION 26

SOLUTION 27 0 5 6 2 0 5 10

SOLUTION 28 0 5 6 9 0 5 1 7

SOLUTION 29 0 7 10 0 7 10

SOLUTION 30 0 7 3 1 0 7 3 1

SOLUTION 3 1 0 7 4 8 0 7 4 8

SOLUTION 32 0 7 6 4 0 7 1 2

SOLUTION 33 0 7 7 1 0 7 1 9

SOLUTION 34 0 7 7 6

6 6

6 6

6 6

6 6

6 6

6 6

6 6

223 15

223 1 5

223 1 5

223 1 5

223 1 5

223 0 7 2 4 15

The Jag-transitive afine planes of order 27 485

133 29

1 3 3 29

1 3 3 29

1 3 3 29

1 3 3 29

1 3 3 29

133 29

261 1

2 6 1 1

2 6 1 1

2 6 1 1

2 6 1 1

2 6 1 1

252 44

70 1 8

141 37

1 9 4 38

1 8 4 28

9 4 4 2

274 14

228 20

98 46

1 9 1 35

2 7 6 1 6

96 44

246 38

84 32

1 6 3 7

1 4 8 4 4

1 1 8 1 4

2 1 1 3

207 5 1

142 38

1 3 9 35

1 0 9 5

3 5 1 39

83 3 1

1 7 8 22

289 29

339 27

257 4 9

7 5 23

1 4 6 4 2

7 9 27

277 1 7

2 8 1 2 1

27 27

1 6 8 1 2

224 1 6

279 1 9

283 23

2 8 1 2 1

295 3 5

8 9 37

1 6 8 1 2

1 5 3 4 9

1 0 2 5 0

1 7 7 2 1

8 8 36

356 44

1 9 7 41

1 2 9 25

179 23

9 0 38

1 4 9 4 5

82 7 3 268 205 30 2 1 8 49

1 8 6 225 282 235 30 1 7 22 27

1 5 1 308 3 2 1 280 47 4 8 9 20

275 202 1 3 6 337 1 5 46 32 25

77 1 2 4 200 3 0 1 25 2 0 44 41

230 240 1 5 8 342 22 32 2 30

1 9 3 1 5 2 345 1 7 1 37 4 8 33 15

1 8 0 298 229 167 2 4 38 2 1 11

308 1 8 9 289 1 3 1 4 8 33 20 27

56 218 336 354 4 10 2 4 4 2

1 5 4 1 9 5 253 296 50 39 45 36

242 232 95 1 5 9 34 2 4 4 3 3

193 92 345 264 37 4 0 33 4

486

SOLUTION 35 0 8 1 5 0 8 1 5

SOLUTION 36 0 8 23 0 8 23

SOLUTION 37 0 8 34 0 8 34

SOLUTION 38 0 8 57 0 8 5

SOLUTION 39 0 8 6 4 0 8 1 2

SOLUTION 4 0 0 8 8 1 0 8 29

SOLUTION 41 0 9 11 0 9 11

SOLUTION 4 2 0 9 1 9 0 9 1 9

SOLUTION 4 3 0 9 28 0 9 28

SOLUTION 44 0 9 32 0 9 32

SOLUTION 45 0 9 39 0 9 39

SOLUTION 46 0 9 7 3 0 9 2 1

SOLUTION 47 0 9 7 6 0 9 24

1 7 5 1 9

1 7 5 1 9

1 7 5 1 9

1 7 5 1 9

1 7 5 1 9

1 7 5 1 9

60 8

60 8

6 0 8

6 0 8

60 8

60 8

60 8

306 46

306 46

306 46

306 46

306 46

306 46

319 7

319 7

319 7

319 7

319 7

319 7

319 7

1 8 1 8

334 22

2 8 4 2 4

272 1 2

276 1 6

1 7 6 2 0

266 6

212 4

252 44

14 5 41

1 8 3 27

205 4 9

246 38

A.R. Prince

35 35

290 30

343 3 1

1 7 2 1 6

8 3 3 1

323 11

297 37

300 4 0

8 4 32

302 42

248 4 0

295 35

289 29

2 3 1 23

26 26

1 1 9 1 5

1 9 2 36

232 2 4

213 5

296 36

237 29

2 2 1 1 3

343 31

329 1 7

285 25

87 35

269 9

4 3 4 3

273 1 3

219 11

344 32

303 4 3

1 3 2 28

148 44

309 4 9

299 39

6 2 10

92 40

342

237 1 4 7 236 1 8 8 29 4 3 28 32

111 1 9 4 265 118 7 38 5 1 4

206 9 1 7 2 1 8 2 50 39 2 0 26

100 2 5 1 249 157 4 8 4 3 41 1

1 5 9 347 9 0 329 3 35 38 17

288 316 1 7 0 1 4 3 28 4 14 39

1 9 5 222 1 7 9 2 9 1 39 14 2 3 3 1

2 3 1 38 236 141 23 38 2 8 37

1 6 0 2 5 0 1 7 3 330 4 4 2 1 7 1 8

34 1 5 5 2 8 4 1 1 6 34 5 1 2 4 1 2

344 9 4 347 207 32 4 2 35 5 1

1 2 1 1 4 9 362 264 1 7 45 5 0 4

245 1 7 7 317 240 30 37 2 1 5 32

SOLUTION 4 8 0 10 17 0 10 17

SOLUTION 4 9 0 10 2 1 0 10 2 1

SOLUTION 50 0 10 25 0 10 25

SOLUTION 5 1 0 10 29 0 10 29

SOLUTION 52 0 10 32 0 10 32

SOLUTION 53 0 10 34 0 10 34

SOLUTION 5 4 0 10 47 0 10 47

SOLUTION 55 0 10 1 5 4 0 10 50

SOLUTION 56 0 11 33 0 11 33

SOLUTION 57 0 11 45 0 11 45

SOLUTION 58 0 11 101 0 11 4 9

SOLUTION 59 0 1 3 32 0 1 3 32

SOLUTION 60 0 1 3 56 0 1 3 4

228 20

228 20

228 20

228 20

228 20

228 20

228 20

228 20

266 6

266 6

266 6

6 1 9

6 1 9

The flag-transitive afine planes of order 27

1 3 9 35

1 3 9 35

1 3 9 35

1 3 9 35

1 3 9 35

1 3 9 35

1 3 9 35

1 3 9 35

297 37

297 37

297 37

204 4 8

204 4 8

346 3 4

305 45

293 33

203 47

14 5 41

2 8 4 2 4

1 3 4 30

253 45

7 0 1 8

5 4 2

348 36

1 4 5 41

224 1 6

3 6 1 4 9

55 3

1 0 7 3

227 1 9

302 4 2

343 3 1

292 32

279 1 9

1 6 3 7

1 0 5 1

1 6 9 1 3

302 42

336 2 4

233 2 5

276 1 6

2 8 28

222 14

272 1 2

1 7 1 1 5

1 4 0 36

219 11

273 1 3

295 35

243 35

225 1 7

20 7 5 1

2 7 1 11

307 47

45 45

310 50

209 1

200 4 4

3 4 1 29

335 23

210 2

3 5 4 4 2

1 7 0 14

313 1

1 3 1 27

58 6

83 3 1

252 44

1 3 2 28

1 7 2 1 6

88 36

1 1 2 8

2 5 1 4 3

9 1 39

7 3 2 1

1 6 4 8

89 37

9 4 4 2

487

234 6 6 2 0 1 26 14 45

1 1 5 6 4 215 11 1 2 7

5 4 8 4 1 0 5 2 32 1

355 2 9 1 5 1 4 3 3 1 5 1

325 57 144 1 3 5 4 0

7 9 152 77 27 4 8 25

267 1 9 6 3 1 1 7 4 0 5 1

1 9 8 157 1 7 4 4 2 1 1 8

1 2 5 1 8 2 99 2 1 26 47

1 2 9 205 218 25 49 10

288 2 4 1 213 28 33 5

304 235 259 44 27 5 1

197 62 1 8 9 41 10 33

488

SOLUTION 6 1 0 1 3 90 0 13 38

SOLUTION 6 2 0 1 5 28 0 1 5 28

SOLUTION 63 0 1 5 4 3 0 1 5 4 3

SOLUTION 6 4 0 1 5 44 0 1 5 44

SOLUTION 65 0 1 5 59 0 1 5 7

SOLUTION 66 0 1 5 62 0 15 10

SOLUTION 67 0 1 5 1 0 3 0 1 5 5 1

SOLUTION 68 0 1 6 27 0 16 27

SOLUTION 69 0 1 6 4 3 0 1 6 43

SOLUTION 7 0 0 1 6 44 0 16 4 4

SOLUTION 7 1 0 1 6 7 3 0 1 6 2 1

SOLUTION 7 2 0 17 25

6 1 9

1 8 1 8

i a 1 8

1 8 1 8

1 8 1 8

1 8 18

1 8 1 8

185 29

185 29

185 29

185 29

346

A.R. Prince

2 0 4 232 1 5 9 92 213 1 4 9 48 2 4 3 40 5 45

35 252 a4 7 6 219 246 35 44 32 2 4 11 38

35 1 9 4 1 1 8 267 9 9 3 1 1 35 38 14 7 47 5 1

35 74 97 218 242 354 35 22 4 5 10 34 4 2

35 80 114 a9 1 1 2 225 35 28 10 37 8 17

35 9 4 207 1 4 9 307 92 35 4 2 5 1 45 47 40

35 326 110 167 1 8 6 298 35 14 6 11 30 38

117 1 8 0 229 282 313 257 1 3 2 4 2 1 22 1 4 9

117 1 9 4 1 1 8 196 245 112 1 3 38 14 4 0 37 8

117 7 4 97 268 100 82 1 3 22 4 5 8 4 8 30

3 6 1 293 107 192 323 100 0 17 25 34 4 9 33 3 36 11 4 8

288 28

2 5 1 4 3

210 2

95 4 3

140 36

1 1 5 11

282 22

304 44

317 5

249 41

1 4 7 4 3

a 1 29

2 6 4 4

289 29

3 4 1 29

1 2 9 25

235 27

2 6 4 4

356 44

1 8 6 3 0

1 4 0 36

339 27

1 7 0 14

249 41

170 14

157 1

1 2 5 2 1

283 23

1 9 6 4 0

215 7

257 4 9

259 5 1

a7 35

192 36

269 9

1 7 6 20

SOLUTION 7 3 0 1 7 26 0 17 26

SOLUTION 7 4 0 1 7 4 1 0 1 7 4 1

SOLUTION 7 5 0 17 4 4 0 1 7 4 4

SOLUTION 76 0 1 7 85 0 1 7 33

SOLUTION 77 0 1 7 1 2 9 0 17 25

SOLUTION 78 0 1 9 24 0 1 9 24

SOLUTION 7 9 0 1 9 26 0 1 9 26

SOLUTION 80 0 1 9 4 1 0 1 9 4 1

SOLUTION 8 1 0 1 9 47 0 1 9 47

SOLUTION 82 0 1 9 48 0 1 9 48

SOLUTION 83 0 1 9 4 9 0 1 9 49

SOLUTION 8 4 0 1 9 5 1 0 1 9 5 1

SOLUTION 85 0 1 9 63


346 361 34 49

346 361 34 49

346 361 34 49

346 361 34 49

346 361 34 49

212 300 4 40

212 300 4 4 0

212 300 4 40

212 300 4 40

212 300 4 4 0

212 300 4 4 0

212 300 4 4 0

212 300

111 265 77 7 5 25

217 1 2 2 1 7 8 9 1 8 22

7 4 97 197 22 45 41

338 239 264 26 3 1 4

218 354 157 10 42 1

1 6 1 1 9 0 25 5 34 25

111 265 242 7 5 3 4

217 1 2 2 2 8 1 9 1 8 2 1

1 3 4 292 2 7 1 30 32 11

1 9 1 3 5 1 222 35 39 14

1 8 4 2 1 1 339 28 3 27

310 355 1 6 4 50 43 8

187 1 6 5 9 3

336 24

207 5 1

246 38

280 20

213 5

152 48

280 20

218 10

103 5 1

246 38

72 20

321 9

116

7 9 27

7 1 1 9

1 8 9 33

92 40

219 11

293 33

95 43

1 9 3 37

2 34 26

132 28

82 30

2 4 1 33

2 94

56 4

9 4 42

76 2 4

168 1 2

288 28

88 36

168 1 2

354 42

326 1 4

76 2 4

1 1 9 1 5

75 23

155

489

200 224 4 4 16

96 62 44 1 0

1 3 1 289 27 29

1 4 9 308 45 48

2 5 1 1 7 0 43 14

107 1 7 1 3 1 5

283 308 23 48

345 1 2 9 33 25

2 0 1 110 45 6

2 9 1 289 3 1 29

268 206 8 50

243 1 5 1 35 47

327 299 0 19 11 4 4 0 3 1 9 4 1 1 2 34 5 1 1 5 39

490

SOLUTION 86 0 1 9 6 4 0 1 9 1 2

SOLUTION 87 0 1 9 9 0 0 1 9 38

SOLUTION 88 0 2 1 23 0 2 1 23

SOLUTION 89 0 2 1 29 0 2 1 29

SOLUTION 90 0 2 1 30 0 2 1 30

SOLUTION 9 1 0 2 1 31 0 2 1 3 1

SOLUTION 92 0 2 1 1 1 3 0 2 1 9

SOLUTION 93 0 22 27 0 22 27

SOLUTION 94 0 22 31 0 22 31

SOLUTION 95 0 22 47 0 22 47

SOLUTION 96 0 22 59 0 22 7

SOLUTION 97 0 22 67 0 22 1 5

SOLUTION 98 0 22 103 0 22 5 1

212 4

212 4

305 4 5

305 45

305 45

305 45

305 45

262 2

262 2

262 2

262 2

262 2

262 2

300 4 0

300 4 0

55 3

55 3

55 3

55 3

55 3

1 9 9 43

1 9 9 43

1 9 9 43

1 9 9 43

199 43

199 43

276 1 6

232 2 4

334 2 2

203 47

320 8

98 4 6

270 10

1 8 0 24

98 4 6

1 3 4 30

80 28

78 26

326 1 4

A.R. Prince

83 31

159 3

290 30

227 1 9

53 1

1 0 9 5

332 20

229 2 1

1 0 9 5

292 32

114 1 0

3 3 3 2 1

110 6

7 3 2 1

115 11

308 48

196 4 0

8 1 29

249 4 1

1 7 0 1 4

28 28

82 30

315 3

152 48

76 24

198 42

1 2 4 20

197 41

1 4 4 40

327 1 5

340 28

1 6 0 4

285 25

155 5 1

3 4 1 29

1 2 9 25

353 4 1

127 23

345 33

205 4 9

215 7

280 20

112 8

176 20

192 36

288 28

252 44

268 8

162 6

88 36

246 38

1 7 4 1 8

1 0 2 50

1 8 9 33

209 1

2 9 4 34

1 6 6 10

2 2 1 1 3

1 2 1 17

299 39

267 7

218 1 0

286 26

363 5 1

281

295 35

307 47

1 6 8 1 2

1 4 0 36

323 11

1 0 0 4 8

213 5

8 4 32

339 27

1 3 5 3 1

1 7 1 1 5

289 29

335 2 1 23

301 4 1

1 3 1 27

325 13

93 4 1

137 33

1 7 3 1 7

362 50

1 1 6 1 2

3 1 1 5 1

354 4 2

255 47

358 46

1 9 3 37

SOLUTION 0 23 0 23

SOLUTION 0 23 0 2 3

SOLUTION 0 2 3 0 2 3

SOLUTION 0 23 0 2 3

SOLUTION 0 2 3 0 23






SOLUTION 0 25 0 25

SOLUTION 0 25 0 25

SOLUTION 0 25


99 25 334 290 25 22 30

100 33 334 290 33 22 30

101 38 334 290 38 22 30

102 45 334 290 45 22 30

1 0 3 62 334 290 10 22 30

1 0 4 45 1 6 1 1 9 0 45 5 34

1 0 5 65 1 6 1 1 9 0 1 3 5 34

106 67 1 6 1 1 9 0 1 5 5 34

107 9 1 1 6 1 1 9 0 39 5 34

108 100 1 6 1 1 9 0

48 5 34

1 0 9 30 293 107 30 33 3

110 4 1 293 107 4 1 33 3

111 4 4 293 107

293 33

70 1 8

141 37

54 2

94 42

54 2

1 8 1 25

78 26

273 1 3

249 4 1

320 8

217 9

7 4

107 310 1 4 6 355 275 5 1 3 50 4 2 43 1 5 5 1

1 6 3 2 5 1 1 6 2 219 315 157 7 4 3 6 11 3 1

148 41 58 217 233 122 4 4 41 6 9 25 1 8

105 285 222 1 2 1 1 3 2 362 1 25 1 4 17 28 50

207 206 271 119 234 72 5 1 5 0 11 1 5 26 20

1 0 5 329 7 9 344 77 347 1 1 7 27 32 25 35

220 2 4 1 146 243 275 1 6 4 1 2 33 42 35 1 5 8

333 218 1 4 2 354 69 1 2 9 2 1 1 0 38 42 17 25

1 8 2 1 0 2 357 1 2 4 254 301 26 5 0 45 20 46 41

1 9 2 270 313 332 304 1 1 3 36 10 1 20 4 4 9

53 3 1 158 98 230 1 0 9 1 3 1 2 46 22 5

1 2 2 210 142 125 69 99 1 8 2 38 2 1 17 47

97 237 325 231 209 236

49 1

136 32

135 3 1

66 1 4

291 3 1

2 0 1 45

200 4 4

136 32

274 1 4

216 8

259 5 1

277 17

2 7 4 1 4

1 4 4 1 28 4 0 0 25 4 4 33 3 22 45 2 9 1 3 23

SOLUTION 112 0 25 112 0 25 8

SOLUTION 1 1 3 0 26 6 4 0 26 1 2

SOLUTION 1 1 4 0 26 82 0 26 30

SOLUTION 115 0 27 32 0 27 32

SOLUTION 116 0 27 57 0 27 5

SOLUTION 117 0 27 6 4 0 27 12

SOLUTION 118 0 27 7 1 0 27 1 9

SOLUTION 1 1 9 0 28 30 0 28 30

SOLUTION 1 2 0 0 28 4 1 0 28 4 1

SOLUTION 1 2 1 0 28 56 0 28 4

SOLUTION 122 0 28 58 0 28 6

SOLUTION 123 0 28 9 1 0 28 39

SOLUTION 1 2 4 0 28 143 0 28 39

293 33

111 7

111 7

180 24

1 8 0 24

180 24

1 8 0 24

252 4 4

252 4 4

252 4 4

252 4 4

252 4 4

252 4 4

107 3

265 5

265 5

229 2 1

229 2 1

229 2 1

229 2 1

84 32

84 32

84 32

84 32

84 32

84 32

1 4 0 36

276 1 6

268 8

1 4 5 4 1

272 1 2

276 1 6

96 4 4

320 8

217 9

224 1 6

233 25

273 1 3

316 4

A.R. Prince

196 4 0

83 3 1

339 27

302 42

172 1 6

83 3 1

178 22

53 1

122 1 8

336 24

66 1 4

182 26

303 4 3

270 1 0

167 11

250 42

33 33

347 35

157 1

1 0 1 49

315 3

89 37

280 20

348 36

215 7

243 35

342 30

1 7 4 1 8

362 50

160 4

80 28

358 46

1 2 4 20

1 5 1 47

232 24

112 8

8 1 29

193 37

335 23

332 20

298 38

330 1 8

7 0 1 8

344 32

219 11

348 36

162 6

225 17

168 1 2

101 49

115 11

164 8

177 2 1

335 23

1 2 1 17

2 2 1 1 3

59 7

363 5 1

1 0 2 50

3 2 1 9

1 5 9 3

1 4 0 36

1 7 6 20

345 33

1 9 8

113 240 9 32

356 198 4 4 42

309 285 4 9 25

1 6 3 173 7 17

329 1 1 4 1 7 10

2 5 1 127 43 23

1 6 9 301 1 3 4 1

135 75 3 1 23

235 9 0 27 38

308 196 4 8 4 0

1 6 9 323 1 3 11

307 2 8 1 47 2 1

2 4 1 1 7 4 42 33 1 8

SOLUTION 1 2 5 0 2 9 3 1 0 2 9 3 1

SOLUTION 1 2 6 0 29 39 0 29 39

SOLUTION 127 0 2 9 57 0 2 9 5

SOLUTION 1 2 8 0 3 0 32 0 3 0 32

SOLUTION 1 2 9 0 3 0 38 0 3 0 38

SOLUTION 1 3 0 0 3 0 5 1 0 30 5 1

SOLUTION 1 3 1 0 3 0 63 0 3 0 11

SOLUTION 1 3 2 0 3 0 6 8 0 3 0 1 6

SOLUTION 1 3 3 0 3 0 92 0 3 0 4 0

SOLUTION 1 3 4 0 32 56 0 32 4

SOLUTION 135 0 32 6 9 0 32 1 7

SOLUTION 136 0 32 7 1 0 32 1 9

SOLUTION 137 0 32 1 2 1 0 32 1 7

203 47

203 47

203 47

320 8

320 8

320 8

3 20 8

320 8

320 8

145 41

14 5 41

14 5 4 1

145 4 1


227 1 9

227 1 9

227 1 9

53 1

53 1

53 1

53 1

53 1

5 3 1

302 42

302 4 2

302 4 2

302 4 2

98 1 0 9 46 5

1 8 3 248 27 4 0

272 1 7 2 1 2 1 6

1 4 5 302 41 4 2

141 1 4 8 37 44

310 355 50 4 3

187 1 6 5 3 1 9

247 263 39 3

1 4 9 2 6 4 45 4

224 336 1 6 2 4

274 1 4 2 14 38

96 1 7 8 4 4 22

362 285 50 25

.316 4

257 4 9

2 8 1 21

317 5

205 4 9

335 23

100 4 8

1 7 1 1 5

1 2 4 2 0

1 9 3 37

1 6 2 6

215 7

257 4 9

1 5 2 4 8

1 6 8 1 2

1 1 3 9

1 5 3 4 9

336 2 4

85 33

193 37

1 7 8 22

237 29

222 14

363 5 1

280 20

267 7

303 43

1 8 6 30

1 9 3 37

245 37

295 35

1 9 8 4 2

249 41

88 36

102 50

345 33

315 3

115 11

186 30

88 36

308 4 8

2 7 0 1 0

202 46

56 4

338 26

345 33

7 1 1 9

2 3 1 23

1 3 2 28

358 46

1 6 8 1 2

3 1 1 5 1

1 4 3 39

282 22

345 33

87 35

7 3 2 1

1 7 4 1 8

1 9 2 36

1 5 2 4 8

3 0 1 41

2 8 1 2 1

1 3 5 3 1

307 47

282 22

493

1 7 1 1 5

280 20

332 20

337 25

224 1 6

239 3 1

2 8 1 2 1

96 4 4

236 28

2 9 1 3 1

1 2 7 23

308 4 8

3 4 1 29

494 A.R. Prince

SOLUTION 0 34 0 3 4

SOLUTION 0 34 0 34

SOLUTION 0 37 0 37

SOLUTION 0 37 0 37

SOLUTION 0 37 0 37

SOLUTION 0 37 0 37

SOLUTION 0 37 0 37

SOLUTION 0 38 0 38

SOLUTION 0 39 0 39

SOLUTION 0 39 0 39

SOLUTION 0 41 0 41

SOLUTION 0 41 0 41


1 3 8 4 3 284 343 1 9 4 4 3 2 4 31 38

1 3 9 92 284 343 1 4 9 4 0 2 4 3 1 45

1 4 0 56 130 331 2 2 4 4 26 1 9 1 6

141 58 130 331 233

6 26 19 25

1 4 2 95 130 331 283 43 26 1 9 23

1 4 3 1 0 2 130 3 3 1 1 2 4

50 26 1 9 2 0

144 1 1 2 130 331 1 4 0

8 26 1 9 36

1 4 5 57 1 4 1 1 4 8 272

5 37 4 4 1 2

146 4 9 1 8 3 248 1 8 4 4 9 27 4 0 28

147 6 4 1 8 3 248 276 1 2 27 4 0 1 6

1 4 8 4 9 217 122 1 8 4 49 9 1 8 28

1 4 9 1 1 2 217 1 2 2 1 4 0

8 9 1 8 36

1 5 0 45 1 9 4 1 1 8 5 4 45 38 14 2

1 1 8 9 4 353 207 286 62 255 14 4 2 41 5 1 26 10 47

2 6 4 267 100 311 249 3 4 1 1 9 2 4 7 4 8 5 1 41 29 36

336 249 337 1 9 2 202 1 0 0 1 5 3 2 4 41 25 36 46 48 4 9

6 6 342 92 177 1 4 9 240 2 6 4 14 30 4 0 2 1 45 32 4

242 270 103 332 326 1 1 3 110 3 4 10 5 1 20 14 9 6

3 0 1 218 257 354 1 8 6 1 2 9 282 41 10 4 9 4 2 30 25 22

1 9 6 1 8 8 358 147 363 269 127 4 0 32 46 4 3 5 1 9 23

1 7 2 250 338 330 239 309 85 1 6 4 2 26 1 8 31 4 9 3 3

2 1 1 267 178 3 1 1 7 1 3 4 1 96 3 7 22 5 1 1 9 29 4 4

83 332 1 4 6 270 275 1 1 3 1 3 6 3 1 2 0 4 2 10 1 5 9 32

2 1 1 216 347 357 344 254 329 3 8 35 45 32 46 1 7

1 9 6 137 219 340 2 5 1 1 6 6 157 4 0 33 11 28 4 3 1 0 1

1 0 5 3 3 0 1 6 6 309 137 250 340 1 1 8 1 0 4 9 3 3 42 28

SOLUTION 1 5 1 0 4 3 48 0 4 3 4 8

SOLUTION 152 0 4 3 7 1 0 4 3 1 9

SOLUTION 1 5 3 0 4 3 100 0 4 3 4 8

SOLUTION 1 5 4 0 44 69 0 44 1 7

SOLUTION 1 5 5 0 44 112 0 44 8

SOLUTION 1 5 6 0 44 1 3 6 0 44 32

SOLUTION 157 0 4 5 56 0 4 5 4

SOLUTION 1 5 8 0 4 5 6 4 0 4 5 1 2

SOLUTION 1 5 9 0 47 63 0 4 7 11

SOLUTION 1 6 0 0 4 8 77 0 4 8 25

SOLUTION 1 6 1 0 4 9 92 0 4 9 4 0

SOLUTION 1 6 2 0 4 9 1 2 1 0 4 9 1 7

SOLUTION 1 6 3 0 5 1 6 4 0 5 1 1 2

1 9 4 38

1 9 4 38

1 9 4 38

7 4 22

7 4 22

7 4 22

5 4 2

5 4 2

1 3 4 30

1 9 1 35

1 8 4 28

1 8 4 28

310 50


118 1 4

118 14

118 1 4

97 45

97 45

97 45

105 1

1 0 5 1

292 32

3 5 1 39

2 1 1 3

211 3

355 4 3

1 9 1 3 5 1 4 9 35 39 4 9

96 1 7 8 295 44 22 35

249 1 9 2 315 41 36 3

274 1 4 2 337 14 38 2 5

1 4 0 1 9 6 2 9 1 36 40 3 1

275 1 4 6 1 9 3 1 5 4 2 37

224 336 3 1 1 1 6 2 4 5 1

276 83 257 1 6 3 1 4 9

187 1 6 5 232 3 1 9 2 4

7 9 200 2 5 1 27 44 4 3

1 4 9 2 6 4 243 45 4 35

362 285 1 6 8 50 25 1 2

276 83 1 1 2 1 6 3 1 8

176, 20

127 23

215 7

1 5 1 47

307 47

308 4 8

342 30

345 33

1 6 4 8

275 1 5

167 11

255 47

255 47

1 8 4 28

7 3 2 1

1 6 2 6

202 46

222 14

345 33

3 4 1 29

1 8 6 30

1 5 9 3

219 11

1 6 4 8

308 4 8

1 4 0 36

8 1 29

363 5 1

1 1 5 11

3 2 1 9

115 11

280 20

177 2 1

2 8 1 2 1

2 4 1 33

1 4 6 4 2

298 38

286 26

286 26

495

2 1 1 323 3 11

205 358 4 9 46

1 3 5 307 3 1 47

1 5 3 7 5 4 9 23

1 3 2 215 28 7

2 8 1 1 6 8 2 1 1 2

267 240 7 32

282 1 9 3 22 37

90 243 38 35

157 1 3 6 1 32

2 4 1 356 33 44

280 353 20 41

1 9 6 353 4 0 41

496

SOLUTION 1 6 4 0 5 1 67 0 5 1 1 5

SOLUTION 1 6 5 0 56 6 3 0 4 11

SOLUTION 1 6 6 0 56 7 5 0 4 23

SOLUTION 167 0 56 85 0 4 33

SOLUTION 1 6 8 0 56 88 0 4 36

SOLUTION 1 6 9 0 56 1 1 5 0 4 11

SOLUTION 1 7 0 0 57 67 0 5 1 5

SOLUTION 1 7 1 0 57 7 2 0 5 20

SOLUTION 1 7 2 0 57 7 6 0 5 2 4

SOLUTION 173 0 57 101 0 5 4 9

SOLUTION 1 7 4 0 58 7 3 0 6 2 1

SOLUTION 1 7 5 0 58 115 0 6 11

SOLUTION 1 7 6 0 5 9 90 0 7 38

310 50

224 1 6

224 1 6

224 1 6

224 1 6

224 1 6

272 1 2

272 12

272 1 2

272 1 2

233 25

233 25

80 28

355 4 3

336 2 4

336 2 4

336 2 4

336 2 4

336 2 4

1 7 2 1 6

1 7 2 1 6

1 7 2 1 6

1 7 2 1 6

66 14

66 14

114

7 8 2 6

187 3 1

1 5 1 47

338 26

1 5 2 4 8

215 7

7 8 26

1 1 9 1 5

246 38

348 36

205 4 9

215 7

232

A.R. Prince

333 2 1

1 6 5 9

3 2 1 9

239 3 1

1 7 1 1 5

307 47

333 2 1

206 5 0

289 29

1 6 9 1 3

295 35

307 4 7

1 5 9

236 28

279 1 9

268 8

259 51

2 0 1 45

1 3 6 32

285 25

7 5 23

269 9

1 3 1 27

76 2 4

330 1 8

157 1 0 2 4 3 1

1 6 8 1 2

317 5

356 4 4

283 23

358 46

1 7 0 14

1 9 6 4 0

92 4 0

357 45

1 5 4 50

93 41

230 22

168

2 3 1 308 237 280 23 4 8 29 2 0

1 5 4 245 2 5 3 87 5 0 37 4 5 35

82 298 339 167 30 38 27 11

304 2 4 2 313 95 44 3 4 1 4 3

234 363 2 7 1 127 26 5 1 11 23

275 288 1 4 6 213 1 5 28 4 2 5

1 2 1 1 1 2 362 140 1 7 8 5 0 36

1 5 1 1 4 9 3 2 1 2 6 4 47 45 9 1.

147 2 5 4 1 8 8 216 4 3 46 32 8

1 9 7 253 1 8 9 279 41 4 5 33 1 9

246 2 9 4 289 327 38 34 2 9 1 5

309 277 250 1 5 8 4 9 1 7 4 2 2

219 308 2 5 1 280 1 2 11 4 8 4 3 2 0

SOLUTION 1 7 7 0 6 2 8 9 0 10 37

SOLUTION 1 7 8 0 6 3 68 0 11 1 6

SOLUTION 1 7 9 0 63 8 2 0 11 3 0

SOLUTION 1 8 0 0 6 4 88 0 1 2 36

SOLUTION 1 8 1 0 65 8 7 0 1 3 35

SOLUTION 1 8 2 0 67 100 0 1 5 4 8

SOLUTION 1 8 3 0 68 7 5 0 1 6 23

SOLUTION 1 8 4 0 69 85 0 1 7 33

SOLUTION 1 8 5 0 7 1 8 1 0 19 29

SOLUTION 1 8 6 0 7 1 1 1 2 0 19 8

SOLUTION 1 8 7 0 7 2 1 3 5 0 2 0 3 1

SOLUTION 1 8 8 0 7 3 1 3 5 0 2 1 3 1

SOLUTION 189 0 7 5 90 0 23 3 8

9 4 4 2

187 3 1

1 8 7 3 1

276 1 6

1 8 1 25

7 8 26

247 39

274 14

9 6 4 4

96 4 4

119 1 5

205 4 9

1 5 1 4 7


207 5 1

1 6 5 9

1 6 5 9

8 3 3 1

220 1 2

333 2 1

263 3

142 38

1 7 8 22

1 7 8 22

206 50

295 35

3 2 1 9

225 235 242 2 9 1 1 7 27 3 4 3 1

247 263 69 1 9 6 39 3 1 7 40

268 339 275 363 8 27 1 5 5 1

1 5 2 1 7 1 225 254 4 8 1 5 1 7 4 6

245 317 327 264 37 5 1 5 4

249 1 9 2 1 3 7 230 4 1 36 33 22

1 5 1 3 2 1 2 9 1 329 47 9 3 1 1 7

338 239 2 5 4 1 8 6 26 3 1 4 6 3 0

1 7 6 323 299 210 20 11 3 9 2

1 4 0 1 9 6 288 1 9 3 36 40 2 8 37

1 6 2 315 259 237 6 3 5 1 29

1 6 2 315 1 6 7 280 6 3 11 20

232 1 5 9 9 3 110 24 3 4 1 6

95 222 4 3 1 4

274 1 1 2 14 8

1 4 6 358 4 2 4 6

89 216 37 8

294 92 3 4 40

340 277 28 1 7

222 3 4 4 14 3 2

216 282 8 22

1 5 5 1 2 5 5 1 2 1

213 345 5 33

304 2 3 1 44 23

298 168 38 1 2

294 1 0 3

497

283 1 3 2 23 28

1 4 2 1 4 0 38 36

1 3 6 1 2 7 32 23

235 357 27 4 5

93 1 4 9 4 1 4 5

1 6 6 1 5 8 10 2

1 3 2 347 28 35

357 257 4 5 4 9

1 1 6 99 1 2 4 7

1 7 0 2 8 1 1 4 2 1

313 236 1 28

356 308 4 4 4 8

327 326 3 4 5 1 1 5 1 4

498 A.R. Prince

SOLUTION 1 9 0 0 76 8 9 246 289 225 235 137 280 340 1 6 8 1 6 6 308 0 2 4 37 38 29 1 7 27 33 2 0 28 1 2 10 48

SOLUTION 1 9 1 0 8 1 88 1 7 6 323 1 5 2 1 7 1 304 342 313 1 7 7 2 5 9 240 0 29 36 20 11 4 8 1 5 44 30 1 2 1 5 1 32

SOLUTION 192 0 8 2 87 268 339 245 317 88 215 1 5 2 1 1 5 1 7 1 307 0 30 35 8 27 37 5 36 7 4 8 11 1 5 47

SOLUTION 1 9 3 0 89 113 225 235 270 332 250 279 330 1 5 4 309 253 0 37 9 17 27 10 20 4 2 1 9 1 8 50 4 9 45

SOLUTION 1 9 4 0 90 1 5 3 232 1 5 9 202 337 277 255 230 286 1 5 8 353 0 38 4 9 2 4 3 4 6 25 17 47 22 26 2 41

SOLUTION 1 9 5 0 1 1 2 1 5 3 1 4 0 1 9 6 202 337 329 2 3 4 344 2 7 1 347 2 0 1 0 8 4 9 36 4 0 46 25 1 7 26 32 11 35 4 5

SOLUTION 1 9 6 0 1 2 9 144 218 3 5 4 209 325 147 1 6 4 1 8 8 2 4 1 269 243 0 25 4 0 10 42 1 1 3 4 3 8 32 33 9 35

APPENDIX 2

In this appendix, we list which of the solutions given in Appendix 1

are in the same <B,a>-orbit. For each orbit table, the number in

brackets to the left of an entry in the first column gives the power of 8

which translates the first entry in the column to that entry. The action

o f a on the entries of the table is also described in each case

the flag-transitive afine planes of order 27 499

Orbit 1

1 0 9 118 1 5 4 157 1 5 9 162

15 45 25 6 4 62 50

115 185 3 1 88 48 130

1 8 9 1 9 3 4 0 105

177 137 188 1 4 9

4 17 1 4 4 7 3 148 12

68 1 5 1 102 1 3 5 1 1 9 1 2 0

58 3 178 6 0 1 9 4 1 4 2

94 1 7 4 5 2 1

1 5 2 77 10 2 4 47 2 9

1 6 4 1 9 5 26 35

196 7 4 11 1 8

126 114

55 4 4 99 65 90

167 1 0 4

For t h i s t a b l e , the ac t ion of a is given by reading along the rows from

l e f t t o r i g h t , the l a s t en t ry being mapped back t o the f i r s t .

Orbit 2

2 1 6 23 63 52 132 85 1 7 3 95 101

103 1 8 4 112 166 117 72 133 39 140 7 9 146 7 1 156 155 192 34

43 57 6 1

1 2 5 1 4

138 1 9 0

8 4 108 1 7 0 1 6 0 1 3 4

49

93 122 100 110 147 7 0 1 8 1 32

27 67 143 36

54 1 2 9 6 2 0

83 172 168 76 111 1 8 3 136 1 8 6 1 3 1 87

1 2 4 8

82 9 1

150 106 176

46 53

1 6 3 59

1 6 9 127

The ac t ion of a is given by the rows of the t a b l e , a s f o r o r b i t 1.

500 A.R. Prince

Orbi t 3 Orbi t 4

7 1 9 4 2 116

1 9 1 1 6 1

28 66 1 5 8 1 5 3

80 9

33 92 179 75

5 1 128

78 1 7 1 1 8 0 1 3 9

86 38

9 8 1 7 5

- - - - - - - -

- - - - - - - -

- - - - - - - -

- - - - - - - -

Orbi t 5

For o r b i t 3 , the ac t ion of a is indica ted by the d iv i s ions with dot ted l i n e s .

Within each group of s i x , the p a t t e r n is i l l u s t r a t e d by

7 -+ 1 9 -+ 42 + 116 -+ 1 9 1 --j 1 6 1 4 7 , while 9 8 and 1 7 5 a r e interchanged.

For o r b i t 4 , t he ac t ion of a is a l s o ind ica ted by the dot ted l i n e s . Within

each group, each en t ry i s mapped t o the one below, except t h a t the l a s t

en t ry i s mapped t o the f i r s t .

Combinatorics '90 A. Barlotti el al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. 50 1

SYMMETRIC FUNCTIONS AND BIJECTIVE IDENTITIES

Domenico Senato* and Antonietta Venezia**

*Dipartimento di Matematica, Universiti della Basilicata, Potenza, Italy

Dipartimento di Matematica, Universiti "La Sapienza", Roma, Italy

* *

Introduction One of the latest interests of Combinatorics is the

development of a systematic theory of bijective proofs. In this context the idea of proving identities among symmetric functions by bijective arguments comes in again. The first attempt of a systematic approach to this point of view is thanks to G.C. Rota ( 3 1 , resumed later by P. Doubilet ( 2 ) and developed by F.Bonetti, G.C. Rota, D. Senato and A. Venezia in a paper which is yet to be published (1). In (1) two complementary notions are introduced: Formal Polynomials and Polynomial Species that are the key of a categorical setting for bijective proofs of identities among symmetric functions.

In this paper a more general notion of Polynomial Species is introduced, which permits us to extend the techniques which we have used in (1) to the symmetric functions i n two sets of indipendent variables. More specifically we give bijective proofs of the following identities:

n l/l-xy = 2 Sil(X) S A XtY I

Y) l / t I l f

1. FORMAL POLYNOMIAL AND POLYNOMIAL SPECIES

Let X be a set. A m u l t i s e t m on X is a pair (X,iii:X--+Z) where 5 is a function with integer values such that fil(x) 2 0 for any xrX. The suppor t of a multiset m on X is the set:

supp(m) = (xex : E ( x ) > O l

finite set. A f i n i t e m u l t i s e t is a multiset whose support is a

The c a r d i n a l i t y o f t h e f i n i t e m u l t i s e t m is the

502 D. Senato, A. Venezia

following integer:

Iml = 2 %(XI xesupp (m)

Let 21x1 be the ring of all polynomials in the variables of the set X with integer coefficients. A basis of the module Z[Xl is given by all monomials

x m = n p (XI xesupp (m)

When F is a finite subset of X , we denote by e F the endomorphism of the ring Z[Xl which is defined as follows:

F(P) = q

where the polynomial q is obtained from the polynomial p by setting at zero all coefficients of all monomials not supported on F.

Let J be a directed set. Let's say that a family:

of polynomials is Cauchy when the family of polynomials

is eventually constant for every finite subset F of X. This condition for the directed sets defines a Hausdorff topology on the ring Z[Xl, this ring therefore becomes a topological ring.

The r i n g of formal polynomials Z[ (XI 1 is defined to be the completion of the topological ring 21x1.

It is possible to prove that every formal polynomial f is a convergent sum of monomials, i.e.

f = B coeff<f,m> xm. m

as m ranges over all finite multisets on X (see.1). We now give the notion of polynomial species. Let Ens

be the category of sets and functions, and let B be the category of the finite sets and bijections.

Set W = X U Y where X and Y are sets of variables such that X f l Y = 0 . We denote the elements of the set X by x and the elements of the set Y by y.

We define a covariant functor HX from B to Ens by setting :

HX[E] = If: A-X : A G E 1

functor from B to Ens that associates the set For all functors M: B-Ens we denote by Pol(M) the

Symmetric functions and bijective identities 503

to every finite set E.

subfunctor of Pol (MI.

define the functor

We shall write P c_ Pol(M) whenever the functor P is a

Let P E Pol(M) and let F be a finite subset of W. We

EF(P): B-Ens

by setting

EF(P)[E] = ((S,f,g)LP[E]: I m f E F fl X, I m g s F fl YI

where Imf denotes the set of all elements f(a), whenever f is a function from A to X.

A po lynomia l spec ie s P w i t h c o e f f i c i e n t s on M is defined to be a subfunctor of the functor Pol(M) such that the set:

is finite for any finite set E and for any finite subset F of w.

A polynomial species P is said to be symmetric i n X and Y when for every bijection 0 : X-X and for every bijection p: Y-Y we have:

(s,aof,pog) L PIE1

whenever (s,f,g) t P[El. We associate a formal power series with coefficients in

the ring Z[ (W)] of formal polynomials to every polynomial species. If f: A-X and A is a finite set, we denote by gen(f) the monomial in Z[Wl :

gen(f) = n f(a) = n ,If'(x) I aLA xrx

Now, set

gen(s,f,g) = gen(f)gen(g)

where (s,f,g) is an ordered triple, with s an element of an arbitrary set.

(gen ( 6 , f , g) 1

as (s,f,g) ranges in P[E], is summable in Z[(W)]. We can therefore set

gen (P [El 1 = 2 gen(s,f ,g). (s,f ,g) rP[El

Note that gen(P[E]) depends only on the cardinality of E. We can therefore write

Let P be a polynomial species. The family:


gen(P[E]) = gen(P,n) with lEl= n.

s p e c i e s P to be the formal power series:

Gen(P,z) = 2 gen(P,n) zn/n!

The c a t e g o r y of polynomial s p e c i e s is defined by setting Hom(P,Q) to be the set of all natural equivalences I between P and Q whose components zE: PIE] --[El are bijections such that

We define the g e n e r a t i n g f u n c t i o n of the p o l y n o m i a l

n2O

rE(s,f,g) = (t,f,g).

We write P = 0 , when P and Q are naturally equivalent in the category of polynomial species. Clearly if P = 0 then

Gen(P,z) = Gen(Q,z).

We define sum and product of polynomial species. Let I be a finite set. The sum

B Pi it1

of the family of polynomial species (Pi)ieI is defined as follows :

B Pi[El = I((s,i),f,g): (s,f,g)tPi[El for some ieII. it1

The product of polynomial species is defined using the notion of composition. A c o m p o s i t i o n of a set E idexed by a set I is a function k : I-P(E) such that:

i) k(i) n k(j) = @ if i # j

ii) u k(i) = E. it1

The p r o d u c t n Pi is defined as follows. For every

finite set E, let n Pi[El be the set of triples (s,f,g) it1

it1 obtained by the following steps:

a) Choose a composition k of E indexed by the elements of I.

b) Choose (si,fi,gi) t Pi[k(i)l, for every ieI.

c) Set s = (k,(si)irI), define f and g to be the functions

whose restrictions to the set k(i) are fi and gi respectively.

Passing to generating functions, we have:


and

Gen( n Pi,z) = n Gen(Pi,z). it1 it1

We denote by 1 the polynomial species defined as follows :

lt0l = I(O,fO:O-X,gO:O-Y)l: 1[El= 0

Let I be an infinite set. A family (Pi)itI of polynomial species is said to be m u l t i b l i a b l e when

i) PiLC1 = 0 for every ieI,

ii) the set I(n,F) = lie1 : EF(Pi)[E] # 01 is finite, for

if E ?L 0.

every F and n = IEl.

Thus the polynomial species:

I'I (l+Pi) [El = it1

u l(s,f,g)e( 17 l+Pi) [El : Imf = FnX, Img = FnYl F it1 (n,F)

as F ranges on the set of all finite subsets of W, is the product of the family (Pi)ieI(see.l).

The generating function of the product I'I (l+Pi) is irI

Gen( n l+Pi,z) = I'I Gen(l+Pi,z). it1 irI

2. THE SYEMETRIC FUNCTIONS kA,hA,SA.

Recall that a partition of a positive integer n is a weakly decreasing sequence:

A = ( A , , A 2 , . . . A , )

+ A 2 L...L 1 , L 0

of positive integers, i.e.:

such that:

A , + A 2 + . . . + A , = n.

The integers A i are called p a r t s of A and the number of

506 D . Senato, A. Venezia

the parts l e n g t h of A . Sometimes it is convenient to use the notation which indicates the number of times each integer occur as part:

1 = (lrl, zr2, . . . 1

means that exactly Let E be a finite set. To any partition n of E, we

associate the partition cl(n) of the integer IEl whose length is In1 and which parts are the cardinalitxes of the blocks of n. Further to any function f from E, we assign a partition Kerf of E, by putting e' and el' in the same block if f ( e ' ) - f (e").

Let A = ( A , ,A2 , ... L q ) = (lrt 25... be a partition of the integer n.

The monomial symmetric f u n c t i o n k;z(X) is defined as follows:

ri parts of A are equal to i.

where the sum ranges over all distinct monomials with distinct indices. The (kA (XI ) ( as A runs through all partitions of the integer n) form a 2-basis of the module of symmetric functions.

Let r be an integer. The complete homogeneous f u n c t i o n hr(X) is defined as follows:

hr(X) = 2 xm

where the sum ranges over all multisets m such that Iml = r. m

The symmetric function hA(X) is defined by:

When A ranges over all partitions of the integer n, (hA(X)) form a 2-basis for the module of symmetric functions.

The power sum symmetric f u n c t i o n sr(X) is defined by:

Sr(X) = B xr xex

and the symmetric function s*(X) :

3. BIJECTIVE IDENTITIES.

We define the polynomial species Hx,yf for every (x,y) in the set XxY, as follows:

Y,,y[Ol = 0 and &,y[El = l(o,f,g)l

where :

i) a is a permutation of E,


ii) f: E-X is the function taking constant value x,

iii) g: E-Y is the function taking constant value y.

The generating function of the polynomial species l+%,y is :

Gen(l+Hx,y,z) = 2 n! xnyn zn/n! = l/l-xyz. n2 0

The family

(%,y) (x,y) rXxY

is multipliable and the product is the species defined by:

l+&,y[El as (x,y) ranges in (FnX)x(FnY): V F XI:

Imf = FnX, Img = FnYl,as F ranges over all finite subset of W.

The generating function of the polynomial species

Gen( n l+%,y,z) = n l/l-xyz. XtY X,Y

It is easy to see that the species n l+&,y is XPY

naturally equivalent to the s p e c i e s B of d i s p o s i t i o n s , defined as follows:

H[#l = uO,fO:O-X1gO:O-Y)l and HIE] = l(d,f,g)l

where :

i) f is a function from E to X I g is a function from E to Y

ii) d = (ax,y)x,y is a family of permutations such that

is a permutation of the set E(x,y) = IerE: f(e)=x,g(e)=yl

Thus we have:

Gen(H,z) = n l/l-XYZ. XIY

The polynomial species S is defined as follows:

s [ 0 ] = i(0,f0:0-x,g0:04Y)1 and S[El = l(a,f,cr)l

where :


i) a is a permutation of the set E,

ii) f is a function from E to X, g is a function from E to Y,

iii) Kerf 2 n ( a ) and Kerg 2 n ( o ) , where n(o) is the partition of E determined by the cycles of 0 .

The generating function of the polynomial species S is:

Gen(S,z) - 1 + 2 2 sI(X) sI(Y) zn / [ A 1

where 121 = lrl r,! 2'2 r$ .. . and n!/[i l l is the number of permutations of E of class I .

3.1. Proposition. S = H.

Proof. The natural equivalence between the species S and B is defined by the following bijection. We associate to the triple (o,f,g) e S[El the disposition (d,f,g) where d = (ax,y) and ax is the permutation of E(x,y) defined by:

oXly(e) = o(e).

Every permutation is well-defined since

~ ( x , y ) = f-l(x) n il(y)

classical identity:

X # Y I

nll Icn

IY

and n ( o ) s Kerf A Kerg.

We have thus provided a bijective interpretation of the

n l/l-xy = 2 S L ( X ) S I ( Y ) l/tIl

In order to proof the identity:

we denote by R the polynomial species defined as follows:

R [ I ] = I(0,f0:@ -X,gz:c -Y)l and RtEI = I(n,v),f,g)l

where :

i) n is a partition of E,

ii) f is a function from E to X and g is a function from E to Y such that Kerg = n

iii) p is a family of permutations each defined on a block of Kerf A n,

The generating function of polynomial species R is:


where :

A ! = A,! A * ! . . ' j l & = r,! r,! ...

and n!/ A ! 111 is the number of partitions of class A . 3.2. Proposition, R = B.

Proof. The map that to every element ((n,v),f,g) of R[El associates the disposition (p,€,g) is a natural bijection.

We have thus obtained a bijective interpretation of the classical identity:

n l/l-xy = 2 hA(X) k A ( Y ) XeY A

4.

1

2

3

4

REFERENCES

Bonetti F., Rota G.C., Senato D., Venezia A.: "On the foundations of combinatorial theory X: A categorical setting for Symmetric Functions", to appear, Studies in App. Math. Doubilet P.: "On the foundations of combinatorial theory VII: Symmetric Functions through the theory of Distribution and Occupancy". Studies in App. Math. v. 51 (1972). Rota G.C.: "Baxter Algebra and combinatorial identities 1,II" Bull. Amer. Math. SOC. 325-329 (19691, 330-334 (1969). Stanley R.: "Theory and Application of Plane Partitions" part 1 and 2, Studies in App. Math. v.L, n.2 and 3 (1971).


Combinatorics '90 A. Barlotti et al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. 5 1 1

On the Existence of Nearly Kirkman Systems

Hao Shen

Department of Applied Mathematics, Shanghai Jiao Tong University, Shanghai 200030, China

Abstract In this paper, existence and construction of nearly Kirkinan systems are discussed.

In the case k = 4, it is proved that there exists a nearly Kirkman system NICS(2,4; u ) if and only if (1 O(mod 12) and 11 2 24, with at most 16 possible esceptions.

1. INTRODUCTION

A balaiiced incomplete block design B(k , A; v ) is an ordered pair (X, B) where S is a finite set containing 21 points a.nd B is a collection of k-subsets (called blocks) of X such tha.t each pair of distinct points of X is contained in exactly X blocks. When X = 1, a B( k , 1; v ) is usually called a Steiner system and denoted S(2, k ; v).

A parallel class in it B(k , A; u ) is a set of blocks which partitions X. A B ( k , X; v ) is called resolvable and clenoted RB(X:, X; v ) if the blocks can be partitioned into parallel classes. A resolvable Steiner system RS(2, k ; v ) is also known as a Kirkman system and denoted IiS( 2. k ; v).

Let A' and hd be given sets of positive integers. A group divisible design GD(IC, M ; 0)

is a n ordered triple ( r . G , B) where -Y is a ?!-set, G is a. set of subset,s (called groups) of X, G partitions X and ]GI E A4 for each G E G , and B is a set of subsets (called blocks) of X such that IBJ E Ii for each B E B, JB n GI _< 1 for every B E B and every G E G , aid each pair of points from distinct groups is contained in a unique block. If I< = { k } and A4 = { n z } , then a G D ( { k } , { m } ; v ) is simply denoted G D ( k , l ? l ; t ~ ) . A GD(IC, M ; 1 1 ) is called resolvable and denoted RGD(I<, M ; v) if the blocks can be partitioned into paallel classes. A transversal design T D ( k , m) is a GD( k , m; v ) with v = k7n. A resolvable T D ( k , m ) is denoted R T D ( k , m ) .

According to the above definitions, a I<irkman system ICS(2, k ; 21) then is a RGD(k, 1; L J ) if we take ea,ch point, as a group or an RGD(k, k ; v ) if we take the blocks of a fixed parallel class as groups. For m = k - 1, an RGD(k , k - 1; v) is also known as a nearly Kirkman system and denoted NIiS(2, k ; u ) .

The existence and construction of Kirkinan systems has been studied extensively since Iiirkman ra.ised his famous schoolgirl problem in 1847 [ S ] . It is well known that the following condition is necessary for the existence of a ICS(2, h ; u ) :

Research supported by the National Natural Science Foundation of China

512 H. Shen

Kirkman's schoolgirl problem was completely settled by Ray-Chaudhuri and Wil- son [9]: There exists a KS(2,3; v ) if and only if v 3(mod6). For k = 4, the existence of KS(2,4; v ) was dttermined by Hanani, Ray-Chaudhuri and Wilson [5]: There exists a KS(2,4; v) if and only if v 4(mod 12). For arbitrary k, Ray-Chaudhuri and Wil- son [lo] established the asymptotic existence of KS(2, t; v): For a given positive integer k , there is a vo = v o ( k ) such that for v > vo, there exists a IiS(2, k ; v ) if and only if 'u z k(modk(k - 1)).

For the existence of a nearly Kirkman system NICS(2, k ; v ) , we have the following necessary condition:

= O(mod k( k - 1)). (2)

In the case k = 3, the existence of nearly Kirkman triple systems was completely solved by the joint work of Kotzig and Rosa [7], Baker and Wilson [2], Brouwer [3] and Rees and Stinson [ll]: There exists an NI<S(2,3; v ) if and only if

v f O(mod6), u 2 18. (3)

The present author [13] studied the existence of NliS(2,4; v ) and gave some constructions. In this paper, we shall discuss the existence and construction of NKS(2, k ; v ) for arbitrary k. Further, in the case k = 4, let

E = {84,120,132,180,216,264,312,324,372,456,552,648,660,804,852,888},

we shall prove the following theorem: Theorem 1. If v $ E , then there exists an NIiS(2,4;v) if and only if

7) E O(mod 12), o 2 24. (4)

2. LABELED RESOLVABLE DESIGNS

To give constructions for nearly Kirkman systems of small orders, we need a kind of

Let (V , B) be an RB(k , A; v ) where X = m and V = {0,1,2, . . . , v - 1). For each resolvable design-labeled resolvable block designs.

block B = { a l , c 1 2 , . . . , n k } , we may suppose tha.t

Let

On the existence of nearly Kirkman systems 513

For an RB(k, i n ; v), if there exists a mapping 'p satisfying the following conditions:

(1) For each pair { a , b } c V with a < b, let { a , b } be contained in the blocks B1, B2, . . ., B,. Let ~ ( u , b); be the values of 'p( a, b ) corresponding to the blocks B,, i = 1 , 2 ,..., m , t h e n f o r 1 5 i , j < m ,

if and only if i = j ; (2) For any block B = {al, a 2 , . . . , a k } with 0 5 a, < a2 < . . . < u k 5 v - 1, we have

then we call i t a labelcd resolvable block design and denote it by LRB(k,m; v). The blocks are denoted in the following form:

For more details of the study and construction of labeled resolvable designs, thc interested reader is refered to [14]. For the application of labeled resolvable Iilock designs in the construction of nearly Kirkman systems, we have the following theorem which is special case of Theorem 6 in [14]:

Theorem 2. If there is an LRB(k, t - 1; v), then there exists an N K S ( 2 , k ; ( k - 1 ) ~ ) . Corollary. There exists an NA'S(2,4; 3v) for v = 8,12, 16 or 24.

Proof. We form an LRB(4,3; 2)) for each v E {8,12,16,24} as follows: (1) v = 8 , V={O,1,2,3,4,5,6,7},

B : (0 ,1 ,3 ,6;0,0, l , O , l , l ) , (2 ,4 ,5 ,7;1,2,0,1,2,1) ; (0 ,1 ,2 ,4; 2 ,2 ,2 ,0 ,0 ,0 ) , (3,5,6,7; 1 ,2 ,0 ,1 ,2 ,1) ; (1 ,2 ,3 ,5;2,2,2,0,0,0) , (0,4,6,7; 1 ,2 ,1 ,1 ,0 ,2) ; (2 ,3 ,4 ,6;2,2,2,0,0,0) , (0,1,5,7; 1,2,2,1,1,0); (0 ,3 ,4 ,5; 1 ,0 ,0 ,2 ,2 ,0 ) , (1,2,6,7; 1,2,2,1,1,0); (1,4,5,G; 1 , 0 , 0 , 2 , 2 , 0 ) , (0,2,3,7; 1 ,2 ,0 ,1 ,2 ,1) ; (0 ,2 ,5 ,6;0,1,0,1,0,2) , (1 ,3 ,4 ,7;1,2,0,1,2,1) .

(2) 2) = 12, v = {0,1,2, . . . ,11}.

B : ( i , ? + 1: i + 4, i + 10;2,2,0,0,1, l), (i +2,i + 5, i + 6 , i t 8;2,0,0,1,1,0),

( i + 3 , i + 7,i + 9,11; 1,2,1,1,0,2) , i = 0,1 ,2 , . . . , l o

where the addition is t,aken modulo 11. (3) = 16, 1' = {0.1,2, . . . ,15},

B : (i,? + 1,i + 3 , i + l l ; O , l l O , l , 0 , 2 ) , ( i + 5 , i t 7 , i t 12, ; t 13;0,2,0,1,0,1) ,

( z +2,z + 4 , i + S , i + 14;2, 1 ,0 ,2 ,1 ,2 ) , ( i t 6 , i + 9 , i + 10,15;2,1,0,2,1,2),

z = O , 1 , 2 , . . . , 14

514 H . Shen

where the addition is taken modulo 15. (4) v = 24, I' = {0,1 ,2 , . . . ,23} ,

B : ( 2 , z + 2,z +9 ,z f 13;1 ,1 ,0 ,0 ,2 ,2) , ( i + 3 , i + 6 , 2 +7,i+ 12;0 ,1 ,2 ,1 ,2 ,1) ,

( i + 5,a + 17,i + 1S,i + 20;2 ,2 ,1 ,0 ,2 ,2) ,

(i + l , i + 8,z + 11,23; 1 ,2 ,0 ,1 ,2 ,1 ) ( i + 4 , i + 10,z + 15,z + 19 ;0 ,0 ,0 ,0 ,0 ,0 ) , ( i f 14,i + 16,z + 21,z + 22 ;0 ,2 ,1 ,2 ,1 ,2 ) ,

2 = 0 ,1 ,2 , . . . ,22

where the addition is taken modulo 23. The conclusion then follows from Theorem 2.

3. KIRKMAN FRAMES

In our construction of nea.rly 1Grkma.n systems, we also need the concept, of I<irkrn;iii frames which was introduced by Stiiison [15].

Let (A', G, B) he a G D ( I i , M ; v ) and P be a subset of B. If P partitions the sc>t X \ G for some group G, then P is called a holey parallel class with hole G. If B c a i i

be partitioned into holey parallel classes, then ( X , G , B) is called a Kirltman I<-frainc. A { 6)-frame is briefly denoted as a k-frame.

A group divisible design (X, G , B) is called of type rni'm? . , . mfi. if G conta.iiis t 1

groups of size m l , t 2 groups of size 1122, . . ., and t , groups of size 112,.

Rees and Stiiison [Ill made essential use of Icirkman 3-frames in the constructioii of resolvable group divisible designs with block size 3.

Since for any group G of size m in a I<irkman k-frame, there are exactly m / ( k - 1) lioley Imrallel classes with hole G and there are exactly nz/(k - 1) parallel classes iii

an NIi 'S(2 , k ; 7)7 + k - I ) , t,lieii we have the following constriictioii for nearly IGrkninii systems:

0(mod k - 1) for 1 5 i 5 s, such that for each i, there exists an NIiS(2, k ; m; + k - I ) , then there exists an NIiS(2, k ; C3=1 tinz, + k - I ) .

Corol1a.q. If there is an NI iS (2 ,4 ; 31n + 3) , then there exists an NIiS(2,4; 31n(4t + 1) + 3) for each t 2 0. Proof. It is well known [5] t,liat there exists a I<s (2 ,4 ; 2,) if and only if I ) = 4( mod 12). Write t~ = 12t + 4 and delete one point, this gives a Iiirkinaii 4-frame of type 341+'. If ' i n 3(111ot~4) and 171 > 3, then there exists an RTD(4, m ) . Thus give each p i n t , weight ni we obtain a IGrkman 4-frame of type (37~2)"~' . The conclusion then follows froin Theorem 3.

Now let, (X, G, A ) be an NIi 'S(2 , k ; 7 4 ) aiid (Y, GI, B) be an NIiS(2, k ; v). If -Y c I,., G c G1 iuid A c B, t,hen (X, G, A ) is called a subdesign of (Y , GI, B) or ( X , G, A ) is said to be einbctlded i n (1'; GI , B). We prove the following generalization of Thcorciii 3, it provides a construction for nearly Kirkman systems containing suhsystems:

Theorem 4. If there exists a Kirkman k-frame of type mi1 ni,? . . , rn.? such that for each i = 1 , 2 , . . . , s, there exists an NIiS(2, k ; mi +tr0) containing an NIiS(2, k ; vo) as i I

subsystem. Then t,here exists a.11 NICS(2, k ; C;'=l t;nzi+u0) containing a.n NIiS(2, k ; t i , , )

aiid an NIiS(2, k ; m, + v o ) for each z . Proof. Let T = ( .uo - k + l ) / (k- 1) and (Vo,Go,Bo) h e a n N I i S ( 2 , k ; v o ) with parallcl

, P,.. Let u = z:= I t2m; and (V, G, B) be a Kirkman k-frame of typc

Theorem 3. If there exists a. I<irknian k-frame of type m:'n?.y . . . mfd with ? i t ;


ini'm? . . . in : . For each group G of size m,, 1 5 z 5 s, there are exactly r , = in,/( k - 1) holey parallel classes with hole G, denoted Gi, Gi, . . ., Gi. For each group G of si/c* m,, form an NlCS(2, k ; m, + W O ) containing (VO, Go, Bo) as a subsystem, on G u VO. 111 this NICS(2, k ; in, + ?lo), there are exactly r , parallel classes containing no blocks of Bu denoted GY, G" . . ., G:, and exactly r parallel classes each contains a parallel class o! Bo, denoted PI u PI, P? U P2, . . ., P," UP,.. Now let GG u Go be the group set of tilts Nl iS(2, k; rn, + U O ) on G u Vo, let

8'

Y = V u V , , G* = ( u G G ) U G ~ GEG

and let A consist of tlie following CB=l t l r l + r parallel classes:

(1) For each group G of size m , , r , parallel classes:

(2 ) r parallel classes of the following form:

( U P F ) U P , , j = 1 , 2 , . . . , r. GEG

Then ( Y , G * , A ) is the desired N I i S ( 2 , I ; ; t i + z ~ ~ )

4. GROUP DIVISIBLE DESIGNS

We shall give a recursive construction for nearly Kirkman systems in this sectioii. For this purpose, the following lemma is needed:

Lemma 1 [15]. Let (X, G , B) be a GD(I<, Ad; v), let to: X + Z+ U (0) ( U J is called i i

weighting). If for eacli B E B, there is a k-frame of type nzEB ~ ( 5 ) . Then there exists a. k-fra,me of type ncEG (EXEC; U J ( T ) ) .

If t h r w exists a GD( li, M ; 21) such that, for each r E I<, there is :I

ICS(2, k ; ( k - 1 ) r + 1) and for each in E M , there is an NI<S(2, k ; ( k - 1 ) m + 210)

containing an NICS(2. k ; U O ) as a subsystem, then there exists an N K S ( 2 , k ; ( k - l ) v + q , ) containing an N I i S ( ' 2 , k ; o o ) and a n N I < S ( S , k ; ( k - 1)m + u o ) for each m E M . Proof: Let (X, G, B) he the GD(Zi, M ; o), give each n' E X weight k - 1. Since for each 1' E li, there is ;I ZCS(2, k ; ( k . - 1 ) r + I), deleting one point then gives a IGrkiiinn k-frame of type ( k - l ) r . Th~ i s , by Leninia 1, there exists a Kirkinan b-franie of t yp ' n c E G ! k - l)IG(. As for each G E G , there is an NIiS(2, k ; ( I ; - 1)IGl + V O ) containills an N h S(2, I ; ; 7 1 0 ) as a subsystem, the conclusion then follows from Theorem 4.

Since the existence of a T D ( k . n t ) is equivalent to tlie existence of I; - 2 mutually orthogonal Latin squares of order i n , then we have the following corollary, which was proved in [2] in the case I; = 3.

Corollary. If there exists an N I < S ( 2 , k ; v ) and kt - 1 mutually orthogonal Latin squares oforder ( u - k + I ) / ( k - I ) , tlieii thereexistsan N K S ( 2 , r ; ; ( k t + l ) ( u - k + l ) + k - l ) containing an NIiS( 2 , k ; u ) .

Theorem 5.

516 H. Shen

The following construction for group divisible designs is a generalization of a result obtained in [2], which is useful in the construction of nearly Kirkman systems:

Theorem 6. Let k and Ic-1 be prime powers. If there exist k 2 - 1 mutually orthogonal Latin squares of order n. Let 0 5 m l , m2 5 n. Then there exists a GD( I{, M ; v ) where

v = ( k 2 + 1 ) n + k ( m 1 +m2), I i - = { k + 1 , k 2 + 1 } , M = { n , n + k m 1 , n + k m z )

Proof. Let ( X , G , B) be a TD(kZ + 1,n) and G1 and Gz be two of the kz + 1 groups. Giving ml points of GI and m2 points of G2 weight k - 1, that is, replace such a point x by k - 1 new elements 21 , 22, . . ., xk-1, and giving every other point weight 1, we obtain a set Y with IYI = (k' + 1)n + k(m1 + m z ) . Now for each block B E B, if B contains no points of weight k-1, then take B as a block of the GD( h', M ; v). If B contains one point n. of weight k - 1 , then forin a projective plane of order k on ( B \ { x } ) ~ { T ~ , x ~ , . . . , x - k - 1 ) such that { ~ ~ , . i - ~ , . . . , ~ k - ~ } is a line of the plane. Delete this line and take each of the remaining lines as a block. If B contains two points x, y of weight k - 1, then forin an affine plane of order k + 1 on ( B \ {I, y } ) U { x1,x2, . . . , ~k - 1, y1, y~ , . . . , yk - 1 } such that { X I , 1 2 , . . . , . z k - l } and {yl , y2 , . . . ,yk-1} are two lines of the plane. Delete these two lines and take each of the remaining lines as a block. For the groups, replace GI by a group with I? + kml elements and G2 by a group with n + kni2 elements. This completes the proof.

5 . EXISTENCE OF NICS( 2,4; v )

The purpose of this section is to prove our main result Theorem 1. The following

Lemma 2. If there exists an NZiS(2,4;u) with u > 24, then there exists an

Lemma. 3. If s l(mod3), r l(mod4), r 2 5, then there exists an

Lemma 4. There doesn't exist an NI<S(2,4; 12).

two constructions for NZ<S(2,4; 1 1 ) can be found in [13].

NI iS (2 ,4 ; 1 1 ) where 1 1 = (3s + I)?(, s 2 0.

NIiS(2 ,4 ; 121-3).

Proof'. The existence of an NILS( 2,4; 12) is equivalent to the existence of an RTD(4,3) and therefore equivalent to the exist,ence of 3 mutually orthogonal La.tin squares of order 3, this is impossible.

We rewrite the corollary of Theorem 3 as the following lemma: Lemma 5. If there exists an NIiS(2 ,4 ; u ) , then there exists an N I i S ( 2 , 4 ; 1 1 ) wherc

Lemma 6. < 972 and 11 @ E , then there exists an

Proof. The existence of ail NICS(2,4;.v) for ZI = 24,36,48 or 72 was shown in the rorolla.ry. For '11 = 96, we form an NI<S(2,4; 24) 011 each group of an RTD(4,24), this gives an NIiS(2 ,4 ; 96). In Lemma. 3, let s = 1 and T = 4t+ l , 1 5 t 5 19, then we obtain an NIiS(2 ,4 ; u ) for 1) E {GO, 108,156,204,252,300,348,396,444,492,540,588,636,684, 732,780,828,876,924). Let s = 4 and T = 5,9 ,13 or 17, we obtain an NIiS(2 ,4 ; 21)

for u E (240,432,624,816). Let s = 7 and = 5 or 9, we have an NII'S(2,4;420) or an NIiS(756). Let s = 10 and r = 5, we have an NI<S(2,4;600). Let s = 16 and r = 5, we have an NI<S( 2,4; 960). In Lemma 5, let u = 24 and t = 2, 3, 4, 6, 7, 8, 10 or 11, we obtain an NI<S(2,4; v) for v E { 192,276,360,528,612,696,864,948). Let

v = (4t + l ) ( , u - 3) + 3, t 2 0.

NIiS(2 ,4 ; v). If 11 E 0(mod 12), 12 <


u = 36 and t = 1 or 4, we obtain an NI<S(2,4; 168) or an NICS(2,4; 564). Let v = 48 and t = 1 or 2, we have an NI<S(2,4;228) or an NI<S(2,4;408). Let u = 60 and t = 2 or 3, we have an NIIS(2,4;516) or an NIiS(2,4; 744). Let u = 96 and t = 2, we have an NKS(2,4;840). Let u = 144 and t = 1, we have an NKS(2,4; 708) where thc existence of an NI<S(2,4; 144) can be proved by Lemma 2 with u = 36 and s = 1. For the remaining 14 values of v , the existence of NKS(2,4; u ) can be proved by Lemma 2. shown in the following table.

Table 1 NIiS(2,4; (3s + 1)u)

'U = (3s + 1)u S 3s + 1 ZI

288 336 384 468 480 504 576 672 720 768 792 900 912 936

1 2 1 4 3 2 1 1 3 5 7 8 1 4

4 7 4

13 10 7 4 4

10 16 22 25 4

13

72 48 96 36 48 71

144 16s 71 48 3G 36

225 72

Lemma 7. There exists an NKS(2,4; v) for u E (2340,2352,2364,2376,2388). Proof. 2340 = 41 ' 5 7 + 3, 2388 = 5 .477 + 3. Since there exists an NKS(2,4; 60) and an NKS(2,4; 480), then there exists an NI<S(2,4; 2340) and an NZ<S(2,4; 2388), by Lemma 5. Since 2352 = 4.588, 2376 = 22.108 and there exists an NI<S(2,4; 588) and an NKS(2,4; 108), then there exists an NI<S(2,4; 2352) and an NKS(2,4; 2376). In Lemma 3, let s = 1 and = 197, we obtain an Nlr'S(2,4; 2364).

Let k = 4 in Theorem 6, we have the following lemma: Lemma 8. If there exist 15 mutually orthogonal Latin squares of order n.

0 5 m l , n22 5 n. Then there exists a G D ( K , M ; v) where Let

Proof of Theorem 1. We prove the theorem by induction. In Lemma 8, let n = 19. 23, 31, 47, 67, 83, 127, 179, 243, 331, and then 323-1 . 17, 325-1 . 25, 323-1 . 37, 3"-l . 49. 323 . 23, 323 . 31, 323 43, for s = 2,3,4, . . .. Then choose ml and 7712 such that 0 5 nil, 1712 5 n and 3(n + 4m1) + 3, 3(n + 4m2) + 3 c$ E . The conclusion then follows from Lemma 6, Theorem 5 and Lemma 7.

Acknowledgement. The author would like to thank Rolf Rees for helpful discussions.

518 H. Shen

REFERENCES

1 R.D. Baker, Whist tournaments, Proc. 6th South-Eastern Conf. Combinatorics, Graph Theory and Computing ( 1975), 89-100.

2 R.D. Baker and R.M. Wilson, Nearly Kirkman triple systems, Utilitas Math. 11

3 A.E. Brouwer, Two new nearly Kirkman triple systems, Utilitas Math. 13 (1078),

4 A.E. Brouwer, The number of mutually orthogonal Latin sclua.res, Math. Ceiit,ruIn Amsterdam Report, ZW 123/79.

5 H. Hanani, D.K. Ray-Chaudhuri and R.M. Wilson, On resolvable designs, Discrete Math. 3 (1972), 343-357.

G C. Huang, E. Mendelsolin and A. Rosa, On partially resolvable t-partitions, Anmls of Discrete Math. 12 (1983), 16’&183.

7 A. IWzig a.iitl A . Rosa, Nearly Kirkman systems, Proc. 5th South-Eastmi C h i f . Combinat., Graph Theory and Coniputing (1974), 607-614.

S T.P. I<irkman. On a. probleni in Combinations, Cambridge a,nd Dumlin Math. J. ’7

9 D.K. Ray-Clia.udliuri and R.M. Wilson, Solution of Kirkman’s schoolgirl ixoblem, Proc. Synip. Pure Math. 19 (1971), Amer. Math. Soc., Providence, R., 187-203.

10 D.K. Ray-Chaudhuri and R.M. Wilson, The existence of resolvable block designs, in: A survey of Coinliina.toria1 Theory, North Holla.nc1, Amsterclmi (1973), 361-375.

11 R. Rees and D.R. Stinson, On resolvable group divisible designs with block size 3. Ars Cornbinatoria 23 ( 1987), 107-120.

12 H. Slieii, On resolvable group divisible designs, Prepriiit Series in Mathematics ; i t

the University of Toronto, No.1 (1987). 13 H. Shcn, On resolvable group divisible designs with block size 4, JCMCC 1 (19S7),

125 130. 14 H. Shen, Constructions and uses of labeled resolvable designs, in: Comhinatoriiil

Designs and Applications, Marcel Dekker Inc., New York, 1990. 15 D.R. Stinson, Frames for ICirkinan triple systems, Discrete Math. 65 (1987), 289

300.

(1977), 289-296.

311-314.

(1847), 191-204.

Cornbinatorics '90 A. Barlotti et al. (Editors)

1992 Elsevier Science Publishers B.V. 519

Codes and semilinear spaces

J. Simonis

Delft University of Technology, Faculty of Technical Mathematics and Informatics, P.O. Box 5031, 2600 GA Delft, The Netherlands

Abstract If C is a binary linear n,k,d] code such that no [n-6,k-4,d] code exists, the coordi-

nate positions and the wor d s of weight 4 in the dual code constitute the points and the lines of a so-called semilinear space. (This is an incidence structure such that any line has at least two points and any two points are on at most one line.) Verhoeff s table 51 on minimum-distance bounds for binary linear codes provides many instances of co d es C satisfying the above condition, e.g. the [25,8,10] codes, whose existence has recently been put into doubt by computer calculations of several authors, and the [16,7,6] codes. We propose to use the latter to demonstrate how geometric techniques can be used in the classification of codes.

1. BINARY LINEAR CODES

Let C be a binary linear [n,k,d] code, i.e. a k-dimensional linear subspace of lFzn of minimum distance d . If S is the set of the n coordinate positions, then each codeword of C , and, more generally, each vector in the ambient Fzvector space lFzn , will be identified with its support in S , i.e. the subset of S consisting of the coordinate positions containing 1 . In other words, we identify Fzn with the power set P ( S ) of S . Thus we can apply sat theoretic notions like union, intersection, inclusion and complement to vectors in Fzn . The complement of X E Fzn in the set S will be denoted by X . The sum of two vectors in Fzn is their symmetric difference as subsets of S . The weight of a vector X E Fzn , i.e. its cardinality as a subset of S , will be denoted by 1x1 *

Let

di(C) := { X E C 1x1 = i } , i = O,1, ...., n, I denote the subsets of C of constant weight. Then the weight distribution of C is defined to be the ordered sequence of the n + 1 non-negative integers Ai(C) := I Ai(C) I .

The dual C' of the code C is the orthogonal complement of the linear subspace C

520 J . Simonis

of lFzn with respect to the standard inner product on Fzn . We occasionally use the, redundant, notation Bi(C) := Ai(C') , Bi(C) := Ai(C*) . The weight distributions of C and C' are connected by the celebrated Mac Williams identities: n 2: Ki(j;n)Bj(C) = 2n-k*Ai(C) j=O

(i=O, ..., n),

with i

a=O Ki(j;n) := C (-1)"[i] [::A] .

2. THE CODES CT AND CT .

Let C be a binary linear [n,k,d] code, with coordinate position set S , and let T c S be any subset, say of cardinality m .

2.1 Definition. i) The binary linear code

c T : = { c f l T I C E C }

ii)

of length m is said to be derived from C by puncturing (with respect to the complement T of T ). The binary linear code

C T : = { C € C I C c T }

of length m is said to be derived from C by shortening (with respect to 'T).

The following proposition shows that puncturing and shortening are intimately connected: 2.2 Proponition.

i) If we interpret CT as a subcode of C , then

T ii) The dual codes (CT)' and (C )* (with respect to the standard inner

product on ?(T) 2 F2m ) satisfy (CT)' = (C')T and (C T )' = (C')T.

Codes and semilinear spaces 52 I

Proof.

i ) The kernel of the surjective linear mapping

7 C d C T , C - C n T I

T isequal to C .

X E (C')T H XEC' A X c T e=, X E (C,)'.

Substitution of C' for C yields the second equality.

ii) Note that

0

2.2.1 Corollary. T If dim(CT) =: p and dim(C ) =: q , then

dim(CT) = k - q , dim((C*)T) = m - q , dim((C')T) = n - m - k + p ,

dim(C ) = k - p , = m - p , dim((C ) ) = n - m - k + q . T ' T

2.2.2 Application. Let C be an [n,k,d] code and suppose that no [n-6,k4,d] code exists. Then any two words in E4C) share at most one coordinate position. Proof. If two words X, Y E &(C) had more than one coordinate position in common, we could choose a 6 a e t T c S such that X c T and Y c T . Hence the dimension of

0 T (C')T would be at least 2 , and C would be an [n-6,bId] code with b 2 k - 4.

2.2.2.1 Remark. So, in this case, the elements of S and the words in E4C) constitute the points and the lines of a so-called semilinear space , i.e. an incidence structure such that any line has at least two points and any two points are on at most one line. Verhoeff s table [5] provides many instances of codes C satisfying the condition of 2.2.2. For instance the [16,7,6] codes, to be discussed in section 111, and the [25,8,10] codes whose existence has recently been put into doubt by computer calculations of several authors (cf. [2] and [S]).

A general semilinear space does not have much structure, but , due to the fact that &(C) consists of all weight four codewords in a linear code, our semilinear space (S,E,(C)) has additional properties, for instance the pentagon property: if four sides of a complete pentagon are present in the geometry, then the fifth side is present as well.

522 J . Simonis

figure 1.

2.3 Weight distributions. The weight distributions of CT , CT and C are linked in the following way:

Introducing the integers

Ai,j := Idi,j I where

di,j := { C E di+j(C) 1 IC n TI = i } , we have

T I: Ai,j = I C I .Aj(CT) and n

i = O ii)

n iii) C Ai,j = Aa(C) .

i+ j=p Proof.

i) The set uy=o di,j is the inverse image of the set di(CT) under the linear

mapping +y used in the proof of the preceding proposition. Interchange T and T in i) . The sets di,j with i + j = /3 partition the set 4.

ii) iii)

Codes and semilinear spaces 5 2 3

2.3.1 Remark. The integers Ai,j(C) and di,j(C') are connected by a set of identities of MacWilliams type. These identities and their uses will be discussed in a forthcoming paper.

2.4 A restriction on the minimum weight of CT .

2.4.1 Definition. The covering radius of a code P c Fan is the integer given by

t(P) := max {d(X,P)

2.4.2 Proposition. For any subset T c S we have

X E Fzn} .

where d(CT) is the minimum weight of CT .

Proof. Choose a codeword X E C such that 1 X n T I = d(CT) . Then

d = d(X,C\{X}) 5 d(X,CT) = IX n TI + d(X fl T,CT) 3 d(CT) + t(CT) 0

2.4.2.1 Example. If

the codeword T (cf. [4]). Since t(C ) 5 k , we have: d(CT) 2 d - k . E di(C) , then CT is called the residual code of C with respect to

T

3.THE [16,7,6] CODES .

3.1 The weight distribution.

(If we claim the nonexistence of a particular code, we implicitly refer to [ 5 ] . )

Let C be a [16,7,] code. Then

(If T E A1 , then CT would be a [9,6,3] code, and such a code does not

(If T E dB , then CT would be a [7,6,3] code, because words of weight 2 in

CT require the existence of words of weight 7 in C . But no [7,6,3] code

exist).

exists) T T T

(An element T E E l would yield the, nonexisting, [15,7,6] code C ).

(An element T E E Z would yield the, nonexisting, [14,6,6] code C ).

(An element T E E3 would yield the, nonexisting, [13,5,6] code C ).

524 J . Simonis

Now the MacWilliams identities allow only three possible weight distributions for C , depending on the values of A14 and A16 . In each case, B16 = 1 , i.e. C is an even weight code.

case 1)

case 3)

A14 = 0, A16 = 1, B4 = 20.

A14 = 0, A16 = 0, B4 = 10.

case 2) A14 = 1, A16 = 0, B4 = 12.

3.2 The semilinear spsces B4 .

constitute the points and the lines of a semilinear space. Let us define

In 2.2.2.1, we have seen, that the 16 elements of S and the B4 elements of E4

vp:= I { L E E 4 1 P E L } [ , p ~ S , a n d

I { P E s I v p = i ) I .

In S , there is no room for more than 5 lines through a given point, so vp < 5. More- over, the value vp = 4 does not occur: if L1 , La , La and L4 pass through p , then S + L1 + - - - 4 L4 is a fifth line passing through p . The usual counting of incidences leads to the following equalities:

I: Cri = 16 (= I S [ ) , C i a i = 4 B q and

C [i] Oi = v : the number of pairs of intersecting lines.

3.3 The description of the [16,7,6] codes.

3.3.1 At4 = 0 , A16 = 1 . Since B4 = 20 , through each pair of points passes exactly one line. Consequently,

the lines are the blocks of the, unique, 2(16,4,1)4esign, better known as the affine plane over the field IF4 (cf. [l], p.390). One may verify that the lines in IF? indeed generate a [16,9,4] code with the required weight distribution. (The 48 words of weight 6 in C are the hyperovals in IF4 .) So C is unique, and its symmetry group, isomorphic to the affine collineation group %rc(2,4) , acts transitively on S .

2

Codes and semilinear spaces 525

3.3.2 Air = 1 , A16 = 0 . Let T

T

T be the unique word of weight 14. Since CT is a [14,5,6] code with A14(C ) = 1 , its weight distribution is fixed. In particular, it turns out that B2(C ) = 1, so exactly one line L E B4(C) is not contained in T . The two points in L \ T each have multiplicity vp = 1 , so al 2 2.

The two points in T n L are the only ones for which vp could be equal to 5, so a5 < 2. Combining this with the three equations in 3.2 and taking into account that v < , the number of allline pairs, we obtain the unique solution

a0 = 0, a1 = 2, 0 2 = 0, a3 = 12 and 05 = 2.

"2"l

Through each of the two points p, q of T n L pass four lines distinct from L , say LI, Lz, Ls, L4 and M1, M?, Ms, M4 respectively. Any pair of lines Li forms a quadrilateral with exactly one pair of lines Mi , and the sum of these four sides must be one of the three lines that do not intersect L . Hence the space B4 is unique (see fig. 2).

figure 2.

The symmetry group of B4 has three orbits in S : L \ T , T n L and S \ L . The dual PA of the [16,8,4] code P spanned by B4 contains one word of weight 14

and six words of weight 12, and these seven words span a six-dimensional subspace & . Three seven4mensional subspaces of P contain & . One of these contains words of weight 2 and 4, but the other two are [16,7,6] codes. They are isomorphic under the mapping induced by the interchanging of p and q .

526 J . Simonis

3.3.3 Air = 0 , Ale = 0 . Let L E E4 be any line. For the [12,7,2] code Ci; , we have:

BdCr;) = 1, Az(Cz) < 1 , and A12(Cz) = 0 .

(If A11(Cr;) = 1 , then all words # L in &(C) would be contained in

B4(Ci;) = !9 . But this is incompatible with the MacWilliams identities for CE).

These restrictions leave us with just two possible weight distributions for CT :

i )

ii)

In other words, L intersects six or eight other lines, depending on whether L is contained in a word of de(C) or not.

Let a be the number of lines for which the former situation holds. If T E da(C) and dirn((C')T) = 1 , then T cannot contain a word of &(C) , for otherwise CT

would be a [10,6,4] code. So a equals the number of words in de(C) for which dim((C'jT) = 1.

Now we count the number of inclusions T c U , with T E da(C) and U E d12(C) :

, i.e.

AZ(CT> = 0 and B4(Cr) = 1, or

Az(Cr) = 1 and B~(CT) = 3.

T! i) If dim(dim((C')T) = l ( 0 ) , then dim(C ) = 2 (1) , and CT! contains three

(one) word of weight six. So a words of da(C) are contained in three words of d12(C) and 44 - a words of da(C) are contained in one word of d12(C) . The three-dimensional code Cu contains six words of weight six. So the 10

words of Alz(C) each contain six words of de(C) . ii)

The equation

3 . 0 4- 1.(44 - a) = 6.10

gives a = 8. Hence the number v of pairs of intersecting lines in B4(C) is equal to 1 Z(a.6 + (10 - a).8) = 32,

and the three equations have only one solution:

a. = 0, a1 = 0, 02 = 8, a3 = 8 and a5 = 0.

The sum of all words in Bq is a word in BS that consists of all points of multiplicity 3. Hence this word must contain the 10 - a = 2 lines L and M that intersect eight other lines. So L and M are disjoint, and all other lines intersect both L and M . Eventually, we are left with just one possible structure for the near-linear space E d

(see figure 3).

Codes and semilinear spaces 527

L figure 9

(The inner and outer squares represent the lines L and M ). The symmetry group of B4 has two orbits in S : L U M and S \(L U M) . The dual P* of the [16,8,4] code P spanned by B4 has the same weight distribution as 3 . As a matter of fact, the codes P and P* are isomorphic! Finally, we can retrieve C as the, unique, seven4imensional subcode of P' that does not contain any word of weight four.

3.3.4 Remark. In [2], Kostova and Manev produced generator matrices for the three non-equivalent [16,7,d] codes. Their result is based on a computer search.

4. THE [15,7,5] CODES .

Since all [15,7,5] codes can be obtained by puncturing a [16,7,6] code, we merely have to consider the orbits of the symmetry group for each of the three equivalence classes established in the preceding section. In all, we find six equivalence classes of [15,7,5] codes which we designate by I , 11, , IIb , IIc , 111, and IIIb . (The roman numerals indicate the corresponding [16,7,6] code in the order of treatment we chose in the preceding section.) For future reference, some information concerning the weight distribution is collected in the table below:

528 J . Simonis

5. References

1. Th. Beth, D. Jungnickel and H. Lena Design Theory. Mannheim: B.1.-Wissenschaftsverla (1985).

2. B.K. Kostova and N.L. hanev, "A [25,8,10] code does not exist," preprint. 3. J.H. van Lint. Introduction to Coding Theory. New York: Springer (1982). 4. F.J. MacWilliams and N.J.A. Sloane. The Theory of Error-Cm-ecting Codes. New

York: North Holland, 1983). 5. T. Verhoeff, "An up 6 ated table of minimum4stance Luunds for binary linear

codes," IEEE Transactions on Infomation Theory, vol. IT-33, pp. 665-680, Sept. 1987. Revised version: Jan. 1989.

6. 8 . Ytrehus and T. Helleseth "There is no binary [25,8,10] code," IEEE Transactions on Infomation Theory, vol. IT-36 (1990), pp. 695-696.

Cornbinatorics '90 A. Barlotti et al. (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved. 529

Old and new results on spreads and ovoids of finite classical polar spaces

J.A. Thas

University of Gent, Seminar of Geometry and Combinatorics, Krijgslaan 281, B-9000 Gent, Belgium

Abstract

This paper surveys the existence and non-existence theorems on spreads and ovoids of finite classical polar spaces. Several of the results are new. Further, partial spreads and partial ovoids of finite polar spaces are introduced and upper bounds for their sizes are obtained.

1. Introduction

1.1. Finite classical polar spaces

Let P be a finite classical polar space of rank T , with T 2 2. We shall use the following notation:

W,,(q): the polar space formed by the absolute points and totally isotropic lines for a given symplectic polarity 0 of PG(n, q ) , n odd and n 2 3: here T = (n + 1)/2;

Q(2n, q ) : the polar space formed by the points and lines of a non-singular quadric Q in PG(2n, q) , n 1 2: here T = n;

Q+(Zn + 1,q): the polar space formed by the points and lines of a non-singular hyperbolic quadric &+ in PG(2n -+ l ,q),n 2 1: here T = n t 1;

Q-(2n -t 1,q): the polar space formed by the points and lines of a non-singular elliptic quadric &- in PG(2n + l ,q),n 2 2: here T = n;

H ( n , q'): the polar space formed by the points and lines of a non-singular hermitian variety H in PG(n, q2)ln 2 3: for n odd T = (n + 1)/2, for n even T = n/2.

Let [PI denote the number of points of PI and let C(P) be the set of all maximal totally isotropic subspaces or maximal singular subspaces of P; all elements of C ( p ) have dimension T - 1. For a proof of the following theorem we refer e.g. to Hirschfeld [1979] and Thas [1981].

5 30 JA. Thas

Theorem 1

The numbers of points of the finite classical polar spaces are given by the formulae:

IWn,q2)1 = (q"+' + (-l)")(q"- ( - l ) " / ( q 2 - 1).

The following theorem is an easy corollary.

Theorem 2

The numbers of maximal totally isotropic subspaces or maximal singular subspaces of the finite classical polar spaces are given by:

1.2. Ovoids and spreads of polar spaces

Let P be a finite classical polar space of rank r 2 2. An 0 of P is a pointset of PI which has exactly one point in common with every maximal totally istropic subspace or maximal singular subspace of P. A spread S of P is a set of maximal totally isotropic subspaces or maximal singular subspaces of PI which constitutes a partition of the pointset. The following theorem is easily proved, cfr. e.g. Thas [1981].

Theorem 3

Let 0 be an ovoid and let S be a spread of the finite classical polar space P. Then

for P = Wn(q), 101 = 1st = q(n+')/2 + 1,

Spreads and ovoidr of finite classical polar spaces 53 1

for P = Q ( 2 n , q ) , 101 5 IS1 = q" + 1,

for P = Q+(2n t l , q ) , 10) = IS1 = q" t 1,

for P = &-(an + 1, q) , 101 = IS1 = q"+' + 1,

for P = H ( 2 n , q 2 ) , 101 = 15') = q2"+l t 1,

for P = H ( 2 n t l , q 2 ) , 101 = IS1 = q2"+l + 1.

2. Useful theorems and observations

2.1. The polar spaces Q(2n,q) and WZ,-l(q), with n 2 2 and q even

Let Q be a non-singular quadric of P G ( 2 n , q ) , n 2 2 and q even, and let k be the kernel or nucleus of Q (see Hirschfeld [1979]). By projection of Q from k onto a P G ( 2 n - 1, q ) c P G ( 2 n , q ) not containing k, one obtains the following theorem.

Theorem 4

For q even and n 2 2, the polar space Q ( 2 n , q ) is isomorphic to the polar space Wzn-l(q).

2.2. Isomorphisms and anti-isomorphisms between the finite classical polar spaces of rank 2; the Klein correspondence

A finite classical polar apace of rank 2 is also called a finite classical generalized quadrangle. For a proof of the following theorem we refer to Payne and Thas [1984].

Theorem 5

(a) Q ( 4 , q ) is isomorphic to the dual of W3(q). Moreover, Q ( 4 , q ) (or W3(q)) is self-dual if and only if q is even.

(b) Q - ( 5 , q ) is isomorphic to the dual of H(3,q ' ) .

Remarks

1. Since W3(q) (or Q ( 4 , q ) ) , q even, is self-dual, with each spread S of W3(q) there corresponds an ovoid 0 of W3(q), and conversely. Clearly S is a linespread of PG(3, q) , and in Thas [1972] it is shown that 0 is an ovoid of P G ( 3 , q) . The spread of P G ( 3 , q ) is regular if and only if the ovoid is an elliptic quadric, and the spreads of P G ( 3 , q ) corresponding to the Tits ovoids were first discovered by Liineburg, see Thas [1972]. It is easy to show that W3(q) always contains a regular spread

532 J.A. Thas

of PG(3,q), and Kantor [1982] proves that for any odd q, q not a prime, W3(q) contains a non-regular spread of PG(3,q).

2. A short proof of (b) and of the first part of (a) can be given using the Klein correspondence between the lines of PG(3, q ) and the points of the Klein quadric Qf(5, q) .

3. By the Klein correspondence, with any ovoid of Q+(5, q ) there corresponds a linespread of PG(3,q), and conversely. The ovoid is an elliptic quadric if and only if the spread is regular. If the ovoid is contained in a non-singular quadric Q c Q+, then the corresponding linespread is a spread of a W ~ ( Q ) .

2.3. Observations on spreads

Let Q+ be a non-singular hyperbolic quadric of PG(2n + 1, q) , n 2 1, and let C1 and Cz be the two families of maximal singular subspaces of &+. If a , ~ ’ are maximal singular subspaces, then a, a’ belong to the same family if and only if dim(a n a‘) has the parity of n. It easily follows that Q+(4n t l ,q) ,n 2 1, has no spread.

Let S+ be a spread of Qf(4n + 3,q),n 2 1, and intersect Qt with a hyperplane PG(4n + 2,q) of PG(4n + 3,q) which is not tangent to Q+. Clearly the intersection of S+ with PG(4n + 2, q ) is a spread S of Q(4n t 2, q) , with Q = Qt n PG(4n t 2,q). Next, let S be any spread of Q(2n,q),n 1 3. Intersect the quadric Q with a hyperplane PG(2n - 1, q ) of PG(2n, q) , where Q n PG(2n - 1, q ) is a non-singular elliptic quadric Q- of PG(2n - 1 , q ) . Then the intersection of S and PG(2n - 1 , q ) is a spread S- of Q-(2n - 1, q).

Let S’ be any spread of Q ( 4 n t 2 , q ) , n 2 1, and let Q be embedded in the non-singular hyperbolic quadric Q+ of PG(4n + 3, q) . The two families of maximal singular subspaces of Q+ are denoted by C1 and Cz. Then one easily shows that the elements of C; containing the respective elements of S’, constitute a spread S,!+ of Qt(4n + 3, q ) , i = 1,2.

Theorem 6

(a) I f Q + ( 4 n t 3 , q ) , n > 1, hasaspread, thenQ(4nt2 ,q) hasaspread. I fQ(2n ,q) ,nL 3, has a spread, then Q-(2n--l,q) has a spread. If Q(4n+2,q),n 2 1, has a spread, then Q+(4n + 3, q ) has a spread.

(b) Q+(4n + l , q ) , n 2 1, has no spread.

Next, let S be a spread of H(2n + 1,q2),n 2 2, and intersect H with a hyperplane PG(2n, q’) of PG(2n + 1, 4’) which is not tangent to H. Then the intersection of S with PG(2n, q 2 ) is a spread S’ of H’(2n, q’), with H‘ = H n PG(2n, 4’).

Theorem 7

If H(2n + 1, a’), n 2 2, has a spread, then also H(2n, 4’) has a spread.

Spreads and ovoids of finite classical polar spaces 533

2.4. Observations on ovoids

Let 0 be an ovoid of Wzn+l(q),n 2 2, and let B be the symplectic polarity of PG(2n+l, q ) defining Wzn+l(q). Further, let z be a point of PG(2n+ 1, q ) not on 0, let a = z’, and let PG(2n - 1, q ) be a hyperplane of a not containing z. The intersections of PG(2n - 1, q ) and the totally isotropic lines (resp. planes) of a containing z are the points (resp. lines) of a polar space W2n-1(q). The intersections of PG(2n - 1 , q ) with the totally isotropic lines joining z to a point of 0 clearly form an ovoid 0’ of Wzn-l(q).

Next, let 0 be an ovoid of the polar space Q+(2n + l , q ) ,n 2 2. Further, let z be a point of PG(2n + 1 , q ) not on 0, let a be the tangent hyperplane of Q+ at z, and let PG(2n - 1, q ) be a hyperplane of a not containing z. The intersections of PG(2n - 1, q ) and the lines on Q+ containing z and a point of 0 clearly form an ovoid 0’ of the polar space Q’+(2n - l , q ) , with Q’+ = Q+ n PG(2n - 1,q) . Similar arguments hold for Q(2n, q)(n 2 3>, Q-(2n + 1, q)(n 2 3) and H ( n , q2)(n 2 5).

BY embedding of Q-(2n + 1, q ) in Q(2n + 2, q ) , of Q(2n, 4) in Q’(2n 1 , q ) and of

H(2n, q 2 ) in H ( 2 n + 1, q 2 ) , we see that the following holds

Theorem 9

If Q-(2n+1, q ) , n 2 2 , has an ovoid, then also Q(2n+2, q ) has an ovoid. If Q(2n, q) , n 2 2 , has an ovoid, then also Q+(2n + 1, q ) has an ovoid. If H(2n, q’), n 2 2 , has an ovoid, then also H(2n + 1, q 2 ) has an ovoid.

2.5. Triality

By triality (see e.g. Tits [1959]) we have the following theorem.

Theorem 10

The polar space Q+(7, q ) has an ovoid if and only if it has a spread.

2.6. Quadratic extensions

Consider the polar space 9-(4n+ l ,q),n 2 1. In the extension PG(4n+1, q 2 ) of PG(4n-f 1, q) , the polar space Q-(4n + 1, q ) extends to the polar space Q+(4n + 1,q’). On Q+ it is possible to choose a projective 2n-space a with a n T = 8, where T is conjugate to a with respect to the quadratic extension GF(qa) of GF(q). The lines of Q- whose extensions intersect a and T form a linespread T of Q-, i.e., a set of lines of Q- which partitions the pointset of Q-. It can be shown that the points common to a and the extensions of the

534 J.A. Thas

lines of T form a hermitian variety H of a. Hence for n 2 2 there arises a polar space

Consider a spread S of H(2n,qa) ,n 2 2. If C E: S, then the union of the lines of T whose extensions contain a point of ( is a projective (2n - 1)-space on Q-, hence a maximal singular subspace (’ of Q-(4n + 1, q) . It is clear that any two such spaces (’ are skew. Since IS1 = qZnf1 + 1, the spaces C’ form a spread S’ of Q-(4n + 1 ,q ) .

Next, let 0 be an ovoid of Q-(4n + l ,q),n 2 2. Let 0’ be the set of all intersections of ?r with the extensions of all lines of T which contain a point of 0. It is easy to show that no two points of 0’ are on a common line of H. Since (0‘1 = pan+’ + 1 , the pointset 0’ is an ovoid of H(2n , 4’).

H(2n1 $1).

Theorem 11

If Q-(4n+lIq),n 2 2, has an ovoid, then also H(2n , q 2 ) has an ovoid. If H(2n , q2) ,n 2 2, has a spread, then also Q-(4n + 1, q ) has a spread.

Remark

From the construction of H(2n , q 2 ) for n = 1, follows that the polar space Q-(5, q ) has a spread.

Next, consider the polar space Qt(4n + 3, q ) , n 2 0. In the extension PG(4n + 3, q 2 ) of PG(4n + 3, q) , the polar space Q+(4n + 3, q ) extends to the polar space Q+(4n + 3 , q 2 ) . It can be shown that the polar space Qf(4n + 3, q a ) contains a maximal singular subspace a with a nF = 0, where F is conjugate to a with respect to the quadratic extension GF(qa) of GF(q) . The lines of Q+(4n + 3, q ) whose extensions intersect a and form a linespread T of Qf(4n + 3 , q ) , i.e., a set of lines of Q+(4n + 3 , q ) which partitions the pointset of Q+(4n + 3, q) . It can be shown that the points common to a and the extensions of the lines of T form a hermitian variety H of a. Hence for n 2 1 there arises a polar space H ( 2 n t 1,qZ).

Similar to the first part of this Section 2.6, we then obtain

Theorem 12

If Qt(4n + 3 , q ) , n 2 1, has an ovoid, then also H ( 2 n t l , q a ) has an ovoid. If H ( 2 n t 1, qa) ,n 2 1, has a spread, then also Q+(4n + 3 , q ) has a spread.

3. Ovoids and spreads of generalized hexagons

3.1 Introduction

A generalized hexagon of order n,n 2 1, is an incidence structure S=(P, B , I ) of points and lines, with an incidence relation satisfying the following axioms:

(i) each point (resp. line) is incident with n + 1 lines (resp. points);

(ii) IPI = IBI = 1 + n + na + n3 + n4 t ns


(iii) 6 is the smallest positive integer k such that S has a circuit consisting of k points

An &d 0 of S is a set of n3 + 1 points any two of which are at distance 6 (in the incidence graph of S); a spread S of S is a set of n3 + 1 lines any two of which are at distance 6 (in the incidence graph of S).

The first part of the next theorem on polarities of generalized hexagons is due to Cameron, Thas and Payne I19761 , the second part to Ott [1981] .

and k lines.

Theorem 13

(a) If 0 is a polarity of the generalized hexagon S of order n, then the set of absolute points of 0 is an ovoid of S, and the set of absolute lines of 0 is a spread of S.

(b) If the generalized hexagon S of order n admits a polarity, then either n = 1 or 3n is a square.

In his celebrated study of triality, Tits [1959] showed that there is a generalized hexagon H ( q ) = (P , B,I), with P the set of all points of &(6,q), B a subset of the lineset of Q(6, q) , and I the incidence of Q(6, q). The generalized hexagon H ( q ) is called the classical generalized hexagon of order q. Dickson’s group GZ(q) acts as an automorphism group on H ( q ) .

Let Q be the non-singular quadric of PG(6, q ) with equation

Then by Tits [1959] H ( q ) is the incidence structure formed by all points of Q and by those lines on Q whose Grassmann coordinates (for the definition of Grassmann coordinates, see Hodge and Pedoe [1947]) satisfy

Further, these lines are also the lines of PG(6, q ) whose Grassmann coordinates satisfy (1) together with

3.2. Ovoids and spreads of H ( q )

The distance of two points 2, y of H ( q ) is at most 4 if and only if 2 and y are on a common line of the polar space Q(6, q) . Now it is easy to show that all the lines of H ( q ) containing a point z are in a plane of the quadric Q (see e.g. Thas [1980]). These (q6 - l ) / ( q - 1) planes of Q will be called the H(q)-planes. If 0 is an ovoid of H ( q ) , then clearly the H(q)-planes containing the elements of 0 form a spread of Q(6,q).

In Thas [1981] it is shown that the ovoids of H ( q ) are the ovoids of Q(6, q) .

536 JA. Thas

Theorem 14

H ( q ) has an ovoid if and only if Q(6, q ) has an ovoid.

Consider a hyperplane PG(5, q ) of PG(6, q ) , for which PG(5, q ) n Q is a non-singular elliptic quadric Q-. Thas [1980] proves that the lines of H ( q ) in Q- form a spread of the polar space Q-(5,q) and a spread of H ( q ) .

Theorem 15

H ( q ) and Q-(5 , q ) have a spread.

For a proof of the following theorem we refer to Salzberg [1984].

Theorem 16

The generalized hexagon H ( q ) is self-dual if and only if q is a power of 3; it admits a polarity if and only if q = 32h+1, h 2 0.

Combined with Theorems 15 and 14 this gives

Theorem 17

For q = 3" the generalized hexagon H ( q ) and the polar space Q(6, q ) have an ovoid.

4. Existence and non-existence of ovoids

4.1. The known results

In Thas [1972] it is shown that W3(q) has an ovoid if and only if q is even. Moreover any ovoid of W3(q), q even, is an ovoid of PG(3,q) . Conversely, any ovoid of PG(3, q) , q even, is an ovoid of some W3(q) (see Hirschfeld [1985], p.36). Further, Thas [1981] proves that Wn(q), with n = 2t -t 1 and t > 1, has no ovoid.

By Theorem 4 and the preceding paragraph, the polar space Q(2n, q ) , with q even and n > 2, has no ovoid. In Thas [1981] it is proved that Q-(Zn + l ,q),n 2 2, has no ovoid. Kantor [1982] shows that there is no ovoid in Q+(2n+ 1 , 2 ) , n 2 4, and Shult [1989] proves that there is no ovoid in Q+(2n+ 1 ,3) , n 2 4. The polar space Q(4, q ) has always an ovoid, see e.g. Payne and Thas [1984]. Clearly Q+(3, q ) has an ovoid and by Remark 3 of 2.2 Q+(5, q ) admits an ovoid. By Theorem 17 also Q(6, q) , with q = 3", has an ovoid, and consequently, by Theorem 9, also Q+(7, q) , q = 3", has an ovoid. Further, Q+(7, q ) has an ovoid in at least the following cases: q even, q an odd prime, and q odd with q 3 2 (mod 3) (see Conway, Kleidman and Wilson [1988], Dye [1977], Kantor ([1982], [1982l], [1982']), Shult [1985]).

By the remark which follows Theorem 11, by Theorem 15, and applying Theorem 5(b), it is clear that H(3,q2) admits ovoids (see also Payne and Thas [1984] and Thas [1983]). Finally, in Thas [1981] it is proved that H(2n , q 2 ) , with n 2 2, has no ovoid.

Spreads and ovoids of finite classical polar spaces 5 3 1

4.2. Open problems

The existence or non-existence of ovoids in the following cases.

(a) Q(6,q) for q odd with q # 3h;

(b) Q(2n, q ) for n > 3 and q odd;

(c) Q+(7, q ) for q odd, with q

(d) Q+(2n + 1, q ) for n > 3 and q > 3;

(e) H(2n + 1,q') for n > 1.

1 (mod 3) and q not a prime;

5. Existence and non-existence of spreads

5.1. The known results

A spread of Wn(q), n = 2t + 1, is also a t-spread of PG(n, q). For every n = 2t + 1, t 1 1, the polar space Wn(q) has a spread which is also a regular t-spread of PG(n, q) , see e.g. Thas [1977]. Many other examples of spreads of W,,(q) are known.

By Theorem 6(b) Q+(4n + l , q ) ,n 2 1, has no spread. Since W3(q), q odd, has no ovoid, by Theorem 5(a) Q(4,q),q odd, has no spread. By the preceding paragraph and Theorem 4 the polar space Q(2n, q) , n 1 2 and q even, has a spread. Now by Theorem 6(a) also Q+(4n + 3, q) , q even and n 2 I , and Q-(2n + 1, q) , q even and n 2 2, have a spread. Clearly Q+(3, q ) has a spread, and, by the remark which follows Theorem 11 and by Theorem 15, also Q-(5, q ) admits a spread (see also Payne and Thas [1984] and Thas [1983]). Applying triality to results of Section 4.1, and then applying Theorem 6(a), we see that Q+(7,q) and Q(6,q) have a spread in at least the following cases: q even, q an odd prime, and q odd with q

Since Q-(5, q ) has no ovoid, by Theorem 5(b) H(3, q') has no spread. Finally, Brouwer [1981] proved that H(4,4) has no spread.

0 or 2 (mod 3).

5.2. New results

In this section we prove the non-existence of two types of spreads.

Theorem 18

The polar space Q(4n, q ) , q odd and n 2 1, has no spread. Proof. Suppose that S is a spread of Q(4n, q) , q odd and n 2 1. Let PG(4n- 1, q ) be a non-tangent hyperplane of Q(4n, q ) , for which PG(4n- 1, q)nQ

is a non-singular hyperbolic quadric Q+. Suppose that S contains T maximal singular subspaces of &+. Counting in two ways the number of ordered pairs ( z , ~ ) , with I E

Q+, z E T , 7~ E S, we obtain

538 J.A. Thas

Hence T = 2. Moreover, these two elements of S belong to the same system of maximal singular subspaces of Q+ (cfr. Section 2.3).

Let a. E S and let ( be a maximal singular subspace of Q intersecting a0 in a (271 - 2)- space. The elements of S - {ao} having a non-empty intersection with (, have exactly one point in common with (; let these elements be denoted by a l , a ~ , . . . ,a,p-~. Let q be a third projective (2n - 1)-space of Q containing a0 n (. In the polar space Q(4n, q ) there is no point of < - a0 collinear with a point of 9 - ao. Hence 7; n 9 = 0, with i = 1 , 2 , . . . , q2"-l. Hence, if 7, is the projective (4n - 1)-space generated by A; and 7, then 7, n Q is a non-singular hyperbolic quadric Qf , i = 1,2, . . . , q2"-l. In the first part of the proof we have shown that Qf contains two elements of S, one of them being a;. Let the other element of S be pi . Also, A, and p; belong to the same system C of maximal singular subspaces of QT. Clearly ( is contained in Q f , and since ( and A; have exactly one point in common, we have ( 4 C . It follows that ( n pi # 0, i = 1,2 , . . . , qZn-l. Consequently pi E {a~,al, . . .ap.-L} - {a,}. Since ( n 7 is contained in exactly two (271 - 1)-spaces of Q: (clearly, < and q), the space a0 is not contained in Qf . Hence pi # TO. Now it is clear that 8 : a; ++ p;, i = 1 , 2 , . . . , q2"-l is an involution without fixed elements in the set {TI,. . . , 7 ~ p - i } . This contradicts q being odd.

For a polar space PI a packing of spreads is a set of spreads which forms a partition of the set of all maximal totally isotropic subspaces or maximal singular subspaces of P.

0

Theorem 19

The polar spaces Q(4n,q) and w4,-1(q),q even and n 1 1, do not admit a packing of spreads.

Proof. Let P = {SI, Sz, . . .} be a packing of spreads of Q(4n, q) , q even and n 2 1. Let PG(4n-1, q ) be a non-tangent hyperplane of Q(4n, q ) for which PG(4n-1, q)nQ is anon- singular hyperbolic quadric Q+. Such as in the first part of the proof of Theorem 18 one shows that S; contains two elements of the same system of maximal singular subspaces of Q+, i = 1 ,2 , . . . . In this way there arises a pairing in each system C; of maximal singular subspaces of Q + , i = 1 ,2 . Since I C; I = ( q + l ) ( q a + 1) . . . (q'"-l + 1) is odd, we have a contradiction.

Now by Theorem 4 also W4,,-1(q),~ even and n 2 1, does not admit a packing of spreads. 0

Lemma

Let all TZ, 7r3 be three mutually disjoint maximal singular subspaces of H(2n + l , q a ) , n > 1. Then the lines of H(2n + 1,q2) intersecting 7r1,~2,73, intersect a, in the points of a polar space &(n, qa),i = 1,2,3.

Proof. Let all a2, 7r3 be mutually disjoint projective n-spaces on H. The unitary polarity defining H will be denoted by 8. If L is a line which intersects ~1,a2,a3, then Le n a, is a (n - 1)-dimensional space (;,i = 1,2,3. Let L n A; = { p ; } . Then L is a line of H(2n + 1, q z ) if and only if pi E <,,i = 1,2,3. If p l , p { , p : / are distinct collinear points of 7 1 , then the corresponding points p 2 , p ~ , p ~ of are collinear, so the (n - 1)-spaces t1 = al n Le = al n p : , ( ; = x1 nL" = aI n p f , ( ; ' = 7r1 = al n p : s belong to a


pencil of hyperplanes of a1. If e.g. (1 = (;, then the (n -+ 1)-space containing ( l , p z , p ' , belongs to H , a contradiction. Hence (1,(;,([ are distinct. It follows that 0, : pi H (, is an anti-automorphism of x,,i = 1,2,3. Let q1 E (1 = p:',pl E XI. Then p! contains q1 and 43, so contains 42. Hence qz E (2. Similarly 43 E (3. Consequently 4142 c L B , so L c (q1qZ)'. It follows that pl E 4:'. We conclude that 6, is a polarity of X,,i = 1,2,3. Moreover the lines L of H intersecting a1, az, x3, intersect x, in the absolute points of the polarity B, , i = 1,2,3. Let the set of these absolute points be denoted by Hi.

Let A 4 1 be a line of x1. Further, let Mz = nznM1a3 and M3 = a3flM1x2. It is clear that the lines L intersecting M ~ , T ~ , T J , intersect a2 in the points of Mz and x3 in the points of M3. It immediately follows that Ml,MZ,M3 are contained in a common PG(3,q2). Suppose that PG(3,q') n H is non-singular. Then by 1.2.4 of Payne and Thas [1984] there are exactly q +- 1 lines of H which interseci MI, M z , M3. Hence IH1 n = q + 1. If PG(3, q z ) n H is a singular hermitian variety, then it consists of q +- 1 planes through a common line N which intersects MI, Mz, M3. In such a case N is the only line of H whichintersects MI, Mz, M3; so IH1nMII = 1. If PG(3,q2) is contained in H , then clearly IH1 nM1 I = q 2 + 1. Hence each line of a1 intersects HI in 1, q + 1 or q2 + 1 points. similarly, each line of a, intersects Hi in 1, q + 1 or q2 + 1 points, i = 1,2,3.

Now clearly B1 is not an orthogonal polarity, i = 1,2,3. Let 6'1 be either a symplectic polarity or a pseudo-polarity. Then in x1 there is at least one hyperplane (1 consisting entirely of absolute points (see e.g. Hirschfeld [1979]). Let (2 and (3 be the corresponding hyperplanes of respectively az and a3, i.e., let (Z = a 2 n (1x3 and (3 = x3 n t 1 a 2 . Let L be any line intersecting ( 1 , ( ~ , ( 3 , and let L n x1 = {z},z E L , z 4 x1,y E (1,y 4 L. Since Izy n Hl I = qz -+ 1, by the preceding paragraph the plane yL belongs to a PG(3, q z ) which is contained in H . Hence zy is a line of H. It follows that all lines intersecting t1 and tZ are contained in H. Hence H contains a projective (271 - 1)-space, a contradiction since n > 1.

We conclude that 6'i is a unitary polarity and that Hi is a non-singular hermitian variety of a,,i = 1,2,3. 0

Theorem 20

The polar space H(2n + 1, q 2 ) , n > 1, has no spread. Proof. Let S be a spread of H(2n + l , q z ) , n > 1. Let a l , ~ 2 E S and count in two

ways the pairs (z ,a) with a E S - (a1,x2},z E x, and where the line L through z and intersecting xll ~2 belongs to H(2n- t 1,q')). By the preceding lemma we obtain the equality

Hence ( q - 1)( ( -l)n - ( -1)"q2"+l +q"(q- 1)) = 0, i.e., (-1)" - ( -l)nqzn+l +q"(q- 1) = 0. Consequently q divides (-l)", a contradiction. 0.

540 J.A. Thas

Remark In Thas [1989] the following short proof for the non-existence of spreads of H(5,qa) is given. If S is a spread of H(5,qa), then the number of planes of H containing a line of an element of S is equal to (q6 + l ) (q4 + qa + l ) q + (q5 -I- l), which is more than the total number of planes of HI a contradiction.

5.3. Open problems

The existence or non-existence of spreads in the following cases.

(a) Q(6,q) for q odd, with q = 1 (mod 3) and q not a prime;

(b) Q(4n + 2, q ) for n > 1 and q odd;

(c) Q+(7,q) for q odd, with q = 1 (mod 3) and q not a prime;

(d) Q+(4n + 3, q ) for n > 1 and q odd;

(e) Q-(2n + 1 ,q ) for n > 2 and q odd;

( f ) H(4,qa) for q > 2;

(9) H(2n , q a ) for n > 2.

6. Partial ovoids and partial spreads

6.1. Definitions

Let P be a finite classical polar space of rank r 2 2. A partial ovoid or cap of P is a pointset of P , which has at most one point in common with every maximal totally isotropic subspace or maximal singular subspace of P. A partial spread of P is a set of mutually disjoint maximal totally isotropic subspaces or maximal singular subspaces of P . In Section 6 we obtain some upper bounds for the size of a partial ovoid, resp. partial spread, in cases where P does not admit an ovoid, resp. spread.

6.2. Known results

It is clear that for any partial spread S of &+(4n + l , q ) , n 2 1, we have IS1 5 2. Tallini ([1989],[19..]) proves that the size of any partial ovoid of Wa(q), q odd, and so of any partial spread of Q ( 4 , q ) , q odd, is at most qa - q + 1. By a theorem of Thas [1989] the size of any partial ovoid of Q-(5, q) , and so of any partial spread of H ( 3 , q'), is at most q3 - qz + q + 1. Finally, in Thas [1989] it is shown that any partial spread of H(5, q a ) has at most qz(qz + q - 1) elements.

Spreaa3 and ovoids of finite classical polar spaces 54 I

6.3. Partial spreads of H ( 2 n + l ,q2),n 2 1 and n odd

Theorem 21

For any partial spread S of H(2n + 1 , q 2 ) , n 1 1 and n odd, we have

Proof. Let S be a partial spread of H ( 2 n + 1, q'), n 2 1 and n odd. By 6.2, for n = 1 we have IS1 5 q3 - q2 + q + 1. So let n > 1. Let ~ 1 ~ x 2 E S and count in two ways the pairs ( z , x ) , with x E

S - {TI, X Z } , z E x, and where the line L through z and intersecting all az belongs to H(2n + l , q 2 ) . By the lemma of 5.2 we obtain the inequality

(1st - 2)(q"+' - l)(q" + 1) (q2 (n+l ) - l ) ( q 2 " - 1)(qZ - 1) qz - 1 (q2 -

I I

i.e.,

6.4. Bounds on caps

Theorem 22

For any partial ovoid 0 of Q-(2n t 1 , q ) , n L 3, we have

Proof. Let 0 be a cap of Q-(2n + 1, q ) , n 2 2. Further, let 0 be the polarity defined by Q-, let z,y E 0 ,z # y, and let x = (zy)'. Then x is a projective (2n - 1)-space and x n Q- is a non-singular elliptic quadric &-. Clearly 7r n 0 = 0. Now we count in two ways the number of ordered pairs (u,v), with u E & - , v E 0 - {z,y}, and uv a line on Q-. For each choice of v E 0 - {z, y} v e nQ- is a non-singular quadric in the (2n - 2)-dimensional space (zyv)'; for each choice of u E &- us n 0 is at most the number of maximal singular subspaces containing a given point of Q- divided by the number of maximal singular subspaces containing a given line on Q-. Hence we obtain the inequality

Hence

542 JA. Thus

i.e.,

Consequently,

for n = 2 101 5 q3 - q2 + q + 1, and

for n > 2 101 5 qn+l - q2 + 2.

Theorem 23

Let 0 be a cap of Wn(q),n = 2 t t 1 and t >_ 1, and let L be a line having m t 1 , g 2 m 2 1, points in common with 0. Then

(a) for n = 3

(b) for n > 3

101 I q2 - mq + q + 1, and

101 I q'+' - mq t m + 1.

In particular, for n > 3

101 5 q'+' - q + 2.

Proof. Let 0 be a cap of Wn(q),n = 2t + 1 and t 2 1, and let L be a line having m + 1,m 2 1, points in common with 0. If 0 is the polarity defining Wn(q), then L' is a (n - 2)-dimensional projective space a, with L n a = 0. Also 0 n a = 0. Now we count in two ways the number of ordered pairs (u,v), with u E a ,v E 0 - L, and uv a line of Wn(q). For each choice of v E 0 - L v' n a is a projective (n - 3)-space; for each choice of u E a ue n 0 is at most the number of maximal totally isotropic subspaces containing a given point divided by the number of maximal totally isotropic subspaces containing a given line of Wn(q). Hence we obtain the inequality

( q + l ) (qa + 1). . . (q t + 1) - m - 1 ] . (101 - m - l)(q"-2 - 1)

q - 1 ( q + l)(qZ + 1). . . (qt-1 t 1)

Hence

i.e.,


Hence, for t = 1 we have 101 5 q2 - mq + q + 1. Now assume t > 1. Then

Consequently 101 5 q'+' - mq + m + 1, and so 101 5 q'+' - q t 2. 0

Remark

Let 0 be an ovoid of W3(q). Then necessarily q is even (cfr. 4.1). By (2) m = 1, SO

each line of PG(3, q ) has at most two points in common with 0. Hence 0 is an ovoid of PG(3, q) .

References

1. Brouwer, A.E. (1981), Private communication, 1981. 2. Cameron, P.J., Thas, J.A., and Payne, S.E.(1976), Polarities of generalized hexagons and perfect codes. Geom. Dedicata, 5:525-528, 1976. 3. Conway, J.H., Kleidman, P.B., and Wilson, R.A. (1988), New families of ovoids in 0:. Geom. Dedicata, 26:157-170, 1988. 4. Dye, R.H.(1977), Partitions and their stabilizers for line complexes and quadrics. Ann. Mat. Pura Appl., 114:173-194, 1977. 5. Hirschfeld, J.W.P.( 1979), Projective geometries over finite fields. Oxford University Press, Oxford, 1979. 6. Hirschfeld, J.W.P.( 1985), Finite projective spaces of three dimensions. Oxford Univer- sity Press, Oxford, 1985. 7. Hodge, W.V.D. and Pedoe, D. (1947) Methods of algebraic geometry (three volumes). Cambridge University Press, Cambridge, 1947. 8. Kantor, W.M. (1982), Ovoids and translation planes. Canad. J . Math., 34:1195-1207, 1982. 9. Kantor, W.M. (1982l) Spreads, translation planes and Kerdock sets. I. SIAM J . Alg. Disc. Meth., 3:151-165, 1982. 10. Kantor, W.M. (19822) Spreads, translation planes and Kerdock sets. 11. SZAM J . Alg. Disc. Meth., 3:308-318, 1982. 11. Ott, U. (1981) Eine Bemerkung uber Polaritaten eines verallgemeinerten Hexagons. Geom. Dedicata, 11:341-345, 1981. 12. Payne, S.E. and Thas, J.A. (1984) Finite generalized quadrangles. Volume 110 of Research Notes in Math., Pitman, Boston-London-Melbourne, 1984. 13. Salzberg, B. (1984) An introduction to Cayley algebra and Ree group geometries. Simon Stevin, 58:129-151, 1984. 14. Shult, E.E. (1985) A sporadic ovoid in R+(8,7). Algebras, Groups and Geometries,

15. Shult, E.E. (1989) Nonexistence of ovoids in R+(10,3). J . Combin. Theory Ser.A, 51:250-257, 1989.

2~495-513, 1985.

544 JA. Thas

16. Tallini, G.(1989) VarietA di sistemi di Steiner. Rend. Mat., 9:545-328, 1989. 17. Tallini, G. (19. . ) Blocking sets with respect to planes in PG(3,q) and maximal spreads of a non singular quadric in PG(4 ,q ) . Proc. of the Conf. on ”Blocking Sets”, Giessen, July 1989. 19.. To appear. 18. Thas, J.A. (1972) Ovoidal translation planes. Arch. Math., 23:llO-112, 1972. 19. Thas, J.A. (1977) Two infinite classes of perfect codes in metrically regular graphs. J. Combin. Theory Ser. B, 23:236-238, 1977. 20. Thas, J.A. (1980) Polar spaces, generalized hexagons and perfect codes. J. Combin. Theory Ser. A , 29237-93, 1980. 21. Thas, J.A. (1981) Ovoids and spreads of finite classical polar spaces. Geom. Dedicata,

22. Thas, J.A. (1983) Semi-partial geometries and spreads of classical polar spaces. J. Combin. Theory Ser.A, 35:58-66, 1983. 23. Thas, J.A. (1989) A note on spreads and partial spreads of hermitian varieties. Simon Stevin, 63:lOl-105, 1989. 24. Tits, J. (1959) Sur la trialitd et certains groupes qui den ddduisent. Znst. Hautes Etudes Sci. Publ. Math., 2:14-60, 1959.

10 ~135- 144, 1981.


Large partial parallel classes in Steiner systems

Zs. Tuza

Computer and Automation Institute, Hungarian Academy of Sciences, H-1 1 1 1 Budapest, Kende u. 13-17, Hungary

Abstract We show that every Steiner system S(k-1, k, v) contains more than v/(k + 1 + l/k) - k

+ 2 pairwise disjoint blocks. This lower bound improves the previously known estimates in the range 3k2 + 3k + c 5 v 5 1p - 2k3 + 2k2 + k, most significantly when v is about k3.

1. INTRODUCTION

In this note we deal with Steiner systems S(k-1, k, v), i.e., families S of k-element subsets (called blocks) B of a v-element set X such that every (k-1)-element subset of X is contained in precisely one block B E S. Apartial parallel class (PPC, for short) is a collection of pairwise disjoint blocks of S.

It is a long-standing open problem to find the best possible general lower bound m(v, k) for the maximum size of a PPC in all systems S(k-1, k, v), as a function of v and k. Two of the unsolved problems in this area are:

(1) Is m(16,4) = 4 ? (2)

The difficulty in answering (1) and (2) in the affirmative is that perfect or nearly perfect packings of blocks should be found in the Steiner systems in question. (A negative answer would require some unusual types of constructions.) Although these basic questions are still open, there are many interesting partial results. For the sake of easier comparison, they are summarized in Tables 1 and 2 for k = 3 (i.e., Steiner Triple Systems) and for k 2 4, respectively. It is remarkable that no combination of those numerous estimates implies any other of them, ie., each of them is best possible for some range of v.

Theorem 1. For all v and k, m(v, k) > v/(k + 1 + l/k) - k + 2.

Is m(v, 3) 2 (v - 4)/3 ?

In this note we prove the following results.

For v relatively small, the lower bound can be improved as follows.

Theorem 2. For k 2 4 and v I (k - 1)2 (k - 2), m(v, k) 2 v/k - 2v/(kt + k) holds, where t is the largest integer not exceeding d(v/(k - 2)) ; hence, m(v, k) > v/k - 2d(v/k).

Recall that a Steiner system S = Sh(h, k, v) of index h is a collection of k-element subsets of a v-element set X such that every h-element subset of X is contained in precisely

546 Z. Tuza

h blocks B E S. It would be of interest to determine the “worst case” with respect to PPCs in the following sense.

Problem 1. Given k, h, and h, determine the largest (or supremum) value of a constant c = c(k, h, h) such that every Steiner system S = Sh(h, k, v) contains a PPC of at least cv blocks; also, characterize those v and S for which c cannot be replaced by a larger constant. Moreover, how much larger can c be if S is supposed to have no repeated blocks?

The simplest particular case is h = 1, i.e., where S is a A-regular k-uniform set system (hypergraph). One might guess that Problem 1 should then become easy, but even this part of the question is unsolved in full generality so far, A complete solution for all k with h = 2 and for all h with k = 2 was given in [8], where all extremal structures were described as well.

2. PROOFS

One of the important steps in the proofs of Theorems 1 and 2 can be formulated as a lemma on unions of bipartite graphs. Recall that a matching in a graph is a collection of pairwise vertex-disjoint edges.

Lemma 1. Let E be the edge set of a bipartite multigraph with vertex partition Y u Z, and suppose that E is decomposed into t matchings El, . . . , E, with the property that each edge e E Ei shares a vertex with each e’ E E, whenever i # j. Then

(i) For t = 1, IEl S IYI. (ii) For t = 2, IEl S 4, with equality if and only if E is the cycle of length four. (iii) For t 2 3, IEl I t + 1, with equality if and only if the underlying graph of E is the path

of three edges, the middle edge having multiplicity t - 1.

Proof. The case t = 1 is trivial. Hence, suppose that t 2 2 and IE I 2 IE I2 . . . 2 IEJ. If IE,I = 1, then IEl = t and we have nothing to prove. Moreover, for Ikll 2 3, no edge of E, could meet all edges of E contradicting the assumed property of the decomposition. Thus, suppose that IEII = 2. In tkis case there are only two (disjoint) pairs e’, e’’ joining Y and Z that meet both edges of El. Consequently, either just one of them is an edge of E, implying (iii), or else both of e’ and e’’ belong to the same class 5, and t = 2 follows since no Y-Z edge can meet all of the four edges of El u E,.

Proof of Theorems 1 and 2. Let (B1,. . . , Bm) be a PPC of maximum size, m, in a Steiner system S = S(k-1, k. v). and set Y‘ = Bl u . . . u B For some integer t, to be specified later, we select t pairwise disjoint (k-2)-element su’6’sets Q1, . . . , Q of X \ Y’. Set Z = X \ (Y’ u Q1 u . . . u Q,). Certainly, this 2 has cardinality v - km - tjk - 2).

Since S is a Steiner system, each (k-1)-element set of the form Qi u (z) (1 I i I t, z E Z) is contained in a unique block B E S. Moreover, by the maximality of the PPC chosen, every such B meets Y’ (in precisely one point, as Q. u (z) is disjoint from Y’). Consequently, each Qi defines a matching Mi from Z to k, and the bipartite multigraph M = Ml u . . . u Mi has precisely t(v - km - t(k - 2)) edges (since each vertex x E Z has degree t in M).

We are going to apply Lemma 1, m times, for Y := B, (1 S j I m). Fix Y (by fixing j) for the moment. and let M i denote the set of edge; in Mi that meet Y. We claim that each edge e E M{ shares a vertex with each e’ E Mi’ whenever i r i’. Indeed, otherwise we could replace B, by the two blocks Q, u e and Qp u e’, obtaining a PPC of size m + 1, a contradiction to the choice of (Bl , . . . , Bm). Thus, Lemma 1 yields that

[I

Large partial parallel classes in Steiner systems 547

lMl’l + . . . + lM(l I max (k, t + 1)

for t 2 3. Summing up this inequality for the t possible choices of Y, and recalling that IMI = t(v - km - t(k - 2)), we obtain

t(v - km - t(k - 2)) I max {mk, mt + m).

If v is large, then we simply put t = k. (If there is no room for that many disjoint (k-a)-element sets, then we already have km + k(k - 2) 2 v, m 2 v/k - k + 2, implying Theorem 1 immediately.) The right-hand side is then equal to m(k + l), and rearranging yields

m(k2 + k + 1) 2 kv - k2 (k - 2),

implying Theorem 1. On the other hand, if v is relatively small ( v I (k - 1)2 (k - 2) ), then we obtain a sharper lower bound on m if t is defined to be the integer part of d(v/(k - 2)) that is less than k. Then max {mk, mt + m) = mk, so that

mk(t + 1) 2 tv - t2 (k - 2) 2 v(t - 1)

holds (as t2 I v/(k - 2) ), and Theorem 2 follows. [I

Table 1 Steiner Triple Systems

m(v, 3) 2 for reference

v 2 4 5 v small

Table 2 Steiner Systems S(k-1, k, v), k 2 4

m(v, k) 2 for reference

(V + 2)/(2k - 2) [41 (2v + 4)/(3k) k 2 5 P I v/(k+ 1 + l / k ) - k + 2 Theorem 1 v/k - 2J(v/k) Theorem 2 (v - k + 2)/(k + 1) P I

v/k - o w v-+- r31

v I (k - 1)2 (k - 2) v 2 1p - k3 - 2k2 + 3k

v/(k + lk) - (2k3 - 5k* + 6k - 1) r91

548 2. Tuza

3.

1

2

3

4

5

6

7

8

9

REFERENCES

A.E. Brouwer, On the size of a maximum transversal in a Steiner triple system, Canad. J. Math. 335 (1981) 1202-1204. F. Eugeni and M. Gionfriddo, On the minimum number of blocks of a maximal partial spread in STS(v) and SQS(v). J. Geometry 36:l-2 (1989) 37-48. P. Frank1 and V. RSdl, Near perfect coverings in graphs and hypergraphs, Europ. J. Combinatorics 6:4 (1985) 317-326. M. Gionfriddo, On the number of pairwise disjoint blocks in a Steiner system, Annals of Discrete Math. 34 (1987) 189-195. C.C. Lindner and K.T. Phelps, A note on partial parallel classes in Steiner systems, Discrete Math. 24:l (1978) 109-112. G. Lo Faro, On the size of parallel classes in Steiner systems STS(19) and STS(27), Discrete Math. 45:2-3 (1983) 307-312. M.C. Marino and R.S. Rees, On parallelism in Steiner systems, Discrete Math., to appear. Zs. Tuza, Matchings and coverings in regular uniform hypergraphs, Ars Combinatoria

D.E. Woolbright, On the size of partial parallel classes in Steiner systems, Annals of Discrete Math. 7 (1980) 203-211.

29C (1990) 122-129.


Minimal Line Distinguishing Colourings in Graphs

B. J. Wilson Department of Mathematics Royal Holloway and Bedford New College Egham Hill Egham Surrey TW20 OEX

Summary

In 141 it is shown that for a graph G of diameter 2 on an even number n of vertices to be minimally line-distinguishing (MU-)colourable we must have A= n/2, where 1 is the line distingushing chromatic number of G. An easy example is the Peterson graph. For the case in which n is odd the condition for a MLD-colouring that every pair of colours occurs exactly

once cannot be met and the best that can be achieved is that A = n12 + I . In [41fig.2 an infinite class of diameter 2 graphs on an odd number of vertices for which 1 = n/2 + I is constructed. In #2 some results for graphs of diameter 2 are found. In particular a class of diameter 2 graphs having A= n12 is exhibited and it is shown that the Peterson graph i s the only strongly regular MLD-colourable graph. In #3 it is shown by construction that a class of regular MLD- colourable graphs of any odd valency p > l exists.

1. Line-Distinguishing Colourings.

A line-distinguishing colouring of a graph G on n vertices is a k-vertex colouring such that each pair of colours appears together on an edge at most once. If there is a line-distinguishing k- vertex colouring of G in which each pair of colours appears exactly once then the colouring is said to be minimal and, for brevity, G is said to be MLD-colourable.

For G to be MLD-colourable using k colours the number m of edges of G must satisfy

m = k(k+1)/2. (1.1)

The smallest value h of k for which G has a k-line-distinguishing colouring is the line- distinguishing chromatic number of G. Clearly if G is MLD-colourable then k = h.

The following lemma is theorem 1 of [3] although it originally stems from [2].

Lemma 1.1. In a line-distinguishing colouring two vertices having the same colow are either adjacent or distance at least 3 apart.

The following results then follow.

550 B.J. Wilson

Corollary 1.2. (a) In a line-distinguishing colouring of a graph of diameter 2 any pair of vertices having the same colour must be adjacent and the edge joining them cannot form part of any triangle.

(b) In a MLD-colouring using n12 colours of a graph G each colour appears exactly twice; the two vertices assigned each colour are adjacent and G is necessarily connected.

(c) In a MLD-colouring of a graph G of diameter 2 necessarily having an even number of vertices the number of colours used is nl2 and each colour is used exactly twice.

can be assigned the same colour. (d) In a line distiguishing colouring of a quadrangle at most 2 of the vertices

It follows from (1.1) that if, and only if, a graph G for which h=n/2 is MLD-colourable then m = (n2+2n)/8. However the following lemma gives a forbidden subgraph, the existence of which in a graph having m = (n2+2n)/8 prevents it from being MLD-colourable.

Lemma 1.3. Let G be a graph with n and m related as above. Then G is not MLD-colourable if there exists in G a pair of non-adjacent vertices u, v with p(u) = p(v) = p to which p vertices are commonly adjacent.

Proof. Suppose that G is MLD-colourable using n/2 colours and that u, v are assigned colours i, j respectively. Then there is a vertex uo assigned colour i and a vertex vo assigned colour j, both of which must be adjacent to both u and v. Then the quadrangle uuowo contradicts corollary 1.2(d).O

An example of a 4-regular graph on 14 vertices which the above lemma shows is not MLD- colourable is shown in fig. 1.

U 5

fig.2

Minimal line distinguishing colourings in graphs 55 1

We call a matching M of the graph G a triangle free matching (TFM) if each edge of M does not belong to any mangle of G. It then follows from corollary 1.2(a) that those edges of a line- distinguishing colouring which join pairs of vertices which have been assigned the same colour form a TFM. We can thus state the following theorem.

Theorem 1.4. A graph G (for which m=(t?+2n)l8 ) is MLD-colourable only ifG has a

The graphs of fig. 1 and fig.2 both have perfect matchings; in the case of that of fig. 1 there is a perfect TFM, so the condition of theorem 1.4 is not sufficient.

perfect TFM.

2.Graphs of Diameter 2.

We have seen that for a graph of diameter 2 we have h 2 n/2, which, for n odd implies h 2 (n+1)/2. A class of graphs for which n is odd and h = (n+1)/2 was given in [3]. We shall describe an infinite class of graphs for which h takes the value n/2,

2.1 colourable graph of diameter 2 on n vertices for which h = n/2. Then we have the following lemmas.

Lemma 2.1. I f n > 2 every vertex of G has valency at least 2 .

Proof. Suppose that G bas a vertex v of valency 1. Then there is, by corollary 1.2(b) a vertex u in G having the same colour as v which must, by corollary 1.2(a), be adjacent to u. Since n > 2 there is at least one more vertex w which, so as not to be any distance further than 2 from v, is adjacent to u ; it follows from corollary 1.2(b) that there is a further vertex x having the same colour as w; by corollary 1.2(a) x is adjacent to w and so that x is distance at most 2 from v it must be adjacent to u. Thus we have found a triangle in G having two vertices of the same colour, contrary to corollary 1.2(b).0

Let u, v be distinct vertices of G which have the same colour. Then u and v must be adjacent and exactly one of u,v must be joined to a vertex of each of the remaining n/2 - 1 colours. Let p(v) denote the valency of the vertex v. Then we have

Graphs of diameter 2 having h = n/2. For this section let G denote a MLD-

p(u) + p(v) = (d2-1) + 2 = n/2 + 1. (2.1).

It follows that since p(u) 2 2 we have

p(v) 5 d2 - 1. (2.2).

i.e. we have proved

Lemma 2.2. The maximum valency of any vertex is ni2 - 1 .

552 E.J . Wilson

r - 1 0 1 1 0 0 0 0 0 1 0 2 1 0 0 0 0 1 0 1 0 0

3 1 0 0 1 1 0 0 0 0 0 4 0 0 1 0 0 0 0 1 0 1

5 0 0 1 0 0 1 1 0 0 0 6 0 1 0 0 1 0 0 0 0 1

7 0 0 0 0 1 0 0 1 1 0 8 0 1 0 1 0 0 1 0 0 0

9 1 0 0 0 00 1 0 0 1 10 0 0 0 1 0 1 0 0 1 0 - -

Lemma 2.3. I f n > 2 then every vertex of G has valency at least 3.

Proof. Suppose that G has a vertex v of valency 2 which is adjacent to vertices u, w. Then any other vertex is adjacent to at least one of u, w. Hence the number of vertices of G is at most 1 + 2 + 2(n/2 - 2) = n - 1, which is a contradiction. 0

Using (2.1) this now gives the following result:

Lemma 2.4. r f n > 2 then every vertex of G has valency at most nl2-2.

It follows that 42-2 2 3 with equality if, and only if, n = 10. In that case the valency of every vertex is 3 and the unique example (of diameter 2) is the Peterson graph. We thus assume from now on that n > 10.

We first look at the case of G being regular of valency p. For such a G we require, by lemma 1.1. that

2m = pn = h(h+l). (2.3)

But also by corollary 1.2(c) h = n/2 so (2.3) leads to pn = n(n+2)/4 whence

4 p = n + 2 (2.4)

Putting h = n/2 gives 2p = h + 1 so h must be odd.

Minimal line distinguishing colourings in graphs 553

fig.3

If A is the adjacency matrix of a graph G which is not a complete graph then to show that G has diameter 2 it is sufficient to show that the matrix

(A + In)2

has a non-zero entry in every position. For the Peterson graph we can write A + 110 in the form

where

Because of the cyclic property of this array the fact that the Peterson graph has diameter 2 is exhibited by observing that every row which begins with a B contains a C which lies under a C in the first row and for every row which begins with a C there is a B which lies under a B in the first row.

We now use these ideas to form a graph Gt on 8t+2 vertices. The first two rows of the mamx A + In are expressed in the form

J B C B C . . . B C C B ... C B C B ,

so that, following the initial J there are t pairs B C followed by t pairs C B. The other rows of the array are formed by cyclically permuting this. Then it is immediately clear that Gt is MLD- colourable using n/2 colours. It remains to show that the diameter of Gt is 2. By the remarks

554 B.J. Wilson

above concerning the Peterson graph it is sufficient to show that each row of the array which begins with a B contains a C under a C in the first row and vice-versa. The following table 1 demonstrates how this is achieved.

row begins with relevant column

i, 2 5 i 5 2 t + l B 2t+2

i, 25i12t+l C 2

i, 2t+2 I i I 4t+l C 4t+l

i, 2t+2 5 i s 4t+l B 2t+l

table 1.

An alternative view of these graphs is that they may be obtained by forming two sets of 4t+l vertices, labelled 1,3, ... ,8t+l and 2,4, ... , 8t+2 respectively. Vertex 2i+l is adjacent to vertex 2i+2 for each i=O,l, ..., 4t and the other adjacencies are as follows (modulo 8t):-

2i+l is adjacent to 2i+3,2i+7, ... ,2i+4t-1

2i+2 is adjacent to 2i+6, 2i+10, ... , 2i+4t+2.

The graph for which t=2 is shown in fig.4.

fig.4


2.3.Strongly Regular Graphs. An infinite class of graphs of diameter 2 of which many examples are known are the strongly regular graphs.

Definition: A graph G is said to be strongly regular if it is not a complete or null graph, is regular and, for each pair of distinct vertices v1, v2 the number of vertices which are adjacent to both v1 and v2 is dependent only upon whether v1 and v2 are adjacent or not.

We use a to denote the number of vertices commonly adjacent to v1 and v2 when they are adjacent and use b to denote the number when they are not adjacent. Then G is connected, of diameter 2, if and only if b > 0. We now assume that this condition holds. If a > 0 it follows that any two adjacent vertices form part of a mangle and hence cannot have the same colour in a line-distinguishing colouring. From corollary 1.2(a) we thus have

Lemma 2.5. I fa > 0 then the connected strongly regular graph G is not MLD-colourable; indeedithas A = n .

Hence to find MLD-colourable graphs of diameter 2 we need to discuss the case in which a = 0. An easy counting argument shows that in this case

p(p-1) = (n-p-1)b. (2.5)

(For a vertex vo count the number of edges (vl , v2) where v1 is adjacent and v2 is not adjacent to vo.121)

From (2.4) and (2.5) we obtain, since p- I# 0,

p = 3b. (2.6)

If b = 1 then the graph has girth 5, the only example being the Peterson graph which we have already seen is MLD-colourable; if b > 1 then the graph has girth 4. The rationality conditions obtained as (2.4) of 121 by considering the eigenvalues of a strongly regular graph show that in our case we require that

( (12b-3)f[(12b-9)b/d(b2+8b)])/2

should be non-negative integers. However the only solution is clearly b=l, this leading to the Peterson graph. The following theorem is thus proved.

Theorem 2.6. The only strongly regular MLD-colourable graph is the Peterson graph.

3. Classes of regular MLD-colourable Graphs using n/t Colours with t 2 2 .

In the next two sections we discuss the construction of two classes of regular graphs for which A=n/t with t 2 2. Both constructions begin by forming a graph G1 in a very similar way to the construction used for GI in #2 of [l]. The proofs that the colourings found here are MLD- colourings are essentially the same as those given in [ 11.

556 B.J. Wilson

3.1 A class of valency 2t-1 on t(2t+l)(t-l) vertices.

The vertex set of G1 is the set Zt x Zzt+1 where Zi denotes the cyclic group of residue classes modulo i. The vertex (i, j) is said to be at level i. Adjacency is defined by

(i) (0, j) is adjacent to (0, j+l) for each j. (ii) (i, j) is adjacent to (i+l, j) for each i and each j. (iii) (1, j) is adjacent to (1, j+t) and to (1, j+t+l) for each j.

We now label the vertices (0, j) with colour j for each j; the vertices at higher levels, i.e. with i z l , are labelled with colours j + (i-I)(i+2)D (modulo 2t+l) for all i, j concerned

Lemma 3.1 The colouring just described is an MLD-colouring of GI, each colour occurring exactly t times, once at each level.

Proof. The proof is essentially the same as that of lemma 2.1 of [ 11; an adjustment has to be made because the vertices at level 1 have been coloured with the same colours as the corresponding vertices in level 0 and there is no level t+l, this being compensated for by the additional adjacencies (iii) above.O

We now form G by first forming t-2 further coloured copies of GI , denoted by G2, G3, ... ,Gt-l and designating the colours on Gk by the labels 0. k) for 0 s j I 2t. The

vertices of Gk are now to be denoted by (i, j, k), the union of these forming the vertex set of G. We define further adjacencies by

(iv) (0, j, k) is adjacent to (0, j-1, k') and to (0, j+ l , k ) for all j and all k' = k

The vertex (0, j) had valency 3 in G1 and now (0, j, k) has valency 3+2(t-2) i.e. 2t-1 in G. The adjacencies so far defined ensure that the graph under construction will be connected. For brevity denote the unique vertex in Gk at level i having colour 0, k) by [i, j , k] and order all the vertices by the rules that [i, j , k] precedes [i, j , k+l], [i, j, k] precedes [i, j+ l , k ] and [i, j , k] precedes [i+l, j', k']. At each stage of the following recursive process call a vertex which has valency less than 2t+l and which may be joined to the vertex under consideration without violating the line discriminating colouring rule "available." Then, beginning at [ l , 1, l,] in turn join each vertex which does not yet have valency 2t-1 to the first available vertices until that valency is reached. This process forms a regular MLD-coloured graph of valency 2t- 1 on t(2t+l)(t- 1) vertices using (2t+l)(t-1) colours each t times.i.e. we can state the following theorem.

Theorem 3.2 For each t 2 2 there exists a regular MLD-colourable graph of valency 2t-1 having n=t(2t+l)(t-l) and A=&.

The case t=2 gives the Peterson graph.

3.2 A class of valency p on t(pt-1) vertices. To justify this section recall that in #2 of [I] it is noted that the increase in the harmonious chromatic number with n whilst the valency is kept constant is O(nln) and a class of graphs of any fixed valency p having the harmonious chromatic number n/t such that each colour is used exactly t times in a minimal harmonious colouring is constructed. A similar construction is now given for a class of MLD-colourable graphs.


The vertex set of G1 is the set Zt x Z2t-1. As before the vertex (ij) is said to be at level i but adjacency is defined only by

(i) (0, j) is adjacent to (0, j+l) for each j. (ii) (i, j) is adjacent to (i+l, j) for each

i and each j.

The colouring of G1 is according to the rule for G1 of #3.1, but taken modulo 2t-1. However since there is now effectively an extra level this is still a MLD-colouring. Now form a graph G2 of which the vertex set is Z, x Zt(,,-2). We use the notation (i,j)2 for vertices of G2 to

distinguish them from vertices of GI. There are now two cases according as t(p-2) is even or odd. We fist consider the even case. The complete graph K Q ~ ) on t(p-2) vertices may be

edge coloured using t(p-2)-1 colours so that each colour appears exactly once at each vertex. Label the vertices of K Q - ~ ) by QJ,. . . ,f(g-2)-1 and suppose such a colowing C to have been

found; partition the set of edge colours into one set So of p-3 colours and t- 1 sets S 1, S2,. . . , S, 2 of p-2 colours. Adjacency in G2 is defined as follows

(i) (0,i)z is adjacent to (OJ2 if and only if 6,j) is an edge assigned a colour of S O in C.

(ii) (O,i)2 is adjacent to (I,i)2 for all i = O, l , ...( t-l)(p-1)-1

(iii) (k,i)2 is adjacent to (k,j)2 if and only if 6,j) is an edge assigned a colour of sk in C.

Colours are assigned to the vertices of G2 such that (i,j)2 is assigned the colour j* for j=O,l, ..., t(p-2)-1. This gives a MLD-colouring of the disconnected graph G2. Of the vertices of G2 those (i,j)2 for which i= 0,2,3, ... t-1 have valency p-2 and those for which i=l have valency p-1. We now connect GI to G2 to form the required graph G. The number of new edges required to raise the valency of the vertices of G1 to p is

(2t-l)[(p-3)+(t-2)(p-2)+(p-l)l = (2t-l)t(p-2)

= M say,

of which t(p-2) are required by the vertices of each colour. The number of new edges needed to raise the valency of each vertex of G2 to p is

2(t-l)t(p-2) + t(p-2) = t(2t-l)(p-2)

= M,

of which 2t-1 are required by the vertices of each colour. Thus the graphs G1 and G2 may now be joined by M suitably chosen edges to form the regular MLD-colourable graph G of valency p

558 B.J. Wilson

on t(pt- 1) vertices.

The details of the case in which t(p-2) is odd are similar to those given in [ 11 for the construction of minimally harmoniously colourable graphs. Therein may also be found details of a systematic method of joining G1 and G2. We thus have the following theorem.

Theorem 3.3 For each t22 andeach p22 there is a regular graph of valency p having a MU-colouring in which each colour occurs t times.

Note that for this graph we have h=O(nln). The case of t=2, p=3 does not give the Peterson graph.

References

[l] D.J.Beane, N.L.Biggs and B.J.Wilson. The Growth Rate of the Harmonious Chromatic Number. J.Graph Theory 13 291-299 (1989)

[2] P.J.Cameron and J.H.van Lint. Graphs, Codes and Designs. LMS Lecture Note Series 43 (1980).

[3] O.Frank, EHarary and M. Plantholt. The line-distinguishing chromatic number of a graph. Ars Combinatoria 14, 241-252, (1982).

[4] B.J.Wilson. Line Distinguishing and Harmonious Colourings. in Graph Colourings, Ed. Roy Nelson and Robin J Wilson, Piman Research Notes in Mathematics Series 218, (1990).


Some Results on -1 Multiplier of Difference Lists

Qing Xiang

Department of Mathematics, Sichuan University, Chengdu 610064, P.R. China

Abstract

strengthen the results in [2]. Also we give some applications of our results. In this paper, we obtain several -1 multiplier theorems for difference lists which

1. INTRODUCTION

Let G = { g l , g z , . . . , g,} be a group of order v, A = CE, aigi be an element of the group ring ZG. For any integer t , and g E G, we define A( t ) = C,"=, uigf , Ag = Ck, nig ig . An integer t , ( t , ~ ) = 1, is called a multiplier of A if A(t) = Ag, for some g E G. Now let p be a positive integer, L = C,"=, sigi , s 1 2 0, is called a (s1, sz,. . . , s,; p ) difference list, if it, satisfies the following:

LL(-1) = c s: - p + pG. 1 = l

Obviously we have following relation: U 1,

I = , := 1

If G is abelian or cyclic, L is called abelian or cyclic difference list respectively. Arasu and Ray-Chaudhuri investigated multiplier theorem and -1 multiplier theo-

rem for difference list,s in [2]. We continue their work in this paper, we will see that the fact L has -1 niultiplier implies severe restrictions on both the parameters of L and structure of G. With these restrictions, we can obtain nonexistence results for (172 , n, k , X I , Xz)-divisible difference sets.

2. KNOWN RESULTS

In this section, we quote some known results. Theorem A [ 2 ] . Lct L be (sl, s 2 , . . . , s , ; p ) difference list of abelian group G of order

v. If t is a multiplier of L , ( t - 1;u) = 1, then there exists a translate of L which can be fixed by t .

Theorem B [ 2 ] . Let L be (s1, sz,. . . , s , ; p ) difference list of abelian group G of order v. If v is odd, -1 is a multiplier of L , then all sir i = 1,2 , . . . ,v but one are congruent to p mod 2.

5 60 Q. Xiang

Theorem C [7 ] . Let D be ( I ) , k , A ) difference set in ahelian group of order 1 ) . If -1 is G / H multiplier of D, whew H is proper subgroup of G, then either n is square or there exists a prime p such tliat:

(1) The square-free part of 11 is p . (2) The order of G / H is a power of p . (3) p = l ( inod4) . Remark: In [?I1 Amsu and Ray-Chaudhuri use Theorem B to obtain nonexistence

results for some familirs of (21, I.', A ) difference set. In fact, if we use Theorem C for this purpose, it will yield better results.

3. MAIN RESULTS

In this section, me assunie G is abelian group, and we use z* to denote the square-free part of z.

Theorem 3.1. Let L be ( $ 1 , s2,, . . , st,; 11) difference list in group G of order w. If 2,

is odd, -1 is a miiltiplier of L , and there is an odd prime p , pl C:='=, sf - p, ( p , w) = 1, then all s,, 2 = I , ? , . . . , t i are congruent mod p . Proof. Since -1 is a multiplier of L , 'u is odd, by Theorem A, we can assume L(-1) = L. By (1.1), we have

i = l

where (1 is an iiitegcr, /i(x;Ll s , ) l n . Since pI C:=, sp - p , we have

L ( p ) E LJ' = nG (inotlp)

Note that ( p , w) = 1, so s Z = a ( n i o c l p ) , z = 1 ,2 , . , . ,o. The proof here is similar to that in [lo]. Theore~n 3.2. Let L (s1, sz, . . . , +,; p ) difference list in G. If -1 is a multiplier of

L , then either C:=, s: - 11 is a square or there exists a prime p such that (1) ( C L l sf - P I * = (2) w is a power of p , (3) p = l( inod4).

Proof. If Cy='=, ~f - 11 is not a square, we assume that pl(Cy=l sf - p ) ' , since -1 is a multiplier of L , we have L( -1) = Lg for some g E G. By (1.1)

1 2 1

Let x be any noiiprincipal character of G. Then

i= 1

Some results on -1 multiplier of dizerence lists 56 I

If there is a prime q, q # p , qlv, then we can choose a nonprincipal character x of order q. Viewing (3.1) as the identity of ideals in Z[(,],

where tQ is q-th root of unity, x(g) is unit in &(I,). Since ( p , q ) = 1, so every prime ideal containing p in Z[( ] is unramified (see [4]), thus p appears even times in the canonical factorization of Xi= ] sf - p, this contradicts with PI(X:=~ sf - p ) * . So v is a power of p , we assume v = p a . Let x be nonprincipal character of G , x: G H & ( I p - )

VQ

If p is odd, then x ( g ) is a square in &(Ip*), so Jj' E &(&,a). Since Q ( d ( - l ) F p ) is the unique squa.re subfield in Q ( t P a ) , thus p = 1mod4. If p = 2, then v is even, we can choose a character x of order 2 such that x ( L ) = x ( L ) , so Cbl s: - p is a perfect square, this is a contradiction. Hence p

We discuss some applications of the above theorem. Let D be an ( m , n , k ,X l ,Xz) - divisible difference set of G relative to N . Then

-

1 mod4.

We choose H such that N 5 H 5 G, let o:G + G / H be the natural homomorphism. Then

a(D)a(D)( -1) = k2 - Xzmn + X21HIG/H

thus a ( D ) is a difference list in G / H . Assuming IG/HI = b, if a is a multiplier of D or a ( D ) and a' E -1 mod b, and k 2 - Xzmn is nonsquare, by above theorem, b must be a power of prime p , p

Example: Let G be abelian group of order v = 45, N be a subgroup of G of order 5 . There is no divisible difference set in G relative to N with parameter (9,5,13,9,3).

We choose H to be a subgroup of order 15, then

1 mod4.

o(D)o(D) ( - l ) = 34 + 45G/H

note that 2 2

every t , ( t , v ) = 1, L ( t ) = i if and only if Cr=l s: - p is square. Proof. By (1.1) and L(-1) = L, we have

17mod3, by the multiplier theorem in [2], 2 is a multiplier of a ( D ) , but

Theorem 3.3. Let L be ( ~ 1 ~ ~ 2 , . . . ,sv; p ) difference list in G. If L(-1) = L , then for -1 mod 3, b = 3, this is a contradiction. Hence our conclusion follows.

v

L2 = c s : - p + pG. i= 1

562 Q. Xiang

Let x be any nonprincipal character of G. Then

If C;=I s: - p is a square, then x ( L ) E Q. Let CT: tu H t i be an automorphism of &((,,), where t, is the v-th primitive root of unity. If ( t , v ) = 1, then CT E GalQ(<,)/Q. So x ( L ) = o ( x ( L ) ) = x ( L ( t ) ) for all characters x of G, hence L ( t ) = L. Conversely, if for every t , ( t , v ) = 1, L ( t ) = L, then a ( x ( L ) ) = x ( L ( t ) ) = x ( L ) for all 0 E GalQ((,,)/Q so k ( L ) E Q, thus Cr=l s: - p is a square. This completes the proof.

Theorem 3.4. Let L be (s1, ~ 2 , . . . , su*; p ) difference list in group G of odd ordcr. If -1 is a multiplier of L , pI(Cy=I s: - p ) , then for every integer t , ( t , v ) = 1, L ( t ) = L if and only if t is a square mod p . Proof. Since -1 is a multiplier of L , v is odd, by Theorem A, we can assume L(-1) = L. By (l.l), we have

z=1

By Theorem 3.2, ifpl(C:', s : -p)* , then 2) = p a , p E l(mod4). Let x be a nonpriricipal character of G. Then

(3.2)

y(L) is contained in square subfield E of Q(tu) , now GI = GalQ(J,)/Q 2 ( Z / ( p * ) ) * is a cyclic group. Assuming H 5 G I , and [GI : H ] = 2, then H is unique. By Galois theory, E is also unique, and E = InvH.

Q ( E u ) 2 E 2 Q 1 1 5 H 5 GI = { 1,g, y 2 , . . . , g ' p ( p u ) - l

where g is primitive root of mod p a . If t is a square mod p , then CT E H . Since x ( L ) E E for all charactrrs of G, thus o ( x ( L ) ) = x ( L ) . That, is to say x ( L ( t ) ) = x ( L ) for all characters of G, so L ( t ) = L. Conversely, if for every integer t , ( t , v ) = 1, L ( t ) = L , then x ( L ( t ) ) = x ( L ) for all cliaracters of G, that is a ( x ( L ) ) = x ( L ) , but x ( L ) E E = InvH, so D E H , t is square mod p . This completes the proof.

The proof here is similar to that in [7]. In the next part of this section, we will deal with the case that C,"=l s f - ti is a

Theorem 3.5. Let e(z) = Crzi a,z* be p difference list in cyclic group of odd order square. The methods we use are siinilm to those in [8].

v. If -1 is a multiplier of Q(z), ,Ll = Cyzi u: - p is a square, then

w - 1

Some results on -1 multiplier of difference lists 563

where n is an integer. Proof. By ( l . l ) , we have

~ ( x ) ~ ( x - ' ) = Cup - p + pT(.r)

where T( x ) = 1 + x +. . . + .rt '-I. Since -1 is a multiplier of O(z), w is odd, by Theorem A, we can assiiiiie H ( x ) = O ( . r - I ) . So we have

1 , - I

(nioc1.r" - 1) 1=0

1'- 1

O(z)2 = Car - 11 + p T ( . r ) (In0d.r" - 1). (3 .4) 1 = 0

Now 6 = C:zt; - p IS ii q u a r e , hy Theorem 3.3, O(.r') f O(s) (Inodz'' - 1) for all 1, ( 1 , ~ ) = 1. Tliis C I , = ( I ! , if ( 2 , ~ ) = (1, u ) , where (a ,b ) is the greatest common divisor of a , b, and [o, b] = o b / ( t r , h ) . So we can write O(.r) as follows:

Let .(w) = p ( d ) h ( f-), W ~ U , where p ( d ) is the Mijhius funct,ion. Then

w - I

U I l ~ 1=0

By (3.4) aiitl (3.5) a i d direct iiiultiplication, we have

w-I

w-1 1 ' - 1

564 Q. Xiang

Particularly, we choose i = 1, then c (v ) = p,

c ( w ) = 0, W I D , w # 1 , v .

Summing up

0, if W I D , w # 1 , v ; p, if w = 1; p , if w = v.

By (3.6), we have

(3.7)

By (3.4) and (3.5), we also have

In (3.7), let w = p , plv, p is a prime, then

2a(l)a(p) + a(P)2P = 0

SO a ( p ) = 0, if p / & and a ( p ) = f 2 f i / p , if p l d . Now we assume for wlv, 1 < w < v, we have

Some results on -1 multiplier of difference lists 565

where n is an integer, U ( W ) = f m ) f l / w . Since U ( W ) is an integer, so 1 + n is a square. Let no = f 1 f + 1 + n. Then U ( W ) = n o f i / w .

Case 2. w / G . Since [wl,wZ] = w , w1,wZ # w , thus at least one of w1, w2 can not divide fl. So a(wl)a(wz)(wl ,wz) = 0. By (3.7), 0 = f 2 f i a ( w ) - W U ( W ) ' , but W I U , v is odd, w x d , so a(@) = 0. This completes the proof.

Corollaq 3.6. Let e(z) = Cyii a;ti be a p difference list in cyclic group of odd order, p = cyzi u: - p = 1. If -1 is a multiplier of e(z), then e(z) a(1) + a ( v ) T ( z ) (mods" - l), where a( 1) and a(.) are two integers. Proof. Since -1 is a multiplier of e(z), by Theorem 3.5, we have

w- 1 - -

e(s) a ( ~ ) 1 x " ~ ' ~ + a ( v ) T ( r ) (modz" - 1)

Now ,B = 1, so e ( r ) a(1) + a ( v ) T ( z ) (niodx" - l) , and 2u(l)a(v) + vu(v)' = p,

difference list of group G of odd prime order. If -1 is a multiplier of e(x), p = Crci a: - p is a square, then e(z) = a(1) + a(v)T(z) (modx" - l), where a ( l ) , a(.) are two integers.

a(1) = fl. Corollary 3.7. Let e( r ) = alxz be

The proof is obvious. We give an application of the above results. Example. Let G be a cyclic group of order 28, N its subgroup of order 2. Then

there exists no (14,2,11,6,4) divisible difference set in G. If there exists such a divisible difference set D, then

DD(-') = 5 + 6 N + 4(G - N ) .

Choosing subgroup H , N 5 H _< G, IG/HI = 7, u :G + G / H is natural homomorphism, then cr(D)o(D)(-1) = 9 + 16GIH. Now CyIi a: - p = 9 = 3', by the multiplier theorem in [2] , we know 3 is a multiplier, but 33 f -1(mod7), so -1 is a multiplier of u ( D ) , by Corollary 3.7, we have o ( D ) = a(1) + a(7)T(z), a(1) = f 3 , 2a(l)a(7) + 747)' = 16, note that 4 7 ) 2 0, so a(1) = -3, 4 7 ) = 3, but a(1) + 4 7 ) = -1, this is a contradiction. Hence our conclusion follows.

Acknowledgement. The author thanks Prof. Wan Di Wei for his helpful discussions.

REFERENCES

1

2

3

4

ICT. Arasu, D. Jungnickel and ,4. Pott, Divisible difference set with multiplier -1, J. Algebra. 133 (1990), 35-62. I<.T. Arasu and D.K. Raychaudhuri, Multiplier theorem for difference list, Ars Com- binatorica 22 (1986), 119-137. T. Beth, D. Jungriickel and H. Lenz, Design Theory, Bibliographisches Institut, Mannheim (1985). I<. Ireland and M. Rosen, A classical introduction to modern number theory, Springer, Berlin-Heidelberg, New York (1982).

566 Q. Xiang

5 H.P. KO and D.K. Raychaudhuri, Multiplier Theorems, J. Comb. Theory ser.A 30

6 H.P. KO and D.K. Raychaudhuri, Intersection theorems for group divisible difference sets, Disc. Math. 39 (1982), 37-58.

7 E.S. Lander, Symmetric Design, An algebraic approach, London Mathematics Soci- ety Lecture Notes Series 70 (1983).

8 S.L. Ma, Partial difference sets, Disc. Math. 52 (1982). 9 S.L. Ma, On divisible difference sets which are fixed by the inverse, Arch. Math. 54

10 A. Pott , On abelian difference sets with multiplier -1, Arch. Math. 52 (1989),

(1981), 134-157.

(1990), 409-416.

51&512.

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