The Fourth Janko Group

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The Fourth Janko Group

A. A. IVANOV

CLARENDON PRESS • OXFORD

2004

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PREFACE

We start with the following classical situation. Let V be a 10-dimensional vectorspace over the field GF (2) of two elements. Let q be a non-singular quadraticform of Witt index 5 on V . Let H ∼= O+

10(2) be the group of invertible lineartransformations of V that preserve q. Let Ω = D+(10, 2) be the dual polargraph associated with the pair (V, q), that is the graph on the set of maximalsubspaces of V which are totally singular with respect to q (these subspacesare 5-dimensional); two such subspaces are adjacent in Ω if and only if theirintersection is of codimension 1 in each of the two subspaces. Then (Ω, H) =(D+(10, 2), O+

10(2)) belongs to the class of pairs (Ξ, X), where Ξ is a graph andX is a group of automorphisms of Ξ satisfying the following conditions (C1)to (C3):

(C1) Ξ is connected of valency 31 = 25 − 1;(C2) the group X acts transitively on the set of incident vertex-edge pairs

in Ξ;(C3) the stabilizer in X of a vertex of Ξ is the semi-direct product with

respect to the natural action of the general linear group in dimension5 over the field of two elements and the exterior square of the naturalmodule of the linear group.

The constrains imposed by the conditions (C1) to (C3) concern only ‘local’properties of the action of X on Ξ and these properties remain unchanged whenone takes suitable ‘coverings’. The constrains are encoded in the structure of thestabilizers in X of a vertex of Ξ and of an edge containing this vertex, and alsoin the way these two stabilizers intersect. Denote the vertex and edge stabilizersby X [0] and X [1], respectively and ‘cut them out’ of X to obtain what is calledthe amalgam

X = X [0], X [1]

(the union of the element-sets of the two groups with group operations coincidingon the intersection X [01] = X [0] ∩ X [1]). Because of (C2) the isomorphism typeof X is independent of the choice of the incident vertex–edge pair. If the pair(Ξ, X) is simply connected which means that Ξ is a tree, then X is the universalcompletion of X which is known to be the free product of X [0] and X [1] amal-gamated over the common subgroup X [01]. A reasonable question to ask is aboutthe possibilities for the isomorphism type of X . The following lemma gives theanswer.

Lemma A. Let (Ξ, X) be a pair satisfying (C1) to (C3) and let X be the amal-gam formed by the stabilizers in X of a vertex of Ξ and of an edge incident to

vii

viii Preface

this vertex. Then X is either the classical amalgam H contained in H = O+10(2)

or one extra amalgam G = G[0], G[1].

In a certain sense the existence of the additional amalgam G in Lemma Ais due to the famous isomorphism of the general linear group in dimension fourover the field of two elements and the alternating group of degree eight.

In order to obtain all the pairs satisfying (C1) to (C3) we should consider thequotients of the universal completions of the amalgams H and G over suitablenormal subgroups. However, there are far too many possibilities for choosingthat normal subgroup and the project of finding them all appears fairly hopeless.Nevertheless, we may still try to find particular examples which are ‘small andnice’ in one sense or another.

It can be shown that for every pair (Ξ, X) satisfying (C1) to (C3) there isa ‘nice’ family of cubic (that is valency 3) subgraphs in Ξ as described in thefollowing lemma.

Lemma B. Let (Ξ, X) be a pair satisfying (C1) to (C3). Then Ξ contains afamily S of connected subgraphs of valency 3. This family is unique subject tothe condition that it is stabilized by X and whenever two subgraphs from S sharea vertex and if the neighbours of this vertex in both subgraphs coincide, the wholegraphs are equal.

The subgraphs forming the family S in Lemma B are called geometriccubic subgraphs. If Ξ is a tree then every geometric cubic subgraph is acubic tree which is ‘large’, even infinite. On the other hand, in the classicalexample (D+(10, 2), O+

10(2)) the geometric cubic subgraphs correspond to the3-dimensional totally singular subspaces in the 10-dimensional orthogonal spaceV . The subgraph corresponding to such a subspace U is formed by the maximaltotally singular subspaces in V containing U . This subgraph is complete bipart-ite on 6 vertices denoted by K3,3. Thus here the geometric cubic subgraphs aresmall and nice.

Let (Ξ, X) be a pair satisfying (C1) to (C3). Let Σ be a geometric cubicsubgraph in Ξ, let S and T be the global and the vertexwise stabilizers of Σ inX. Let Σ be the graph on the set of orbits of the centralizer CS(T ) of T in S onthe vertex set of Σ in which two orbits are adjacent if there is at least one edgeof Σ which joins them. Then the natural mapping ψ : Σ → Σ turns out to be acovering of graphs commuting with the action of S. Put

S = S/(TCS(T ))

which is the image of S in the outer automorphism group of T .Direct but somewhat tricky calculation in the amalgams H and G give the

following.

Preface ix

Lemma C. In the above terms the following hold:

(i) if X = H then S ∼= Sym3 Sym2 and Σ is the complete bipartite graphK3,3:

(ii) if X = G then S ∼= Sym5 and Σ is the Petersen graph:

By the remark after Lemma B if the pair (Ξ, X) is (D+(10, 2), O+10(2)) then

the mapping ψ : Σ → Σ is an isomorphism. The following characterization hasbeen established by P. J. Cameron and C. E. Praeger in 1982.

Proposition D. Let (Ξ, X) be a pair satisfying (C1) to (C3) with X = H andsuppose that ψ : Σ → Σ is an isomorphism. Then (Ξ, X) = (D+(10, 2), O+

10(2)).

It is natural to ask what happens when Σ attains the other minimal possibilityin Lemma C, that is the Petersen graph. The main purpose of this book is toanswer this questing by proving the following

Main Theorem. Let (Ξ, X) be a pair satisfying (C1) to (C3) with X = G andsuppose that ψ : Σ → Σ is an isomorphism. Then the pair (Ξ, X) is determineduniquely up to isomorphism. Furthermore

(i) X is non-abelian simple;(ii) |X| = 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43 =

86, 775, 571, 046, 077, 562, 880;(iii) X contains an involution z such that CX(z) ∼= 21+12

+ · 3 · Aut (M22).

The group X in the Main Theorem is the fourth Janko sporadic simple groupJ4 discovered by Zvonimir Janko in 1976 and constructed in Cambridge in 1980by D. J. Benson, J. H. Conway, S. P. Norton, R. A Parker, J. G. Thackray as asubgroup of L112(2).

x Preface

The book was derived from research which originated almost exactly twentyyears ago. This was the golden age of the research seminar on algebra and geo-metry at the Institute for System Studies (VNIISI) in Moscow. Some of itsmembers were examining a special case of the quasithin groups’ classification(and J4 is a quasithin group); others were searching for new distance-transitivegraphs. The latter activity led me to the discovery of the geometry F(J4). Atthis stage our quasithin experts suggested that the discovery might eventuallyemerge into a geometric construction and uniqueness proof for J4. The projectthat seemed preposterous at that time has been fully realized here.

My foremost gratitude goes to Vladimir L’vovich Arlazarov and IgorAlexandrovich Faradjev, who both founded and maintained the VNIISI seminar.The crucial ingredient in the characterization is the simple connectedness of thegeometry F(J4) which I proved in the summer of 1989 within the joint projectwith Sergey Shpectorov on the classification of the Petersen and tilde geometries.I am very thankful to Sergey for his long term friendship and cooperation andhope this will last. When the proof was discussed in (then still West) Berlin inJune 1990, Geoff Mason noticed that the simple connectedness formed a basis forthe uniqueness proof for J4. At the Durham Symposium in July 1990, togetherwith Ulrich Meierfrankenfeld, we found a way of transferring the simple connec-tedness into a computer-free construction. It took a further ten years to see theconstruction published. Many original ideas here and indeed almost the wholeof chapter 8 are due to Ulrich.

The main layout of the book was designed when I gave a series of lectures atthe University of Tokyo in autumn 2002. I am very thankful to Atsushi Matsuowho organized these lectures and to the audience for many stimulating questionsand discussions. My special thanks go to Satoshi Yoshiara and Hiroki Shimakurawho took the notes of the lectures; I used these notes as a draft for the firsttwo chapters of the book. Hiroki wrote the notes in Japanese and now theyare published as (Ivanov 2003). In order to proceed with the project I neededa transparent description of the pentad subgroup in J4. This was achieved inOberwolfach in summer of 2003, again, thanks to a fruitful cooperation withSergey Shpectorov. The writing up began in September 2003 and took aboutsix months. Half way through I received useful comments on the draft fromErnie Shult. His note: ‘nice’ against Lemma 4.5.5 was particularly inspiring.Thank you, Ernie. Corinna Wiedorn is one of the very few who read through ourconstruction paper with Ulrich. So even during the early stages of working onthe book, I was sure that it will have at least one dedicated reader. Her incisivecomments on the final draft exceeded my best expectations. Antonio Pasini notonly kindly offered to read the final draft but also assured me that this wasnot a favour at all and that this was solely for his own pleasure. His thoughtfulcomments are of great value to me.

I am pleased to acknowledge that my the insight in J4 was gradually built upthrough discussions with experts including Michael Aschbacher, John Conway,Wolfgang Lempken, Simon Norton, Richard Parker, Gernot Stroth and RichardWeiss. Whenever I got stuck, Dima Pasechnik was always ready to help

Preface xi

by performing another ingenious computer calculation. I am thankful to mycolleagues and friends including Colin Atkinson, Michael Deza, Martin Liebeck,Jan Saxl and Leonard Soicher for encouragement to write a book whose titlewill be just ‘J4’. The reader can judge to what extent I have achieved this goal.Last, but certainly not least, my sincere gratitude goes to my wife Lena, my sonDenis and my daughter Nina for support, understanding and immense patienceover the years.

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CONTENTS

1 Concrete group theory 11.1 Symplectic and orthogonal GF (2)-forms 11.2 S2m(2), O+

2m(2), and O−2m(2) 4

1.3 Transvections and Siegel transformations 61.4 Exterior square via symplectic forms 81.5 Witt’s theorem 91.6 Extraspecial 2-groups 101.7 The space of forms 121.8 S4(2) and Sym6 131.9 Ω+

6 (2), Alt8, and L4(2) 151.10 L2(7) and L3(2) 16

2 O+10(2) as a prototype 18

2.1 Dual polar graph 182.2 Geometric cubic subgraphs 222.3 Amalgam H = H [0], H [1] 232.4 The universal completion 252.5 Characterization 27

3 Modifying the rank 2 amalgam 303.1 Complements and first cohomology 303.2 Permutation modules 333.3 Deck automorphisms of 26 : L4(2) 373.4 Automorphism group of H [01] 403.5 Amalgam G = G[0], G[1] 433.6 Vectors and hyperplanes in Q[0] 443.7 Structure of G[1] 453.8 A shade of a Mathieu group 47

4 Pentad group 23+12 · (L3(2) × Sym5) 494.1 Geometric subgroups and subgraphs 494.2 Kernels and actions 514.3 Inspecting N [2] 554.4 Cohomology of L3(2) 574.5 Trident group 604.6 Automorphism group of N [2] 644.7 D12, D8-amalgams in Sym6 69

xiii

xiv Contents

4.8 The last inch 714.9 Some properties of the pentad group 74

5 Towards 21+12+ · 3 · Aut (M22) 76

5.1 Automorphism group of N [3] 765.2 A Petersen-type amalgam in G[3] 775.3 The 12-dimensional module 805.4 The triviality of N [4] 82

6 The 1333-dimensional representation 846.1 Representations of rank 2 amalgams 846.2 Bounding the dimension 866.3 On irreducibility of induced modules 886.4 Representing G[0] 896.5 Restricting to G[01] 926.6 Lifting Π[01]

11 946.7 Lifting Π[01]

22 966.8 Lifting Π[01]

12 ⊕ Π[01]21 99

6.8.1 24-dim representations of CG[1](h) 1006.8.2 Structure of Π(h)

12 1046.8.3 Structure of Π(h)

21 1086.8.4 Gluing 111

6.9 The minimal representations of G 1126.10 The action of G[2] 1146.11 The centralizer of N [2] 1196.12 The fundamental group of the Petersen graph 1206.13 Completion constrained at level 2 126

7 Getting the parabolics together 1317.1 Encircling 21+12

+ · 3 · Aut (M22) 1317.2 Tracking 211 : M24 1337.3 P -geometry of G[4] 1367.4 G[4] = G 1387.5 Maximal parabolic geometry D 1397.6 Residues in D 1417.7 Intersections of maximal parabolics 143

8 173,067,389-vertex graph ∆ 1478.1 Defining the graph 1478.2 The local graph of ∆ 1508.3 Distance two neighbourhood 154

Contents xv

8.3.1 Analysing 2-paths 1568.3.2 Calculating µ2 1608.3.3 Calculating µ3 1638.3.4 The µ-subgraphs 166

8.4 Earthing up ∆13(a) 168

8.5 Earthing up ∆23(a) 174

8.6 |G | = 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43 1828.7 The simplicity of G 1868.8 The involution centralizer 188

9 History and beyond 1909.1 Janko’s discovery 1909.2 Characterizations 1929.3 Ronan–Smith geometry 1939.4 Cambridge five 1949.5 P -geometry 1949.6 Uniqueness of J4 1959.7 Lempken’s construction 1969.8 Computer-free construction 1969.9 The maximal subgroups 1969.10 Rowley–Walker diagram 1979.11 Locally projective graphs 1979.12 On the 112-dimensional module 2009.13 Miscellaneous 201

10 Appendix: Terminology and notation 20210.1 Groups 20210.2 Amalgams 20310.3 Graphs 20410.4 Diagram geometries 208

11 Appendix: Mathieu groups and their geometries 21311.1 Witt design S(5, 8, 24) 21311.2 Geometries of M24 21511.3 Golay code and Todd modules 21811.4 Shpectorov’s characterization of M22 21911.5 Diagrams of H(M24) 22311.6 Diagrams of H(M22) 225

References 227

Index 232

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1

CONCRETE GROUP THEORY

According to the current state of art, the existence of large sporadic simplegroups (and J4 is certainly one of them) is explained by exceptional propertiesof concrete small groups. We invite the reader to join the devious trail to J4,which wends its way through peculiar features of common first members of other-wise disjoint families of finite simple groups. As a basic unifying concept we havechosen the class of symplectic and orthogonal forms over the field of two elements.For basic definitions, notation and background results in this and further chaptersthe reader is advised to consult the Appendices and well as standard literature.

1.1 Symplectic and orthogonal GF (2)-forms

Let V be an n-dimensional GF (2)-vector space. A mapping

f : V × V → GF (2)

is a symplectic form on V if it is bilinear and for all u, v ∈ V we have f(u, v) =f(v, u) and f(u, u) = 0. For a symplectic form f a mapping

q : V → GF (2)

is an orthogonal form on V associated with f if

f(u, v) = q(u) + q(v) + q(u + v)

for all u, v ∈ V .With V as above let f be a symplectic form on V and let q be an ortho-

gonal form associated with f . Then the pairs (V, f) and (V, f, q) are calledsymplectic and orthogonal spaces, respectively (although f determines q in thelatter case, we still prefer to keep the explicit reference to f).

For S ⊆ V by f |S and q|S we denote the restrictions of f and q to S ×S andS, respectively. It is useful to keep in mind the following easy observation.

Lemma 1.1.1 Let (V, f) be an n-dimensional symplectic space and B be a basisof V . Then for every function h : B → GF (2) there is a unique orthogonal space(V, f, q), such that q|B = h. In particular there are exactly 2n orthogonal formsassociated with a given symplectic form.

Let (V, f) be a symplectic space. Two vectors u, v ∈ V are said to be per-pendicular if f(u, v) = 0 (particularly every vector is perpendicular to itself).The set of vectors perpendicular to a given vector u ∈ V is called the perp ofu denoted by u⊥. If S ⊆ V then S⊥ consists of the vectors perpendicular to

1

2 Concrete group theory

every vector of S. It is clear that S⊥ is a subspace and S⊥ = 〈S〉⊥ (here andelsewhere 〈S〉 is the subspace in V spanned by S). The symplectic space (V, f)is non-singular if V ⊥ = 0 and singular otherwise.

Lemma 1.1.2 Let (V, f) be a non-singular symplectic space and U be a subspaceof V . Then

dim U + dim U⊥ = dim V.

Proof For w ∈ V the map lw : u → f(u, w) is a linear function on U , that isan element of the dual space U∗ of U . Since (V, f) is non-singular, the mappingν : V → U∗, where ν(w) = lw is a surjective homomorphism of vector spaces,and by the definition, U⊥ is the kernel of ν. Now the result is immediate fromelementary linear algebra.

Two symplectic spaces (V, f) and (V ′, f ′) are isomorphic if there is an iso-morphism α : V → V ′ of vector spaces such that f(u, v) = f ′(α(u), α(v)). Theisomorphism of orthogonal spaces is defined in the obvious similar way. Let (U, g)and (W, h) be symplectic spaces. The direct sum of these spaces is a symplecticspace (V, f), where V is the direct sum of U and W ; f |U = g and f |W = h,and (with respect to f) every vector from U is perpendicular to every vectorfrom W . In the above terms if p and r are orthogonal forms associated with gand h, respectively, then the direct sum of (U, g, p) and (W, h, r) is an orthogonalspace (V, f, q) such that (V, f) is the direct sum of (U, g) and (W, h), and q theonly orthogonal form associated with f such that q|U = p and q|W = r.

We use the symbol ⊕ to denote the direct sum operation on symplectic andorthogonal spaces. When the spaces are considered up to isomorphism, thisoperation is commutative and associative. The following two lemmas are ratherstraightforward.

Lemma 1.1.3 The direct sum (U, g)⊕ (W, h) is non-singular if and only if boththe summands are such.

Lemma 1.1.4 Let (V, f, q) be an orthogonal space. Suppose that V is the directsum of its subspaces U and W with U ≤ W⊥ and W ≤ U⊥. Then (V, f, q) ∼=(U, f |U , q|U ) ⊕ (W, f |W , q|W ).

Now we are well prepared to present the well-known classification of thenon-singular symplectic and associated orthogonal spaces.

Lemma 1.1.5 Let (V, f) be an n-dimensional non-singular symplectic space.Then

(i) n is even, say n = 2m;(ii) the isomorphism type of (V, f) is uniquely determined;(iii) (V, f) is the direct sum of m hyperbolic planes.

Proof We proceed by induction on n. If n = 1, then the only non-zero vectoris perpendicular to itself and V = V ⊥. Thus n ≥ 2 and there must be a pair

Symplectic and orthogonal GF (2)-forms 3

of non-perpendicular vectors. For n = 2 this implies any two non-zero vectorsare non-perpendicular and hence (V, f) is uniquely determined and called thehyperbolic plane. Let us turn to the case n ≥ 3. Suppose that f(u, v) = 1 andset U = 〈u, v〉. Then (U, f |U ) is a hyperbolic plane; U ∩ U⊥ = 0 and by (1.1.2)V is the direct sum of U and U⊥. By (1.1.4)

(V, f) ∼= (U, f |U ) ⊕ (U⊥, f |U⊥)

and by (1.1.3) the second summand is a non-singular (n − 2)-dimensionalsymplectic space to which the induction hypothesis applies.

The direct sum decomposition in (1.1.5 (iii)) will be called a hyperbolic decom-position. Let (V, f) be a non-singular n-dimensional symplectic space. By (1.1.5)n = 2m and we can chose a basis (e1, e2, . . . , en) called the hyperbolic basis suchthat for 1 ≤ i < j ≤ n, we have f(ei, ej) = 1 if j − i = m and f(ei, ej) = 0otherwise. Then the subspaces 〈ei, ei+m〉 taken for i = 1, 2, . . . , m are pairwiseperpendicular hyperbolic planes.

Let q be an orthogonal form associated with f . A vector u ∈ V is said to besingular or non-singular depending on whether q(u) = 0 or q(u) = 1. A subspaceU of V is said to be totally singular if q(u) = 0 for every u ∈ U . A subspace Uis said to be totally isotropic if f(u, v) = 0 for every u, v ∈ U .

In the case n = 2 it is immediate that the isomorphism type of an orthogonalspace (V, f, q) is determined by the number of non-singular vectors which isone or three. These orthogonal spaces of dimension 2 we denote by V +

2 or V −2 ,

respectively. If (e1, e2) is a hyperbolic basis of V then V +2 is obtained by setting

q(e1) = q(e2) = 0, while V −2 by setting q(e1) = q(e2) = 1.

Lemma 1.1.6 Up to isomorphism there are exactly two orthogonal spacesassociated with a non-singular 2-dimensional symplectic space, which are V +

2and V −

2 .

By (1.1.1), (1.1.5), (1.1.6) given a non-singular symplectic space of dimensionn = 2m in order to obtain an orthogonal space it is sufficient to define theorthogonal form on each plane in a hyperbolic decomposition by turning thisplane into V +

2 or V −2 . This immediately shows that the number of isomorphism

types of orthogonal spaces is at most m. This number is dramatically reducedby the following.

Lemma 1.1.7 V −2 ⊕ V −

2∼= V +

2 ⊕ V +2 .

Proof Let (e1, e2, e3, e4) be a hyperbolic basis of V −2 ⊕ V −

2 , so that 〈e1, e2〉 and〈e3, e4〉 are perpendicular hyperbolic planes and q(ei) = 1 for 1 ≤ i ≤ 4. Putd1 = e1 + e3, d2 = e1 + e4, d3 = e2 + e3 + e4, d4 = e1 + e2 + e3 + e4. Then(d1, d2, d3, d4) is another hyperbolic basis and q(di) = 0 for 1 ≤ i ≤ 4. Therefore

α : ei → di

defines the required isomorphism.


Define V +2m to be the direct sum of m copies of V +

2 and V −2m to be the direct

sum of m − 1 copies of V +2 and one copy of V −

2 . By (1.1.7) and the paragraphbefore that lemma we obtain the following.

Lemma 1.1.8 An orthogonal space associated to a non-singular symplectic spaceof dimension n = 2m is isomorphic to either V +

2m or V −2m.

In the next section among other things we will see that V +2m and V −

2m arenon-isomorphic.

1.2 S2m(2), O+2m(2), and O−

2m(2)

Let n = 2m be a positive even integer and (V, f) be a non-singularn-dimensional symplectic space which is uniquely determined up to isomorphismby (1.1.5 (ii)). Define the symplectic group S2m(2) to be the group of invertiblelinear transformations α of V which preserve f in the sense that

f(u, v) = f(α(u), α(v))

for all u, v ∈ V .

Lemma 1.2.1 The following assertions hold:

(i) S2(2) ∼= L2(2) ∼= Sym3;(ii) S2m(2) acts transitively on the set of ordered hyperbolic decompositions

of (V, f);(iii)

|S2m(2)| = 2m2 ·m∏

i=1

(22i − 1).

Proof (i) is immediate from the description of the hyperbolic plane while (ii)follows from commutativity and associativity of the direct sum operation. Let ube a non-zero vector of V . Then by (1.1.2) u⊥ is a hyperplane in V and hencethere are 2n−1 vectors generating with u a hyperbolic plane U . The stabilizer inS2m(2) of the ordered decomposition

(V, f) = (U, f |U ) ⊕ (U⊥, f |U⊥)

is S2(2) × S2m−2(2) and (iii) comes by inductive calculations.

Now we turn to the orthogonal groups and start with the following.

Lemma 1.2.2 Let (V, f, q) ∼= V ε2m and

kεδ(m) = |v | v ∈ V, q(v) = δ|.

Then

k+0 (m) = k−

1 (m) = 22m−1 + 2m−1, k+1 (m) = k−

0 (m) = 22m−1 − 2m−1.

S2m(2), O+2m(2), and O−

2m(2) 5

Proof We know that k+0 (1) = k−

1 (1) = 3, k+1 (1) = k−

0 (1) = 1. For m ≥ 2 werepresent V ε

2m as direct sum U ⊕W , where U ∼= V ε2m−2, W ∼= V +

2 . Then a vectorfrom V (which might or might not be zero) is singular if and only if its projectionsinto U and W are of the same type (both singular or both non-singular). Thisprovides us with a recurrence relation for the kε

δ(m)’s which in turn gives theresult.

By (1.2.2) the spaces V +2m and V −

2m are indeed non-isomorphic.Let (V, f, q) ∼= V ε

2m be an orthogonal space. The orthogonal group associatedwith this space is the group of invertible linear transformations of V whichpreserve q. This group is denoted by Oε

2m(2). Since f is uniquely determinedby q, both O+

2m(2) and O−2m(2) are contained in S2m(2).


(i) O+2 (2) ∼= Sym2, O−

2 (2) ∼= S2(2) ∼= L2(2) ∼= Sym3;(ii) if m ≥ 2 then Oε

2m(2) acts transitively on the set of hyperbolic pairs (thepairs u, v ∈ V such that q(u) = q(v) = 0, f(u, v) = 1);

(iii)

|O+2m(2)| = 2m2−m+1(2m − 1) ·

m−1∏i=1

(22i − 1)

|O−2m(2)| = 2m2−m+1(2m + 1) ·

m−1∏i=1

(22i − 1).

Proof (i) is immediate. If (u, v) is a hyperbolic pair and U = 〈u, v〉, then

(V, f, q) = (U, f |U , q|U ) ⊕ (U⊥, f |U⊥ , q|U⊥) ∼= V +2 ⊕ V ε

2m−2

and (ii) follows. The stabilizer of the above decomposition in Oε2m(2) is Sym2 ×

Oε2m−2(2). Arguing as in (1.2.1) and making use of (1.2.2) we perform inductive

calculations to obtain (iii).

Next we would like to discuss all the orthogonal forms q associated with agiven non-singular symplectic form (V, f). Such a form is said to be of plus orminus type depending on whether (V, f, q) is isomorphic to V +

n or to V −n .

Lemma 1.2.4 Let Q+ and Q− be the sets of quadratic forms of plus and minustype respectively associated with f . Then

(i) the orthogonal group Oε2m(2) permutes transitively both the non-zero

singular and the non-singular vectors;(ii) the symplectic group S2m(2) acts transitively both on Q+ and Q−;(iii) |Q+| = 22m−1 + 2m−1 and |Q−| = 22m−1 − 2m−1.

Proof (i) and (ii) are easy consequences of (1.2.1 (ii)) and (1.2.3 (ii)). Weobtain (iii) comparing the orders of S2m(2) and Oε

2m(2) given in (1.2.1 (iii)) and(1.2.3 (iii)).


By (1.2.4 (iii)) we have |Q+| + |Q−| = 22m = 2n which perfectly agreeswith (1.1.1).

The following well-known result is also easy to establish using induction.Recall that a subspace U of an orthogonal space (V, f, q) is totally singular ifq(u) = 0 for every u ∈ U .

Lemma 1.2.5 Let (V, f, q) ∼= V ε2m be an orthogonal space and let U be a maximal

totally singular subspace in this space. Then dim U = m if ε = + and dim U =m − 1 if ε = −.

1.3 Transvections and Siegel transformations

In this section we discuss some specific automorphisms of orthogonal spaces.Recall some basic terminology. If U is a GF (2)-vector space, u is a non-zerovector in U and W is a hyperplane containing u. Then the transvection withcentre u and axis W is the element of the general linear group of U which fixesW vectorwise and maps v ∈ U \ W onto v + u. It is well-known that Ln(2) isgenerated by its transvections for n ≥ 2.

Lemma 1.3.1 Let (V, f, q) ∼= V ε2m. Then the following hold:

(i) If u is a non-singular vector in V , then the orthogonal transvection

tu : v → v + f(u, v)u

(which is the transvection with centre u and axis u⊥) is an element ofOε

2m(2);(ii) If u, v ∈ V and T = 〈u, v〉 is a totally singular 2-space then the Siegel

transformation

sT : w → w + f(u, w)v + f(v, w)u

is an element of Oε2m(2).

Proof It can be checked directly that tu and sT preserve the orthogonal form.

Notice that the Siegel transformation sT does not depend on the choice ofthe generating vectors u, v. Whenever T ⊆ w⊥ the vector w is fixed by sT

and if T ⊆ w⊥ then sT shifts w by the unique non-zero vector in T which isperpendicular to w.

If m ≥ 3, then Oε2m(2) is generated by the transvections tu taken for all the

non-singular vectors u, while the Siegel transformations sT taken for all totallysingular 2-dimensional subspaces, generate the commutator subgroup Ωε

2m(2)which has index 2 in Oε

2m(2) (cf. Taylor (1992) for details). It is well known andeasy to check that the orthogonal transvections form a class of 3-transpositionsin Oε

2m(2). More precisely, the following lemma holds.

Transvections and Siegel transformations 7

Lemma 1.3.2 Let u and v be non-singular vectors in V . Then tu and tvcommute whenever u and v are perpendicular; otherwise their product is oforder 3.

Now we turn to the Siegel transformations.

Lemma 1.3.3 Let U be a totally singular subspace in V of dimension k ≥ 2.Then

(i) if w is a singular vector outside U⊥ and u ∈ w⊥ ∩ U , then s〈u,w〉 induceson U the transvection whose centre is u and whose axis is w⊥ ∩ U ;

(ii) if W is a totally singular subspace which complements U⊥ in V thenthe Siegel transformations s〈u,w〉, taken for all perpendicular u ∈ U andw ∈ W , generate in Oε

2m(2) a subgroup L ∼= Lk(2) for which U and Ware the natural and the dual natural modules.

Proof Statement (i) follows directly by the definition. Notice that L acts trivi-ally on 〈U, W 〉⊥, so it is a subgroup of the stabilizer in O+

2k(2) of two disjointmaximal totally singular subspaces in the orthogonal 2k-space. Since there is aduality between U and W this stabilizer acts faithfully on each of U and W .Therefore L ≤ Lk(2). Now (ii) follows from (i) and the fact that Lk(2) isgenerated by its transvections.

In order to establish one further property of Siegel transformations we needto recall a well-known characterization of the exterior squares (cf. Lemma 3.1.3in Ivanov and Shpectorov (2002)).

Lemma 1.3.4 Let U be a k-dimensional GF (2)-space, k ≥ 2, let L ∼= Lk(2) bethe general linear group of U and let

[U2

]be the set of 2-subspaces in U . Let W

be a non-zero GF (2)-module for L which satisfies the following:

(i) there is a mapping d :[U2

]→ W which commutes with the action of L

and the image of d spans W ;(ii) whenever T1, T2, T3 ∈

[U2

]are pairwise distinct with dim 〈T1, T2, T3〉 = 3

and dim (T1 ∩ T2 ∩ T3) = 1, the equality d(T1) + d(T2) + d(T3) = 0 holds.

Then dim W = k(k−1)/2, W is irreducible and W is isomorphic to the exteriorsquare

∧2U of U .

Lemma 1.3.5 Let U be a totally singular subspace in V of dimension k ≥ 2.Then the Siegel transformations sT taken for all 2-subspaces T in U generate inOε

2n(2) an elementary abelian subgroup Q of rank k(k − 1)/2. If L ∼= Lk(2) as in(1.3.3) then Q and the exterior square of U are isomorphic as L-modules.

Proof We apply (1.3.4) for d : T → sT . It is an easy exercise to check that theequality in (1.3.4 (ii)) holds, hence the result.


1.4 Exterior square via symplectic forms

Let U be a GF (2)-space and let f be a symplectic form on U which is no longerassumed to be non-singular. Then

U⊥ = u | f(u, v) = 0 for every v ∈ U

is the radical of the symplectic space (U, f). Consider the factor space U = U/U⊥

and the induced form f on U defined by

f(u, v) = f(u, v),

where u = u + U⊥, v = v + U⊥.


(i) f(u1, v1) = f(u, v) whenever u1 ∈ u, v1 ∈ v, so that f is well defined;(ii) (U , f) is a non-singular symplectic space;(iii) (U, f) is uniquely determined by U⊥ and (U , f).

Proof By bilinearity and since u + u1, v + v1 ∈ U⊥ we obtain (i). The form fis symplectic because f is such. Finally (iii) is immediate from (i).

The codimension of the radical is called the rank of the form. When the non-singularity condition is dropped the set of all the symplectic forms on a givenspace U is closed under pointwise addition. This is stated in the following lemmawhose proof is immediate.

Lemma 1.4.2 Let f1 and f2 be symplectic forms on U . Then f1 + f2 defined by

f1 + f2 : (u, v) → f1(u, v) + f2(u, v)

is a symplectic form on U .

The space S(U) of all the symplectic forms on a k-dimensional space U is(k2

)-dimensional since having a fixed basis B of U in order to define such a form

it is necessary and sufficient to assign the value 0 or 1 to every pair of distinctvectors in B.

Lemma 1.4.3 The space S(U) as a module for GL(U) is isomorphic to theexterior square of the dual U∗ of U .

Proof By (1.1.5) the radical of a symplectic form has even codimension. Fur-thermore by the proof of (1.1.5) there is only one non-singular 2-dimensionalsymplectic space. Therefore the symplectic spaces of rank 2 are indexed by thecodimension 2 subspaces in U (that is by the 2-dimensional subspaces in U∗).Let U1 be a hyperplane in U , let U3 be a codimension 3 subspace in U and letf1, f2, f3 be the symplectic forms of rank 2 whose radicals contain U3 and arecontained in U1. Then it is straightforward to check that

f1 + f2 + f3

Witt’s theorem 9

is the zero form and the result follows from the characterization of the exteriorsquares in (1.3.4), since the forms of rank 2 span the space of all the symplecticforms.

We conclude this section with another well known result.

Lemma 1.4.4 The spaces∧2

U and∧2

U∗ are dual to each other asGL(U)-modules.

Proof By (1.3.4) there are mappings

d1 :[U

2

]→∧

2U, d2 :[U∗

2

]→∧

2U∗

With a 2-dimensional subspace V ∗ in U∗ we bijectively associate a codimension2 subspace K(V ∗) in U which is the intersection of the kernels of the linear formsin V ∗. It is easy to check that there is a unique bilinear mapping

δ :∧

2U ×∧

2U∗ → GF (2)

such that δ(d1(W ), d2(V ∗)) = 0 if and only if W ∩ K(V ∗) = 0. Then δ providesus the required GL(U)-invariant duality.

1.5 Witt’s theorem

Let U be an n-dimensional GF (2)-space and let f be a symplectic form on U(which might or might not be non-singular). Let Sf be the symplectic group ofthe space (U, f), which is the group of invertible linear transformations of Upreserving the form f . Then Sf

∼= S2m(2) if n = 2m and f is non-singular. LetW be a complement to U⊥ in U (by (1.1.5) the dimension of W is even, say 2k).Then (W, f |W ) is a non-singular symplectic space and

(U, f) = (W, f |W ) ⊕ (U⊥, f |U⊥),

where the form f |U⊥ is identically zero. Since the isomorphism type of (U, f) isuniquely determined by the dimensions of the subspaces in the above direct sumdecomposition, we have the following.

Lemma 1.5.1 In terms introduced in the paragraph before the lemma thefollowing assertions hold:

(i) Sf induces the full linear group of U⊥;(ii) Sf permutes transitively the complements W to U⊥ in U ;(iii) the stabilizer in Sf of a complement W to U⊥ induces on W the

symplectic group S2k(2), where 2k = dim W .

The following two lemmas are special cases of what is called Witt’s theorem(compare Aschbacher (1986) and Taylor (1992)).


Lemma 1.5.2 Let (U, f) be a non-singular symplectic space of dimensionn = 2m. Then two subspaces U (1) and U (2) of U are contained in the same orbitof the symplectic group S2m(2) if and only if (U (1), f |U(1)) and (U (2), f |U(2)) areisomorphic.

Proof The ‘only if’ part is obvious. Suppose that the restricted symplecticspaces are isomorphic, let l be the dimension and let 2k be the rank of eachof them. We claim that for i = 1 and 2 there exists a decomposition of U intopairwise perpendicular hyperbolic planes such that (a) k planes are containedin U (i); (b) l − 2k planes intersect the radical of (U (i), f |U(i)) in 1-dimensionalsubspaces; (c) the remaining pairs are disjoint from U (i). The claim is easy toestablish arguing inductively as in the proof of (1.2.1). It is also clear that theclaim implies the assertion.

Combining (1.5.1) and (1.5.2) it is easy to establish the following.

Lemma 1.5.3 Let (U, f) be an arbitrary symplectic space. Then the orbit of asubspace W of U under the group Sf is uniquely determined by the isomorphismtype of (W, f |W ) and by dim (W ∩ U⊥).

1.6 Extraspecial 2-groups

Let E be a 2-group with centre Z and put E = E/Z. Then E is said to be anextraspecial 2-group if |Z| = 2 and E is elementary abelian.

A beautiful relationship between the extraspecial 2-groups and the orthogonalspaces over GF (2) is established by the following result.

Proposition 1.6.1 Let E be an extraspecial 2-group with centre Z and E =E/Z. Consider Z as the field GF (2) of two elements and E as a GF (2)-vectorspace. Define the mappings f : E × E → Z and q : E → Z by

f(uZ, vZ) = [u, v], q(uZ) = u2.

Then

(i) f and q are well defined;(ii) f is a non-singular symplectic form;(iii) q is an orthogonal form associated with f .

Proof Since Z is the centre of E we have (i). The basic relation for thecommutator of a product reads as follows:

[uv, w] = [u, w]v[v, w] = [u, w][v, w]

and therefore f is linear in the first coordinate. Clearly [u, u] = 1 and [u, v] =[v, u] since Z has order 2. Therefore (ii) is established. Finally

Extraspecial 2-groups 11

(uv)2 = u2v2[u, v]

and (iii) follows.

By (1.6.1) s(E) := (E, f, q) is a non-singular orthogonal space and by (1.1.5)it is isomorphic to V ε

2m, where ε ∈ +,− and m is a positive integer (noticethat |E| = 22m+1). It turns out that the isomorphism type of s(E) determinesthat of E. One way of proving this is to show that s(E) uniquely determinesthe homology class of 2-cocycles which specifies E as an extension of Z by E(cf. chapter 5 in Frenkel et al. (1988)). Here we follow a more geometric approachand refer to section 23 in Aschbacher (1986) for the missing proofs.

Let E1 and E2 be extraspecial 2-groups whose centres Z1 and Z2 are gen-erated by elements z1 and z2, respectively. Then the central product E1 ∗ E2 ofthese groups is the quotient of the direct product

E1 × E2 = (e1, e2) | e1 ∈ E1, e2 ∈ E2over the normal subgroup of order 2 generated by the element (z1, z2). Thecentral product operation is commutative and associative (provided that theresult is considered up to isomorphism) and therefore one can consider the centralproduct of any number of extraspecial groups.

Lemma 1.6.2 If E1 and E2 are extraspecial 2-groups, then E1 ∗ E2 is alsoextraspecial and

s(E1 ∗ E2) ∼= s(E1) ⊕ s(E2),

where ⊕ stands for the orthogonal direct sum.

Lemma 1.6.3 Let E be an extraspecial group with centre Z, let Z ≤ U ≤ E, letU be the image of U in E, and let s(E) = (E, f, q). Then

(i) U is abelian if and only if U is totally isotropic with respect to f ;(ii) U is elementary abelian if and only if U is totally singular with respect

to q;(iii) U is extraspecial if and only if (U , f |U ) is non-singular.

The following lemma supplies us with the classification of the smallestextraspecial 2-groups.

Lemma 1.6.4 An extraspecial group of order 8 is isomorphic either to thedihedral group D8 or to the quaternion group Q8.

The next lemma forms the inductive basis for the general classification.

Lemma 1.6.5 Let E be an extraspecial group and suppose that s(E) is theorthogonal sum of two non-singular subspaces V and U . Let V and U be thepreimages in E of V and U , respectively. Then E ∼= V ∗ U .

Now combining (1.6.2), (1.6.3), (1.6.4), (1.6.5) with (1.1.6), (1.1.7) and (1.1.8)we obtain the main result of the section.


Proposition 1.6.6 Let E be an extraspecial 2-group and s(E) = (E, f, q). Thenone of the following holds.

(i) s(E) ∼= V +2m for some positive integer m, E is isomorphic to the central

product of m copies of D8, and |E| = 22m+1;(ii) s(E) ∼= V −

2m for some positive integer m, E is isomorphic to the centralproduct of m − 1 copies of D8 together with one copy of Q8, and |E| =22m+1.

As a by-product of the above proposition we obtain the isomorphism D8 ∗D8 ∼= Q8 ∗ Q8.

The extraspecial 2-groups of order 22m+1 corresponding to the possibilities(i) and (ii) in (1.6.6) are denoted by 21+2m

+ and 21+2m− , respectively and are said

to be of type + and −, respectively. Thus for every odd power of 2 greater than2, up to isomorphism there are exactly two extraspecial groups of that order.

Lemma 1.6.7 Let E ∼= 21+2mε be an extraspecial group of type ε ∈ +,− with

centre Z and U be a maximal elementary abelian subgroup in E. Then

(i) U ∼= 2m+δ, where δ = 1 if ε = + and δ = 0 otherwise;(ii) U contains Z;(iii) U is normal in E and if ε = + then CE(U) = U ;(iv) the group E acting on U by conjugation induces the elementary abelian

group of order 2m−1+δ generated by the transvections with centre z (whichis the generator of Z).

Proof (i) follows from (1.2.5) and (1.6.3 (ii)) while (ii) follows from the definitionof U . Since U contains Z, which is also the commutator subgroup of E, it isnormal in E. This and (1.1.2) give (iii). Let x ∈ E \ U . Then by (iii), x acts onU as a transvection with axis z and (iv) follows by the order consideration.

Lemma 1.6.8 Let E ∼= 21+2mε . Then Aut (E) is an extension of Inn (E) ∼=

E/Z = E ∼= 22m by the automorphism group Oε2m(2) of the orthogonal space

(E, f, q), where f and q are defined as in (1.6.1).

Proof It is clear that an automorphism of E preserves f and q. On the otherhand every automorphism of V +

2 or V −2 is realized as an automorphism of D8

and Q8, respectively.

The extension of 22m by Oε2m in (1.6.8) does not split when m ≥ 3

(Griess 1973).

1.7 The space of forms

As in the previous section let (V, f) be a non-singular symplectic space of dimen-sion n = 2m, m ≥ 1, let Q+ and Q− be the sets of orthogonal forms of plus andminus types associated with f .

Lemma 1.7.1 Let V ∗ be the dual space of V (which is the space of linearfunctions on V ). Then V ∗ ∪ Q+ ∪ Q− is closed under the pointwise addition.

S4(2) and Sym6 13

Proof A GF (2)-valued function p on V belongs to V ∗ if and only if

p(u) + p(v) + p(u + v) = 0

for all u, v ∈ V . Comparing this with the definition of an orthogonal form weimmediately get the result.

For every w ∈ V the mapping lw : v → f(v, w) is an element of V ∗ and sincef is non-singular, the mapping ν : w → lw is an isomorphism of V onto V ∗.

Lemma 1.7.2 Let q ∈ Q+ ∪ Q− and w ∈ V . Then q + lw ∈ Q+ ∪ Q− andthe types (which are plus or minus) of q and of q + lw coincide if and only ifq(w) = 0.

Proof Let q ∈ Q− and F ∼= O−2m(2) be the corresponding orthogonal group. By

(1.2.4 (i)) F permutes transitively the 22m−1 + 2m−1 vectors w in V such thatq(w) = 1. Therefore the type of q + lw is independent of the choice of such w.Since by (1.2.4 (iii)) there are only 22m−1 − 2m−1 forms of minus type, q + lw isof plus type and the result follows.

1.8 S4(2) and Sym6

There is a simple way to associate a non-singular 2m-dimensional symplecticspace s(P2m+2) = (V, f) with a set P2m+2 of 2m + 2 elements. The vectors inV are the unordered partitions of P2m+2 into two even subsets (with P2m+2, ∅being the zero vector); the sum of two partitions A1, A2 and B1, B2 is thepartition C1, C2, where

C1 = (Ai ∪ Bi) \ (Ai ∩ Bi) for i ∈ 1, 2,

C2 = (Ai ∪ Bj) \ (Ai ∩ Bj) for i, j = 1, 2.

It is straightforward to check that this operation turns V into a GF (2)-vectorspace of dimension 2m. Define f(A1, A2, B1, B2) to be the size of |Ai ∩ Bj |taken modulo 2. This definition is independent on the choice of 1 ≤ i, j ≤ 2 andprovides us a non-singular symplectic form. The symmetric group Sym2m+2 ofP2m+2 acts naturally on V preserving both the vector space structure and thesymplectic form f which gives the following.

Lemma 1.8.1 Sym2m+2 ≤ S2m(2).

We check using (1.2.1) that the order of S4(2) is exactly 6!, which is of coursethe order of Sym6 and the above inclusion gives the famous isomorphism.

Lemma 1.8.2 S4(2) ∼= Sym6.

The above consideration also provides us with a useful combinatorial models(P6) of the non-singular 4-dimensional symplectic space (V, f). We modify thisdescription to obtain a new one in terms of the group Sym6. The non-trivialeven partitions of P6 are identified with the set of 2-element subsets of P6 andin turn the latter set is identified with the set T of transpositions in Sym6 (with


respect to the action on P6). Then for t1, t2 ∈ T with t1 = t2 we have

(∗) f(t1, t2) = 0 if and only if t1 and t2 commute and t3 = t1 + t2 is definedso that 〈t1, t2, t3〉 = 〈t1, t2〉.

This gives a model s(T ) of the non-singular 4-dimensional symplectic spacein terms of the transpositions in Sym6. This model is merely the reformulationof the structure of s(P6) along with the canonical correspondence between thetranspositions and the non-trivial even partitions of P6.

We know that Sym6 possesses an outer automorphism which maps T onto theset D of products of three pairwise commuting elements of T . Therefore there isa model s(D) of the non-singular 4-dimensional symplectic space where the sym-plectic form and the vector space structure on D ∪ 0 are defined by the rule (∗).By (1.1.5) we have the isomorphism between s(T ) and s(D) which one can useto reprove the existence of the outer automorphism of Sym6. Notice that s(D)is canonically isomorphic to s(R6), where R6 is a 6-element set on which Sym6acts in such a way that the elements from D are transpositions. Consider Sym6as an abstract group and let P6 and R6 be two sets of size 6 on which Sym6 actsin non-equivalent ways permuted by an outer automorphism. Then we have twonon-singular symplectic 4-spaces

s(P6) = s(T ) and s(R6) = s(D)

with Sym6 inducing the full symplectic group on each of the spaces.The following lemma describes the orthogonal spaces associated with

s(P6) = s(T ).

Lemma 1.8.3 Let (V, f) = s(P6) = s(T ) and let q be an orthogonal formassociated with f . Then one of the following holds:

(i) q is of plus type and there is a unique partition A, C of P6 into twotriples such that q(t) = 1 for t ∈ T if and only if t stabilizes the partition;

(ii) q is of minus type and there is a unique element a ∈ P6 such that q(t) = 1for t ∈ T if and only if t stabilizes a.

The orthogonal spaces in (1.8.3 (i)) and (1.8.3 (ii)) will be denoted byQ(P6, A, C) and Q(P6, a), respectively. By elementary calculations one cansee that there are ten forms of plus type, each having exactly six non-singularvectors and six forms of minus type, each having exactly ten non-singular vectors.This is certainly consistent with (1.2.4 (iii)). The stabilizer in Sym6 of an ele-ment a ∈ P6 is the symmetric group Sym5 while the stabilizer of a partition ofP6 into two triples is the wreath product Sym3 Sym2. This gives two furtherexceptional isomorphisms.

Lemma 1.8.4 O+4 (2) ∼= Sym3 Sym2 and O−

4 (2) ∼= Sym5.

The stabilizer in Sym6 of an element a ∈ P6 is not conjugate to the stabilizerof an element b ∈ R6 (they are conjugate in the automorphism group of Sym6).

Ω+6 (2), Alt8, and L4(2) 15

On other hand the stabilizer of a partition of P6 into two triples is the normalizerof a Sylow 3-subgroup and hence it is conjugate to the stabilizer of a partitionof R6 into two triples.

Lemma 1.8.5 Let X ∼= Sym6. Then

(i) X contains two classes of subgroups Sym5 with representatives Fa and Fb

which are the orthogonal groups of minus type for the symplectic spacess(P6) and s(R6), respectively;

(ii) X contains a unique class of subgroups Sym3 Sym2 with representativeE which is the orthogonal group of plus type for both the symplectic spacess(P6) and s(R6);

(iii) Fa acting on the non-zero vectors of s(P6) has two orbits with lengths5 and 10 formed by vectors which are singular and non-singular withrespect to the quadratic form stabilized by Fa;

(iv) the action of Fa on the set of non-zero vectors in s(R6) is transitive.

There is an outer automorphism α of Sym6 which permutes Fa and Fb and anouter automorphism β which normalizes E ∼= Sym3 Sym2.

1.9 Ω+6 (2), Alt8, and L4(2)

In the case when 2m+2 is divisible by 4 (that is when m is odd, say m = 2k+1)the inclusion in (1.8.1) can be refined. Let V (P4k+4) = (V, f) be the symplecticspace defined in the beginning of Section 1.8 and put

q(A1, A2) =|A1|2

taken modulo 2. It is easy to check that q is an orthogonal form associated withf . Furthermore, q is of plus or minus type depending on whether k is odd or even,respectively. This form is clearly preserved by the symmetric group Sym4k+4 andwe have the following.

Lemma 1.9.1 For every positive integer k the inclusion

Sym4k+4 ≤ Oε4k+2(2)

holds, where ε = + or − according to whether k is odd or even.

By (1.2.3 (iii)) the order of O+6 (2) is 8! which is of course the order of Sym8

and hence

Lemma 1.9.2 O+6 (2) ∼= Sym8.

There is yet another model of the orthogonal 6-dimensional space of plustype which leads to the next exceptional isomorphism. Let U be a 4-dimensionalGF (2)-space, let V be the exterior square of U and

(u, v) → u ∧ v


be the natural bilinear mapping of U ×U onto V . Define a GF (2)-valued functionq on V by setting q(w) = 0 if and only if w = u ∧ v for some u, v ∈ U . Thefollowing result is well-known and easy to check.

Lemma 1.9.3 In the above terms q is an orthogonal form of plus type on Vassociated with a non-singular symplectic form.

Since U is 4-dimensional, V is isomorphic to the exterior square of the dualof U . Hence along with L4(2) the contragredient automorphism τ of that groupacts on V and preserves the above defined orthogonal form q. Since the order ofL4(2).〈τ〉 is also 8! we have the next isomorphism.

Lemma 1.9.4 O+6 (2) ∼= L4(2).〈τ〉.

Since Ω+6 (2), Alt8, and L4(2) are the only index 2 subgroups in O+

6 (2), Sym8,and L4(2).〈τ〉, respectively, we also have the following.

Lemma 1.9.5 Ω+6 (2) ∼= Alt8 ∼= L4(2).

1.10 L2(7) and L3(2)

Let P8 = ∞, 0, 1, . . . , 6 be the projective line over the field GF (7) of sevenelements. The stabilizer of the projective line structure in the symmetric groupof P8 is PGL2(7) and the latter contains the group L2(7) with index 2. Themodel Q(P8) of 6-dimensional orthogonal space in Section 1.9 provides us withan embedding

PGL2(7) ≤ O+6 (2).

In this section we refine this embedding geometrically.

Lemma 1.10.1 Let (V, f, q) = Q(P8) and let O+6 (2) be the corresponding ortho-

gonal group. Then there is a pair I1, I2 of disjoint maximal totally singularsubspaces in V such that

(i) L2(7) is the joint stabilizer in O+6 (2) of I1 and I2;

(ii) PGL2(7) is the stabilizer in O+6 (2) of the unordered pair I1, I2.

In particular

(iii) L2(7) ∼= L3(2);(iv) PGL2(7) ∼= L3(2).〈τ〉, where τ is the contragredient automorphism.

Proof Let Q = 1, 2, 4 and N = 3, 5, 6 be the set of non-zero squares andthe set of non-squares in GF (7), so that

P8 = ∞ ∪ 0 ∪ Q ∪ N.

Put u1 = ∞ ∪ Q, 0 ∪ N and u2 = ∞ ∪ N, 0 ∪ Q and define I1and I2 to be the subspaces in V generated by the images under L2(7) of u1

L2(7) and L3(2) 17

and u2, respectively. Then direct check shows that I1 and I2 are as required.Furthermore an element from PGL2(7) \ L2(7) permutes I1 and I2. Standardcalculations show that

|L2(7)| = |L3(2)| = 23 · 3 · 7.

Since L3(2) is the stabilizer in O+6 (2) of the (ordered) direct sum vector space

decomposition V = I1 ⊕ I2, we obtain (i), (ii), and (iii). Since I1 ∩ u⊥2 is a

hyperplane in I1, there is a duality between I1 and I2, which gives (iv).

Exercises

1. Calculate the number of totally singular k-dimensional subspaces in V ε2m.

2. Show that the actions of S2m(2) on O+ and O− are doubly transitive.3. Check that the orthogonal transvections in Oε

2m(2) form a class of3-transpositions.

4. Generalize (1.3.4) to the case of an arbitrary exterior power of U .5. Show that the orthogonal form q in (1.6.1) determines the homology class of

2-cocycles which specifies E as an extension of Z by E.6. Apply the isomorphism Sym6

∼= S4(2) to recover an outer automorphism ofSym6.

2

O+10(2) AS A PROTOTYPE

We will construct J4 as a completion group which is constrained at level 2 of acertain amalgam G = G[0], G[1]. In fact J4 is the only such completion, whichproves the uniqueness of J4. The amalgam G is a modification of a classicalamalgam H = H [0], H [1] contained in the orthogonal group O+

10(2). In thischapter we illustrate our strategy by showing that O+

10(2) is the only completiongroup of H which is constrained at level 2. This reproves a special case of aclassical result by P. J. Cameron and C. E. Praeger.

2.1 Dual polar graph

Let (V, f, q) ∼= V +10 be a 10-dimensional orthogonal space of plus type. This means

that V is a 10-dimensional GF (2)-space, f is a non-singular symplectic form onV and q is an orthogonal form of plus type associated with f . Let H ∼= O+

10(2)be the automorphism group of (V, f, q). We have mentioned (referring to Taylor(1992) for a proof) that H is generated by the transvections

tu : v → v + f(u, v)u

taken for all non-singular vectors u ∈ V , while the commutator subgroup H ′ ∼=Ω+

10(2) (which is non-abelian simple with index 2 in H) is generated by the Siegeltransformations

sU : w → w + f(u, w)v + f(v, w)u

taken for all totally singular planes (2-subspaces) U = 〈u, v〉 in V .Let O = O+(10, 2) be the dual polar space associated with the orthogonal

space under consideration. Then for 0 ≤ i ≤ 4 the elements of type i in O arethe (5 − i)-dimensional totally singular subspaces in V (with respect to q) withtwo such subspaces being incident if one of them contains the other one. Thegeometry O belongs to the following diagram:

0

1 1

2 2

2 3

2 4

2

As usual above a node on the diagram we put the type of the correspondingelements of the geometry. The leftmost edge is the geometry of the vertices andedges of the complete bipartite graph K3,3.

Let Ω = D+(10, 2) be the dual polar graph. The vertices of Ω are themaximal (5-dimensional) totally singular subspaces, the edges are the premax-imal (4-dimensional) totally singular subspaces in V with the incidence relationdefined via inclusion. If U is an edge of Ω then U⊥/U ∼= V +

2 and the latter

18

Dual polar graph 19

contains exactly two 1-dimensional totally isotropic subspaces whose preimagesare precisely the vertices incident to U . Therefore Ω is indeed an undirectedgraph. Clearly the intersection of two distinct 5-dimensional subspaces is atmost 4-dimensional, therefore Ω contains no multiple edges.

It is well-known and easy to prove (cf. Brouwer et al. (1989)) that Ω isdistance-transitive with respect to the action of H with the following intersectionarray:

i(Ω) = 31, 30, 28, 24, 16; 1, 3, 7, 15, 31.

In particular Ω is connected bipartite of valency 31 (the stabilizer of thebipartition is exactly H ′ ∼= Ω+

10(2)).Let x ∈ Ω be a vertex and Ω(x) be the set of neighbours of x in Ω. Then Ω(x)

is in a canonical correspondence with the set of hyperplanes of x (treated as a5-subspace) that is with the non-zero vectors of the dual of x. Therefore there isa natural structure Πx of the projective space of rank 4 over GF (2) defined onΩ(x). This space is canonically isomorphic to the residue of x in O.

The geometry O possesses a nice reformulation in terms of specific subgraphsin Ω called geometric subgraphs. For an element U of type i in O let Ω[i] denote thesubgraph in Ω induced by the vertices which contain U (as subspaces of V ). For0 ≤ i ≤ 4 let S [i] denote the set of all the subgraphs Ω[i]. The following statementis an easy consequence of the theory of locally projective graphs exposed inchapter 9 in Ivanov (1999). In the considered situation everything can be checkeddirectly by a straightforward calculation in the underlying orthogonal space.


(i) O is canonically isomorphic to the geometry in which S [i] is the set ofelements of type i for 0 ≤ i ≤ 4 and the incidence is by the symmetricinclusion;

(ii) S [0] is the set of vertices while S [1] is the set of edges of Ω;(iii) if 2 ≤ i ≤ 4 then

(a) the subgraph Ω[i] is connected of valency 2i − 1;(b) Ω[i] isomorphic to the dual polar graph D+(2i, 2);(c) if x ∈ Ω[i] then Ω[i](x) (the set of neighbours of x in Ω[i]) corresponds

to a subspace in Πx of projective dimension i;(iv) whenever Ω[i],Ξ[i] ∈ Si, x ∈ Ω[i] ∩ Ξ[i] and Ω[i](x) = Ξ[i](x), the equality

Ω[i] = Ξ[i] holds;(v) if x, y ∈ Ω and dΩ(x, y) = i, for 1 ≤ i ≤ 3, then there is a unique

subgraph Ω[i] in S [i] which contains both x and y;(vi) if Ω[2] ∈ S [2] and y ∈ Ω then there is a unique vertex in Ω[2] closest to y

in Ω.

Let Φ be a maximal flag in O. Then Φ can be treated as a chain

0 < U1 < U2 < U3 < U4 < U5

20 O+10(2) as a prototype

of totally singular subspaces, where Uj is j-dimensional, or as a chain

Ω[0] = x ⊂ Ω[1] = x, y ⊂ · · · ⊂ Ω[4]

of the geometric subgraphs in Ω where Ω[i] is induced by the vertices containingU5−i. Notice that the geometric subgraphs are convex in the sense that wheneverit contains a pair of vertices it contains all the shortest paths joining them.

Let H [i] be the stabilizer of U5−i (equivalently of Ω[i]) in H, which is amaximal parabolic subgroup associated with Φ. Let Q[i] be the kernel of theaction of H [i] on V/U5−i and Z [i] be the kernel of the action of Q[i] on U5−i. LetW2i be a complement to U5−i in U⊥

5−i, so that dim W2i = 2i and

(W2i, f |W2i , q|W2i) ∼= V +2i

and let T5−i be a (5−i)-dimensional totally singular subspace in W⊥2i disjoint from

U⊥5−i. We assume that U5 ∩ W2i is i-dimensional and that U5 ∩ W2i < U5 ∩ W2j

for 1 ≤ i < j ≤ 4.Then

V = U5−i ⊕ W2i ⊕ T5−i.

Let C [i] be the stabilizer of T5−i in H [i] (since W2i is the perp of 〈U5−i, T5−i〉the subgroup C [i] stabilizes every term in the above direct sum decomposition),finally let L[i] and R[i] be the kernels of the actions of C [i] on W2i and U5−i,respectively.

The following proposition is a specialization of the structure of maximalparabolics in groups of Lie type (Azad et al. 1990).

Lemma 2.1.2 For 0 ≤ i ≤ 4 the following assertions hold (with obviousinterpretation of the degenerate cases):

(i) if i = 1 then Q[i] = O2(H [i]) and O2(H [1]) = Q[1]R[1];(ii) both Q[i]/Z [i] and Z [i] are elementary abelian 2-groups and unless one

of them is trivial, Z [i] is the centre of Q[i];(iii) H [i] is the semidirect product of Q[i] and C [i];(iv) C [i] is the direct product of L[i] and R[i];(v) L[i] ∼= L5−i(2) and R[i] ∼= O+

2i(2);(vi) Z [i] is generated by the Siegel transformations sW taken for all

2-subspaces W contained in U5−i and Z [i] is isomorphic to the exteriorsquare of U5−i;

(vii) Q[i]/Z [i] is generated by the images of the Siegel transformations sW

taken for all totally singular 2-subspaces W = 〈u, v〉 such that u ∈ U5−i,v ∈ W2i and (as a module for L[i] × R[i]) it is isomorphic to the tensorproduct

U5−i ⊗ W2i.

Proof By (1.2.3 (iii)) we know the order of O+10(2) and it is easy to get from

(1.2.2) the number of totally singular subspaces in V +10(2) of dimension i for

Dual polar graph 21

1 ≤ i ≤ 5. Thus the order of H [i] is known and all we have to do is to pro-duce enough elements of the parabolics. The assertion (vi) as well as the wholestructure of H [0] are immediate from (1.3.3) and (1.3.5).

Therefore the structure of the parabolics is as follows:

H [0] ∼= 210 : L5(2), H [1] ∼= 26+8 : (L4(2) × 2),

H [2] ∼= 23+12 : (L3(2) × O+4 (2)), H [3] ∼= 21+12

+ : (L2(2) × O+6 (2))

H [4] ∼= 28 : O+8 (2).

Now we are ready to establish some important properties of the action of Hon Ω. As usual for a vertex x ∈ Ω, by Ω(x) we denote the set of neighbours of xin Ω, by Ω2(x), the set of vertices at distance 2 from x in Ω. Furthermore, H(x)is the stabilizer of x in H and H1(x) is the kernel of the action of H(x) on Ω(x).


(i) the action of H on Ω is arc-transitive, that is transitive on the incidentvertex–edge pairs;

(ii) H(x) = H [0] ∼= 210 : L5(2), H1(x) = Q[0] ∼= 210 and H(x) induces thefull automorphism group L5(2) of the projective space structure Πx onΩ(x);

(iii) for y ∈ Ω(x) there is a unique bijection ψx,y of the set Lx(y) of lines inΠx containing y onto the set Ly(x) of lines in Πy containing x;

(iv) Q[0] acts faithfully on Ω2(x); for y ∈ Ω(x) and for l ∈ Ly(x) the set l\xis an orbit of H1(x) on Ω(y)\x;

(v) H acts transitively on the set S [i] of geometric subgraphs of valency 2i −1for every 2 ≤ i ≤ 4;

(vi) for Ω[i] ∈ S [i] we have H [i] = HΩ[i] and H [i] induces on Ω[i] theorthogonal group O+

2i(2) ∼= R[i] and the kernel of the action is K [i] :=Q[i]L[i].

Proof The assertion (i) can be seen inductively using (1.2.3 (ii)), while (ii) fol-lows from (2.1.2). The group H [01] := H(x, y) = H [0] ∩ H [1] induces on each ofthe sets Ω(x)\y and Ω(y)\x the maximal parabolic subgroup 24 : L4(2) ofL5(2) (the kernels of the actions are subgroups O2(H(x)) and O2(H(y)) inter-secting in Z [1] ∼= 26). In terms of (2.1.2) R[1] ∼= O+

2 (2) is of order 2 generated bythe orthogonal transvection tu, where u is the only non-singular vector in W2.Then tu ∈ H [1]\H [01] and tu commutes with the quotient H [01]/Q[1] (it evencommutes with a complement L[1] ∼= L4(2) to Q[1] in H [01]). The action of tu onthe set Lx(y) ∪ Ly(x) establishes the required bijection ψx,y. The assertion (iv)possesses a direct check and also follows from the irreducibility of the action ofL5(2) on the exterior square of its natural module. The assertions (v) and (vi)about the geometric subgraphs are rather straightforward.


2.2 Geometric cubic subgraphs

We continue with the notation from the previous section, in particular H ∼=O+

10(2) and Ω = D+(10, 2). In the next section we will see that much of thecombinatorial structure of Ω possesses a natural reformulation in terms of thegroup H. Here we are making a shortcut to this reformulation for the cubicgeometric subgraphs. This provides us with a very simple way to refer to suchsubgraphs as we have done in the Preface.

We start with the following.

Lemma 2.2.1 Πx is the unique projective space structure over GF (2) preservedby the action of H(x) on Ω(x).

Proof Let y and z be distinct vertices from Ω(x). Then the elementwise sta-bilizer of y, z in H(x) stabilizes a unique further point which we denote by u.Then y, z, u is the unique line in Πx containing y and z. It is well known thata projective space is uniquely determined by its line-set, hence the result.

Let l be a line in Πx. Then by the definition of Πx we know that l is formedby the maximal totally singular subspaces intersecting U5 = x in hyperplanescontaining a given 3-dimensional subspace W in U5. Therefore (compare (2.1.1(iv))) there is a unique Ω[2] in S [2] containing x such that Ω[2](x) = l. Supposethat the edge x, y is contained in Ω[2]. By (2.1.3 (ii)), H(x, y) induces on Lx(y)and Ly(x) two isomorphic actions of L4(2) of degree 15. Therefore the stabilizerS of l in H(x, y) stabilizes a unique line m in Ly(x) which is m = ψx,y(l).Since S stabilizes Ω[2], by the uniqueness condition (2.1.1 (iv)) we conclude thatΩ[2](y) = m. Since any two distinct points in Πx determine a unique line, wearrive with at following result.

Lemma 2.2.2 Let Ω[2] ∈ S [2] and let (u, v, w) be a 2-path in Ω[2]. ThenΩ[2](w) = ψv,w(l) where l is the unique line in Πv containing u and w.

We are ready to prove the following characterization.

Lemma 2.2.3 Let S be a non-empty family of connected cubic subgraphs in Ωsuch that

(i) S is stable under H;(ii) whenever Ω[2],Ξ[2] ∈ S, x ∈ Ω[2] ∩ Ξ[2] and Ω[2](x) = Ξ[2](x), the equality

Ω[2] = Ξ[2] holds.

Then S = S [2].

Proof Let Σ ∈ S. Let x, y be an edge of Σ. Since H1(x) fixes Σ(x), byhypothesis (ii) Σ must be stable under H1(x). Hence Σ(y)\x is a union ofH1(x)-orbits on Ω(y)\x. By (2.1.3) this means that Σ(y) is a line l from Ly(x).By the obvious symmetry Σ(x) is a line m from Lx(y). We claim that ψx,y(m) = l.This must be true, since ψx,y(m) is the only line in Ly(x) which is stabilized by thestabilizer of m in H(x, y). Now we apply induction on the distance from x to show

Amalgam H = H [0], H [1] 23

that Σ is uniquely determined by Σ(x) and that Σ coincides with the subgraph Ω[2]

from S [2] which contains x and such that Ω[2](x) = m.

2.3 Amalgam H = H[0], H[1]We would like to characterize the pair (Ω, H) where H ∼= O+

10(2) and Ω =D+(10, 2) in terms of certain combinatorial properties of Ω and of the action ofH on this graph. Towards this end we follow the standard procedure of redefiningΩ in terms of the group H and certain of its subgroups. The procedure is verygeneral and only requires the arc-transitivity of the action of H on Ω.

As above H [0] = H(x) is the stabilizer in H of the vertex x and H [1] =Hx, y is the stabilizer of the edge x, y which contains x. Recall that V (Ω)is the vertex-set of Ω while E(Ω) is the edge-set.

Since H acts transitively on V (Ω), for every z ∈ V (Ω) there is an elementhz in H which maps x onto z and the set of all such elements forms a left cosethzH

[0] of H [0] in H (we adopt the convention that elements acts from the left).Thus there is a mapping

ϑ[0] : V (Ω) → H/H [0]

defined by ϑ[0](z) = hzH[0]. It is immediate that different cosets correspond to

different vertices; therefore ϑ[0] is a bijection and clearly it commutes with thenatural action of H. Similarly we define a bijection

ϑ[1] : E(Ω) → H/H [1]

commuting with the action of H. This provides us with a bijection ϑ betweenthe unions V (Ω) ∪ E(Ω) and H/H [0] ∪ H/H [1] whose restrictions to V (Ω) andE(Ω) are ϑ[0] and ϑ[1], respectively. The only question left is how the incidencebetween vertices and edges is formulated in terms of the cosets. The answer isthe rather simple one given in the following lemma.

Lemma 2.3.1 Let z ∈ V (Ω), s, t ∈ E(Ω). Then z and s, t are incident (i.e.z ∈ s, t) if and only if

ϑ(z) ∩ ϑ(s, t) = ∅.

Proof Suppose first that z ∈ s, t, say z = s. By arc-transitivity there is h ∈ Hwhich maps x onto s and y onto t. Then h ∈ hzH

[0] ∩ hs,tH [1]. On the otherhand if h is in the intersection then it sends the incident pair (x, x, y) ontothe pair (z, s, t). Since h is an automorphism of Ω the latter pair also must beincident.

The next step is to consider H := H [0], H [1] as an abstract amalgam of twoindependent groups (H [0], ∗0) and (H [1], ∗1) whose element sets intersect in H [01]

and the group operations ∗0 and ∗1 are the restrictions of the group operationin H to H [0] and H [1], respectively. Clearly in this case ∗0 and ∗1 coincide whenrestricted to H [01].


We introduce the following construction. Let F = F [0], F [1] be an amalgamof rank 2 such that [F [1] : F [01]] = 2 and [F [0] : F [01]] ≥ 2

ϕ : F → F

be a faithful generating completion of F in F . Recall that by this we mean that ϕis an injection; its restrictions to F [0] and F [1] are homomorphisms and F is gen-erated by the image of ϕ. Define Λ = Λ(F , ϕ, F ) to be a graph such that V (Λ) =F/ϕ(F [0]), E(Λ) = F/ϕ(F [1]) with v ∈ V (Λ) and e ∈ E(Λ) incident if and onlyif their intersection (as cosets) is non-empty. By (2.3.1) we have the following

Lemma 2.3.2 Let id be the identity mapping of H = H [0], H [1] into H. ThenΩ is isomorphic to Λ(H, id, H).

It turns out (cf. Table 1) that quite a few properties of Ω can be already seeninside the amalgam H.

Let α : H → A be an arbitrary faithful generating completion of H. Thenit can easily be checked (cf. Exercises 3 and 4 at the end of the chapter) thatΛ(H, α, A) has no multiple edges. Furthermore, the properties in the secondcolumn of Table 1 hold with (A, Λ(H, α, A)) in place of (H, Ω).

There is an important class of morphisms in the category of faithful generat-ing completions of H. Let α :H → A and β :H → B be two such completions andχ : A → B be a homomorphism. Then χ is called a morphism of completions if

χ(α(h)) = β(h)

for every h ∈ H [0] ∪ H [1].

Table 1. Properties of (H, Ω) and H

N Properties of (H, Ω) Properties of H

1 Ω is an undirected graph [H [1] : H [01]] = 2

2 Ω is connected H = 〈H [0], H [1]〉3 H acts faithfully on Ω If N ≤ H [01] is normal in

H [0] and H [1] then N = 1

4 The valency of Ω is 31 [H [0] : H [01]] = 31

5 H(x)Ω(x) ∼= L5(2) H [0] induces L5(2)

acting on the cosets of H [01]

6 H1(x) ∼= 210⋂

h∈H[0]/H[01] (H [01])h ∼= 210

7 There is a canonical bijection H [1]/O2(H [01]) ∼= L4(2) × 2

ψx,y of Lx(y) onto Ly(x)

The universal completion 25

Lemma 2.3.3 Let α : H → A and β : H → B be two faithful generatingcompletions and χ : A → B be a morphism of completions. Then χ induces acovering

ν : Λ(H, α, A) → Λ(H, β, B)

of graphs.

Proof We define ν(α(aH [i])) = χ(a)β(H [i]) for every a ∈ A and i = 0, 1.This mapping clearly preserves the incidence relation defined in terms of non-emptiness of intersections. Furthermore, since both α and β are faithful

|α(H [0])/α(H [01])| = |H [0]/H [01]| = |β(H [0])/β(H [01])|

and therefore ν is a covering of graphs.

2.4 The universal completion

As is common, I ignore certain technicalities concerning universal covers anduniversal completions. The standard reference for a detailed treatment of themis Serre (1977).

Let ρ : H → H be the universal completion of H. Then ρ is generating, Hpossesses a homomorphism onto any other generating completion group of H, inparticular it possesses such a homomorphism

µ : H → H

onto H ∼= O+10(2). The universal completion group is the free product of the

groups H [0] and H [1] amalgamated over their common subgroup H [01] while thegraph

Ω = Λ(H, ρ, H)

is the regular infinite tree of valency 31.Let ν : Ω → Ω be the covering of graphs as in (2.3.3) induced by the com-

pletion homomorphism µ. Since ν is a covering of graphs, for every u ∈ Ω therestriction of ν to u ∪ Ω(u) is a bijection onto ν(u) ∪ Ω(ν(u)).

Referreing to the definition of Λ(H, ρ, H) in the paragraph before (2.3.2) letx be the vertex of Ω which is the cosets of ρ(H [0]) containing the identity andlet x, y be the edge which is the coset of ρ(H [1]) containing the identity. Then

ν(x) = x and ν(y ) = y

where H [0] = H(x) and H [1] = Hx, y. The natural action of H on Ω is locallyprojective of type (5, 2) and this action clearly commutes with ν. Furthermore,according to our notation

H(x) = ρ(H [0]) and Hx, y = ρ(H [1]).


There is a standard way to describe Ω and ν in terms of arcs in Ω originating atx which works as follows.

Let π = (x0 = x, x1, . . . , xn) be an n-arc originating at x and terminating atxn. Then (because of the covering property of ν) there is a unique n-arc π in Ωoriginating at x which maps onto π. Namely

π = (x0 = x, x1, . . . , xn),

where for 1 ≤ i ≤ n the vertex xi is the unique preimage of xi in Ω(xi−1). SinceΩ is cycleless, π is the only arc which joins x0 and xn. On the other hand, since Ωis connected, its every vertex is joint with x by an arc of some length. Thereforewe can identify the vertex-set of Ω with the set of arcs in Ω originating at x. Inthese terms two vertices in Ω are adjacent if one of the corresponding arcs is acontinuation of the other one by a single edge. The covering ν sends an arc ontoits terminal vertex.

Now let us discuss what happens with cubic geometric subgraphs under thecovering ν. Arguing as in Section 2.2 it is easy to show that Ω contains a familyof such subgraphs. Using the above construction we can provide a more explicitdescription.

Let Ω[2] be the geometric cubic subgraph as in (2.1.3) which is stabilized byH [2]. Recall that Ω[2] is the complete bipartite graph K3,3, K [2] ∼= 23+12 : L3(2)is the vertexwise stabilizer of Ω[2] in H and

H [2]/K [2] ∼= R[2] ∼= O+4 (2) ∼= Sym3 Sym2

is the action induced by H [2] on Ω[2] (this action coincides with the automorphismgroup of Ω[2]). The vertex x and the edge x, y are contained in Ω[2] and

H[2] = H [02], H [12]

is the amalgam formed by the vertex and edge stabilizers in the action of H [2]

on Ω[2]. Notice that K [2] is the largest subgroup in H [012] = H [02] ∩ H [12] whichis normal in both H [02] and H [12].

Let Ω[2] be the geometric cubic subgraph in Ω which contains x and whichmaps onto Ω[2] under the covering ν. We can describe Ω[2] in two different ways.First, when the vertices of Ω are treated as the arcs in Ω originating at x, thevertices in Ω[2] are precisely the arcs all whose vertices are contained in Ω[2].Alternatively, if we put

H [2] = 〈ρ(H [02]), ρ(H [12])〉

then Ω[2] is induced by the images of x under H [2].The group H [2] is the free product of its subgroups ρ(H [02]) and ρ(H [12])

amalgamated over ρ(H [012]); the graph Ω[2] is the infinite cubic tree. The restric-tion νr of ν to Ω[2] is a covering onto Ω[2] and the restriction of µ to H [2] is clearlya homomorphism onto H [2]. Let M be the kernel of µ and put

M [2] = M ∩ H [2],

Characterization 27

so that M [2] is the kernel of the restriction of µ to H [2]. Since µ maps theamalgam ρ(H [0]), ρ(H [1]) isomorphically onto H = H [0], H [1], the intersec-tion M ∩ ρ(K [2]) is trivial. Thus M [2] and ρ(K [2]) are two normal subgroups inH [2] which have trivial intersection. Hence M [2] is contained in the centralizerof ρ(K [2]) in H [2]. On the other hand, it is easy to deduce from (2.1.2) thatCH[2](K [2]) = 1. This gives the following.

Lemma 2.4.1 Let M be the kernel of the completion homomorphism µ : H → Hand M [2] = M ∩ H [2]. Then M [2] = CH[2](ρ(K [2])).

Since ρ(K [2]) is the vertexwise stabilizer of Ω[2] in H we have M [2]∩ρ(K [2]) =1, since the action of M [2] on Ω[2] is faithful. On the other hand, M [2] is thegroup of deck automorphisms with respect to the covering

νr : Ω[2] → Ω[2],

which is universal. Then M [2] is a free group, whose rank is equal to the numberof fundamental cycles in Ω[2] ∼= K3,3 which is

|E(Ω[2])| − |V (Ω[2])| + 1 = 4.

Also because of the universality of νr the group M [2] acts regularly on ν−1r (w)

for every vertex w of Ω[2]. Let Ω[2] be the graph on the set of orbits of M [2] onV (Ω[2]) in which two orbits are adjacent if there is at least one edge which joinsthem. Then by the above discussion we have the following.

Lemma 2.4.2 The graphs Ω[2] and Ω[2] are isomorphic.

Thus the graph Ω[2] is already visible in the universal completion of H. Noticealso that the automorphism group of Ω[2] is the subgroup in Out K [2] generatedby the natural images of H [02] and H [12].

2.5 Characterization

Let ξ : H → X be an arbitrary faithful generating completion of H, Ξ =Λ(H, ξ, X) be the corresponding coset graph. Let

λ : H → X

be the completion homomorphism and

ω : Ω → Ξ

be the graph covering as in (2.3.2). Let Y be the kernel of λ and put Y [2] =Y ∩ H [2]. Since Y [2] intersects ρ(K [2]) trivially, Y [2] ≤ M [2] = CH[2](ρ(K [2])).


Let Ξ[2] = ω(Ω[2]) be the geometric cubic subgraph in Ξ which is the image ofΩ[2]. Then the orbits of M [2] on Ξ[2] have length

d = [M [2] : Y [2]]

Then Ξ[2] is a d-fold covering of the graph Ξ[2] ∼= Ω[2] ∼= K3,3 defined as inthe paragraph before (2.4.2). We find it appropriate to call d the defect of thecompletion ξ. Thus the geometric cubic subgraph is the smallest possible if andonly if the defect of the completion is 1.

We are going to show that H is the only completion group with defect 1.Thus we assume that Y [2] = M [2] = CH[2](ρ(K [2])) or, equivalently that Ξ[2] ∼=Ω[2] ∼= K3,3. We further assume that X is the largest completion with defect 1,which means that Y is the smallest normal subgroup in H which intersects H [2]

in M [2]. Therefore Y is the normal closure in H of M [2] = CH[2](ρ(K [2])). Underthis assumption there is a completion homomorphism

α : X → H

and a covering

β : Ξ → Ω

of the corresponding coset graphs, such that µ is the composition of λ and α,while ν is the composition of ω and β.

In order to establish the stated uniqueness of (H, Ω) it is sufficient to showthat at least one (and hence both) of α and β is bijective. We prefer to dealwith β. By the assumption the restriction of β to any geometric cubic subgraphin Ξ is an isomorphism onto a geometric cubic subgraph in Ω. On the otherhand, every cycle of length 4 in Ω is contained in some geometric cubic subgraph.Therefore every cycle of length 4 in Ω is contractible with respect to β (this meansthat for a 4-cycle δ in Ω the preimage β−1(δ) is a disjoint union of 4-cycles in Ξ).Now in order to conclude that β is an isomorphism it is sufficient to establishthe following.

Lemma 2.5.1 The fundamental group of Ω is generated by the cycles of length 4.

Proof We have to show that an arbitrary cycle in Ω can be decomposed intoquadrangles. Let σ = (x0, x1, . . . , xn) be such a cycle of length n, say. Since Ω isbipartite, n is even, say n = 2m. If m = 2, σ is itself a 4-cycle, so we assume thatm ≥ 3. One can observe that it is sufficient to decompose the cycles such that

dΩ(x0, xm) = m.

The 2-arc π = (xm−1, xm, xm+1) satisfies dΩ(x0, xm−1) = dΩ(x0, xm−1) = m−1.Let Σ be the unique geometric cubic subgraph in Ω containing π. Then by (2.1.1(vi)) there is a unique vertex u in Σ nearest to x0. Since Σ is of diameter 2 andit contains a vertex xm at distance m from x0 and at least two vertices xm−1and xm+1 at distance m − 1 from x0, we conclude that dΩ(x0, u) = m − 2. Let

Characterization 29

(x0 = y0, y1, . . . , ym−2 = u) be a shortest arc which joins x0 and u. Then σdecomposes into a 4-cycle δ and two (2m − 2)-cycles

(x0, x1, . . . , xm−1, u = ym−2, ym−3, . . . , y0 = x0)

and

(x0, x2m−1, . . . , xm+1, u = ym−2, ym−3, . . . , y0 = x0).

Now it only remains to apply the induction on m.

By (2.5.1) whenever all 4-cycles are contractible with respect to a covering ofΩ, this covering is necessarily an isomorphism and we are ready to formulate themain result of the chapter originally established in Cameron and Praeger (1982).

Proposition 2.5.2 Let H = H [0], H [1] be the amalgam formed by the vertexand edge stabilizers of H ∼= O+

10(2) acting on the dual polar graph Ω = D+(10, 2).Let ξ : H → X be a faithful generating completion of H, let Ξ = Λ(H, ξ, X) be thecorresponding coset graph, let Ξ[2] be the geometric cubic subgraph in Ξ and X [2]

be the stabilizer in X of Ξ[2] as a whole. Suppose that the following equivalentconditions holds where K [2] ∼= 23+12 : L3(2) is the vertexwise stabilizer of Ξ[2] inX:

(i) Ξ[2] ∼= K3,3;(ii) CX[2](K [2]) = Z(K [2]) = 1.

Then (X, Ξ) = (H, Ω).

Exercises

1. The natural action of the orthogonal group F ∼= O+2n(2) on the dual polar

graph Φ = D+(2n, 2) is locally projective of type (n, 2). Let u, v be an edgeof Φ and F = F (u), Fu, v. Show that F is the only completion of Fsubject to the condition that CF [2](K [2]) = 1.

2. Show that the dual polar space O+(2n, 2) is simply connected for n ≥ 3.3. Let F = F [0], F [1] be an amalgam of rank 2 such that [F [1], F [01]] = 2 and

[F [0] : F [01]] ≥ 2. Let α : F → F be a faithful generating completion of F .Suppose that (a) F [01] is a maximal subgroup in F [0] (equivalently F [0] actsprimitively on the cosets of F [01]); (b) the automorphism of F [01] inducedby f ∈ F [1] \ F [01] is not a restriction of an automorphism of F [0]. ThenΛ(F , α, F ) has no multiple edges.

4. The conclusion of exercise 3 holds if the condition (b) is substituted by thefollowing two conditions: (c) no non-trivial normal subgroup in F [01] is normalin both F [0] and F [1]; (d) F [01] contains a non-trivial normal subgroup of F [0].

3

MODIFYING THE RANK 2 AMALGAM

In this section we modify the classical amalgam H to obtain the amalgam Gwhich will eventually lead us to J4. The core of the modification is the tripleisomorphism

Ω+6 (2) ∼= Alt8 ∼= L4(2),

which explains the non-vanishing cohomology of the natural 6-dimensional ortho-gonal module of the triform group. Although the modifications looks like aninnocent move, almost immediately exceptional structures like Mathieu groupsappear.

3.1 Complements and first cohomology

Let V be an elementary abelian 2-group, let C be a group, let σ : C → Aut V bea homomorphism (which turns V into a GF (2)-module for C) and let G = V : Cbe the semidirect product of V and C (with respect to σ). We identify V and Cwith their images under the natural injections into G.

In this section we recall some standard tools for calculating the automorphismgroup A of G = V : C. The inner automorphisms are well understood. Somefurther automorphisms are those of C normalizing σ. The tricky part is

D(C, V ) = CA(V ) ∩ CA(G/V ).

The elements of D(C, V ) will be called the deck automorphisms of the semidirectproduct G = V : C. It is known that the deck automorphisms are controlledby the first cohomology group H1(C, V ). We are going to discuss outer deckautomorphisms and relate them to the classes of complements to V in G and alsoto indecomposable extensions of V by trivial C-modules. We follow section 17in Aschbacher (1986).

If d ∈ D(C, V ) is a deck automorphism, then d centralizes V . For everyc ∈ C there is v ∈ V (depending on d and c) such that d(c) = cv and d(C) is acomplement to V in G. Any complement to V in G is a transversal of the leftcosets of V in G. Considering C as a canonical transversal, any transversal canbe represented as

Cγ = cγ(c) | c ∈ C for a function γ : C → V.

Lemma 3.1.1 The transversal Cγ is a complement to V in G if and only if γis a cocycle in the sense that

γ(ab) = γ(a)bγ(b)

for all a, b ∈ C.

30

Complements and first cohomology 31

Proof The transversal Cγ is a complement to V in G if and only if it is closedunder multiplication. The multiplication law in the semidirect product gives thecocycle condition.

Let Γ(C, V ) denote the set of V -valued functions on C which satisfy thecocycle condition in (3.1.1). By the definition every cocycle γ ∈ Γ(C, V ) mapsthe identity element of C onto the identity element of V .

The image d(C) of C under a deck automorphism d ∈ D(C, V ) is a com-plement to V in C and by (3.1.1) d(C) = Cγ for a cocycle γ ∈ Γ(C, V ). Thecorrespondence works in the other direction as well.

Lemma 3.1.2 Let γ ∈ Γ(C, V ) be a cocycle, so that Cγ is a complement to Vin C. Then

dγ : cv → cγ(c)v

is a deck automorphism which maps C onto Cγ .

Proof It is straightforward to use the multiplication law in the semidirectproduct G = V : C to check that dγ is in fact an automorphism. It is alsoclear that dγ centralizes both V and G/V and maps C onto Cγ .

Thus we can define a mapping ϕ : Γ(C, V ) → D(C, V ) by setting ϕ(γ) = dγ .Since D(C, V ) is a group of automorphisms of G, it carries a group structureunder composition. We define a group structure on Γ(C, V ) via pointwise multi-plication (for γ, δ ∈ Γ(C, V ) we define γδ(c) = γ(c)δ(c) for all c ∈ C). It is easyto check that Γ(C, V ) is a group with respect to this multiplication.

Lemma 3.1.3 The mapping ϕ : Γ(C, V ) → D(C, V ) is an isomorphism ofgroups.

Proof Observe that ϕ is surjective by (3.1.1) and (3.1.2). It is immediate to checkthat ϕ respects the above defined group structures. Finally, dγ is the identity auto-morphism if and only if Cγ = C, that is, if γ is the trivial cocycle which maps everyelement of C onto the identity of V . Thus ϕ is an isomorphism.

Since V is an elementary abelian 2-group, so is the group of V -valuedfunctions on C (with respect to the pointwise multiplication).

Lemma 3.1.4 The group D(C, V ) of deck automorphisms is an elementaryabelian 2-group. It acts regularly on the set of complements to V in G.

Proof By the remark before the lemma Γ(C, V ) is an elementary abelian2-group and by (3.1.3) so is D(C, V ). By (3.1.1) every complement to V inG is of the form Cγ for some γ ∈ Γ(C, V ) and by (3.1.3) ϕ(γ) is the unique deckautomorphism which maps C onto Cγ .

32 Modifying the rank 2 amalgam

Let ιg denote the inner automorphism of G induced by g ∈ G acting viaconjugation, so that

ι : g → ιg

is the natural homomorphism of G into its automorphism group A.

Lemma 3.1.5 If u ∈ V then

(i) ιu is a deck automorphism;(ii) if γu = ϕ−1(ιu) then γu : c → ucu for every c ∈ C.

Proof Since V is an abelian normal subgroup in G we have that ιu centralizesboth V and G/V , which gives (i). Furthermore, ιu(c) = ucu = cucu (since u isof exponent 2 we suppress the inverses) and (ii) follows.

In terms introduced in the proof of (3.1.5) let

B(C, V ) = γu | u ∈ V ,

so that B(C, V ) is the image of ι(V ) under ϕ−1.The first cohomology group of the representation σ of C on V is defined as

H1(C, V ) ∼= Γ(C, V )/B(C, V ) ∼= D(C, V )/ι(V ).


(i) H1(C, V ) acts regularly on the set of conjugacy classes of the complementsto V in C;

(ii) H1(C, V ) is isomorphic to the image of D(C, V ) in the outer automorph-ism group of G = V : C.

Proof Since V and C factorize G, ι(V ) acts regularly on set of complementsconjugate to C in G, (i) is by (3.1.4) and the definition of H1(C, V ). (ii) followsdirectly from the definition.

Since both ι(G) and D(C, V ) are normal in NA(V ), the elements ιc for c ∈ Cact on D(C, V ) via conjugation. This action turns D(C, V ) into a GF (2)-modulefor C. Pushing this action through the isomorphism ϕ−1 : D(C, V ) → Γ(C, V )we also turn Γ(C, V ) into a GF (2)-module for C. With respect to this definition,B(C, V ) is a submodule in Γ(C, V ) isomorphic to ι(V ) (notice that the latter isisomorphic to V/CV (G)).

Lemma 3.1.7 With respect to the GF (2)-module structure of Γ(C, V ) for Cdefined above the following equivalent assertions hold:

(i) [Γ(C, V ), C] ≤ B(C, V );(ii) C centralizes Γ(C, V )/B(C, V ).

Proof We calculate in A = Aut G. Since ι(V ) centralizes D(C, V ), we have[D(C, V ), ι(C)] = [D(C, V ), ι(G)] and since D(V, C) ∩ ι(G) = ι(V ), (i) follows.

Permutation modules 33

By (3.1.6) the elements of Γ(C, V )/B(C, V ) ∼= H1(C, V ) are indexed by theconjugacy classes in G of the complements to V in G. Clearly ι(G) acts triviallyon these classes, which gives (ii).

Suppose now that CC(V ) = 1, so that ι(V ) ∼= B(C, V ) ∼= V . In this case by(3.1.7) Γ(C, V ) is an indecomposable extension of V by trivial C-modules. Thenext lemma shows that this extension is the largest one.

Lemma 3.1.8 Suppose that CV (C) = 1 and let W be a GF (2)-module for Cwhich contains V as a submodule. Suppose further that [W, C] ≤ V (this meansthat W is an indecomposable extension of V by a trivial module). Then there isan injection ψ : W → Γ(C, V ) which sends V onto B(C, V ).

Proof Consider the semidirect product H = W : C of W and C with respectto the natural action. Then H contains G = V : C as a normal subgroup and wecan consider λ : H → A = Aut G via conjugation in H. Clearly the restrictionof λ to G coincides with ι. Since CV (C) = 1 and [W, C] ≤ V the restriction ofλ to W is an isomorphism. Furthermore, since V is an abelian normal subgroupin W , λ(W ) ≤ CA(V ) and since W/V is a trivial C-module, λ(W ) ≤ CA(G/V ).Hence λ(W ) ≤ D(C, V ) and composing the restriction of λ to W with ϕ−1 weobtain the required injection ψ.

We will also need the following straightforward generalization of (3.1.2).

Lemma 3.1.9 Let G = Q : C be a semidirect product of Q and C, let γ : C → Qbe a function satisfying the cocycle condition in (3.1.1), so that Cγ = cγ(c) |c ∈ C is a complement to Q in G. Suppose further that γ(c) ∈ Z(Q) for everyc ∈ C. Then

dγ : cq → cγ(c)q

is a deck automorphism of G which centralizes Q and maps C onto Cγ .

We conclude this section a special case of Schur–Zassenhaus Theorem (cf.(17.10) in Aschbacher (1986)) which we present without a proof.

Lemma 3.1.10 Suppose that C is a group of odd order and V is a GF (2)-modulefor C. Then H1(C, V ) is trivial, so that all complements to V in G = V : C areconjugate in G.

3.2 Permutation modules

We have seen in the previous section that the first cohomology group H1(C, V )can be understood through indecomposable extensions of V by trivial C-modules.An important source of such extensions are the GF (2)-permutation modules.

Let Pn be a set of n elements and Pn be the space of GF (2)-valued functionson Pn. Then Pn carries a GF (2)-vector space structure with respect to thepointwise addition. Identifying a function with its support, we treat Pn as the


power set of Pn. In these terms the addition is performed by the symmetricdifference operator defined by

AB = (A ∪ B)\(A ∩ B)

for A, B ⊆ Pn. For any group R of permutations of Pn the space Pn carries theGF (2)-module structure called the permutation module over GF (2).

Let h : Pn × Pn → GF (2) be defined by

h(A, B) = |A ∩ B| mod 2

and put

Pcn = ∅, Pn, Pe

n = A | A ⊆ Pn, |A| = 0 mod 2.

The following result is standard.

Lemma 3.2.1 Let R be the symmetric or alternating group of Pn. Then

(i) h is bilinear, R-invariant and non-singular;(ii) Pc

n and Pen are the only proper R-submodules in Pn.

By (3.2.1) Pn is self-dual, (Pcn)⊥ = Pe

n and (Pen)⊥ = Pc

n. Furthermore, whenn is odd Pn = Pc

n ⊕ Pen while

0 < Pcn < Pe

n < Pn

is the only R-invariant composition series of Pn when n is even, in particular theheart

Hn = 〈Pen,Pc

n〉/Pcn

of the permutation module is always irreducible for Altn. If n is even then Pn/Pcn

is an indecomposable extension of Hn by the trivial 1-dimensional module.

Lemma 3.2.2 Let C(1) = Ω+6 (2), V (1) = V +

6 ; C(2) = L4(2), V (2) be the exteriorsquare of the natural module of C(2); C(3) = Alt8, V (3) = H8. Then

(i) the groups V (i) : C(i) are pairwise isomorphic for 1 ≤ i ≤ 3;(ii) an element from P8/Pc

8 : Alt8 acting on the subgroup V (3) : C(3) byconjugation induces a deck automorphism of the latter;

(iii) H1(C(i), V (i)) ∼= 2 for 1 ≤ i ≤ 3.

Proof (i) is directly by (1.9.5) and its proof while (ii) is by (3.1.8) in view of theremark before that lemma. By (ii) we know that H1(C(i), V (i)) is non-trivial.The exact order is known in the literature (cf. exercise 6.3 in Aschbacher (1986)in terms of Alt8 and Bell (1978) in terms of L4(2)).

We can reformulate the discussions in Section 1.10 to observe the following.Let L ∼= L2(7) act on P8 as on the projective line over GF (7). Then

H8 ∼= I1 ⊕ I2

where I1 and I2 are the natural and the dual natural L3(2)-modules of L.

Permutation modules 35

Lemma 3.2.3 Let J2 be the preimage of I2 in Pe8 and put I1 = P8/J2. Then

(i) I1 is an indecomposable extension of I1 by a 1-dimensional trivial module;(ii) H1(L, I1) ∼= 2.

Proof Let S be a Sylow 2-subgroup in L. Then |S| = 8, S intersects triviallythe stabilizer of a point from P8 and hence S acts transitively on P8. Since Smust stabilize a non-zero vector in each L-submodule, we conclude that Pc

8 isthe unique minimal L-submodule and dually Pe

8 is the only maximal submodulewhich gives (i).

By (i) H1(L, I1) is non-trivial in view of (3.1.8) and I1 : L contains atleast two conjugacy classes of complements (compare (3.1.6)). Hence in orderto establish (ii) it is sufficient to show that I1 : L contains at most two classesof complements. Let L(1) be a complement which is not in the class of L. Thenby (3.1.10) and the Sylow theorem we may assume without loss that L ∩ L(1) isthe normalizer F of a Sylow 7-subgroup in L (which is the Frobenius group oforder 21). Let T be a Sylow 3-subgroup in F . Then NL(T ) ∼= Sym3 and since Fis maximal in L we conclude that L is generated by F together with an element aof order 2 inverting T . Clearly L(1) is generated by F and a similar element a(1).We assume that aI1 = a(1)I1, so that aa(1) is a non-identity element in I1 com-muting with T . Since CI1(T ) ∼= 2 this element is uniquely determined and theresult follows.

We will make use of the following result, which is a special case ofproposition 3.3.5 in Ivanov and Shpectorov (2002).

Lemma 3.2.4 Let M ∼= L4(2) and let U be the natural 4-dimensional GF (2)-module for M . Let P15 be the set of non-zero vectors of U on which Macts in the natural way and let P15 be the corresponding GF (2)-permutationmodule. Then

P15 = Pc15 ⊕ Pe

15

and Pe15 possesses the unique M -invariant composition series

0 < R1 < R2 < Pe15,

where

Pe15/R2 ∼= U, R2/R1 ∼=

∧2U ∼=

∧2U∗, R1 ∼= U∗.

We use the above lemma to prove yet another fact on first cohomology whichwe will need later.

Lemma 3.2.5 Let M and U be as in (3.2.4). Then H1(M, U) is trivial.

Proof If H1(M, U) is non-trivial, then by (3.1.7) there exists an indecomposableextension W of U by a 1-dimensional trivial module. Consider the dual W ∗ of W .Then [M, W ∗] =W ∗ (since the extension is indecomposable), W ∗/CW ∗(M) ∼=U∗

and CW ∗(M) is 1-dimensional (trivial). The group M ∼= L4(2) acts on the set


of non-zero vectors in U∗ as it does on the cosets of a subgroup K ∼= 23 : L3(2).The preimage in W ∗ of a vector from U∗, stabilized by K is of size 2. SinceK has no subgroups of index 2, it fixes every vector in the preimage and hencethere is a 15-orbit of M on W ∗ and since the extension is indecomposable, thisorbit generates the whole of W ∗. Thus W ∗ is a quotient of the dual of P15.By (3.2.4) the latter involves only one 1-dimensional composition factor and thisfactor splits as a direct summand. Hence there are no indecomposable extensionsW ∗ as required and the result follows.

Let S ∼= S4(2) ∼= Sym6, let P6 be a set of six elements on which S inducesthe symmetric group and let P6 be the corresponding GF (2)-permutation mod-ule. By the results in Section 1.8, the heart H6 of the permutation module isisomorphic to the 4-dimensional symplectic module of S (recall that S possessestwo such modules which are permuted by the outer automorphism group). By(3.2.1) Pe

6 is an indecomposable extension of a trivial 1-dimensional module byH6 while P6/Pc

6 is an indecomposable extension of H6 by a trivial 1-dimensionalmodule.

Lemma 3.2.6 Let S ∼= S4(2) ∼= Sym6, let V4 be the natural symplectic modulefor S and let V5 be an indecomposable extension of a trivial 1-dimensional moduleV1 by V4. Then

(i) H1(S, V4) ∼= 2;(ii) V5 ∼= Pe

6 ;(iii) H1(S, V5) ∼= 22;(iv) if T ∼= Alt6 is the commutator subgroup of S, then H1(T, V4) ∼=

H1(T, V5) ∼= 2.

Proof By the paragraph before the lemma and (3.1.8), the first cohomologyof S on V4 is non-trivial. For the exact order in (i) we refer to exercise 6.3in Aschbacher (1986). Assertion (i) and (3.1.8) (in view of the duality) give theuniqueness of V5 and hence (ii). Since P6 is an indecomposable extension of Pe

6 bya trivial 1-dimensional module, H1(S, V5) is non-trivial. If S(1) is a complementto V5 in the semidirect product R of V5 and S, then NR(S(1)) = S(1) ×V1, whereV1 = CR(S(1)), contains two subgroups isomorphic to S(1). This constructiondoubles the number of classes of S4(2)-complements in R/V1 and (iii) follows.Now (iv) must also be rather clear.

Lemma 3.2.7 Let V5 be a 5-dimensional GF (2)-space and L ∼= L5(2) be the gen-eral linear group of V5. Let X be the stabilizer in L of a 1-dimensional subspaceV1 of V5 together with a non-singular symplectic form f on V5/V1. Then

(i) X ∼= 24 : S4(2) is the semidirect product of S4(2) and its naturalsymplectic module;

(ii) X contains two classes of S4(2)-complements with representatives S(1)

and S(2), say;

Deck automorphisms of 26 : L4(2) 37

(iii) the representatives can be chosen so that the action of S(1) on V5 issemisimple while that of S(2) is indecomposable;

(iv) there is a deck automorphism of X which transposes S(1) onto S(2).

Proof Statement (i) is clear while (ii) is by (3.2.6 (i)). We know that thereare two extensions of a trivial 1-dimensional module by the natural symplecticmodule: the semisimple and the indecomposable ones. Hence there are twoS4(2)-subgroups S(1) and S(2) in L acting on V5 in these two ways, preserving a1-dimensional subspace V1 and a fixed non-singular symplectic form f on V5/V1.Then both S(1) and S(2) are contained in X and it is easy to check that they arenot conjugate in X. Hence (iii) follows. Now (iv) is immediate from (3.1.6).

3.3 Deck automorphisms of 26 : L4(2)

Let Z [1]L[1] ∼= 26 : L4(2) be a subgroup of H [1] (compare (2.1.2)). In this sectionwe analyse the deck automorphisms of Z [1]L[1] (which are the automorphismsacting trivially on both Z [1] and Z [1]L[1]/Z [1]).

By (1.9.5) and (3.2.2) Z [1]L[1] can be treated in the following four differentlanguages (notice that Z [1] is self-dual as a module for L[1]).

Linear: L[1] is the general linear group of U4 and Z [1] ∼=∧2

U4 is the exteriorsquare of U4.Symplectic: L[1] is the general linear group of U4 and Z [1] ∼= S(U4) is the spaceof symplectic forms on U4.Orthogonal: L[1] ∼= Ω+

6 (2) is the commutator subgroup of the orthogonal groupof plus type in dimension 6 over GF (2) and Z [1] ∼= V +

6 is the natural orthogonalmodule.Permutation: L[1] ∼= Alt8 is the alternating group of a set P8 of eight elementsand Z [1] ∼= H8 is the heart of the GF (2)-permutation module P8 of L[1] on P8.

The following lemma (which is a direct consequence of (1.9.5), (3.2.2) and(1.4.3) gives a classification of the non-zero vectors in Z [1] in the above fourlanguages.

Lemma 3.3.1 The group L[1] acts by conjugation on the set of non-identityelements of Z [1] with two orbits Υ0 and Υ1 of length 35 and 28, respectively.Furthermore,

(i) if z ∈ Υ0 then

L[1](z) ∼= 24 : (Sym3 × Sym3) ∼= (Sym4 Sym2)+

(where + indicates the intersection with the alternating group);(ii) if y ∈ Υ1 then

L[1](y) ∼= S4(2) ∼= Sym6;

(iii) z ∈ Υ0 if and only if the following equivalent conditions hold:(a) z = u ∧ v ∈

∧2U4 for some linearly independent u, v ∈ U4;

(b) z ∈ S(U4) is a form with 2-dimensional radical;


(c) z is singular in V +6 ;

(d) z is a partition of P8 into two 4-element subsets;(e) z is the Siegel transformation sW for a 2-dimensional subspace W

in U4;(iv) y ∈ Υ1 if and only if the following equivalent conditions hold:

(f) y = u ∧ v + w ∧ r ∈ ∧2U4 for some linearly independent u, v, w, r ∈ U4;

(g) y ∈ S(U4) is a non-singular symplectic form;(h) y is non-singular in V +

6 ;(i) y is a partition of P8 into a 2-element subset and its 6-element

complement.

Let h ∈ S(U4) be a symplectic form on U4 considered as an element of Z [1].Let Zh be the orthogonal perp of h with respect to the non-singular symplecticform f on Z [1] preserved by L[1] (so that Z [1] ∼= V +

6∼= (V, f, q) in the orthogonal

language). Since f is non-singular, every hyperplane in Z [1] arises in this way.In the permutation terms two even partitions

P8 = A1 ∪ A2 and P8 = B1 ∪ B2

(considered as elements of Z [1]) are perpendicular with respect to f if and onlyif |Ai ∩ Bj | is even for 1 ≤ i, j ≤ 2. This gives the following.


(i) if h ∈ Υ0 then Zh contains 19 vectors from Υ0 and 12 vectors Υ1;(ii) if h ∈ Υ1 then Zh contains 15 vectors from Υ0 and 16 vectors from Υ1.

The deck automorphisms of Z [1]L[1] are best understood in the permutationlanguage, but we also will deal with them in the other languages.

Let Z [1]u be the indecomposable extension of the module Z [1] by the trivial1-dimensional submodule. Then by (3.2.2), Z [1]uL[1] ∼= (P8/Pc

8) : Alt8. An ele-ment d ∈ P8/Pc

8 naturally corresponds to a partition of P8 into two subsets.Furthermore, d ∈ Z [1] if and only if the partition is even. Whenever d ∈ Z [1]

the subgroup L[1] and its image (L[1])d under d represent different classes ofcomplements to Z [1] in Z [1]L[1]. For i = 1 and 3 let di correspond to a partitionof P8 into two subsets of size i and (8 − i), respectively.

Lemma 3.3.3 There are two classes of elements in Z [1]u \ Z [1] with represent-atives d1 and d3. Furthermore

CL[1](d1) ∼= Alt7, CL[1](d3) ∼= (Sym3 × Sym5)+

where + indicates the intersection with the alternating group Alt8.

In what follows we will need some further information on the action of ele-ments from Z [1]u on the preimages of maximal parabolic subgroups of Z [1]L[1]/Z [1] ∼= L[1] ∼= L4(2).

Let u ∈ U#4 and let L[1](u) ∼= 23 : L3(2) be the stabilizer of u in L[1].

Deck automorphisms of 26 : L4(2) 39

Lemma 3.3.4 Let

(i) K(1) be the stabilizer in L[1](u) of a hyperplane V in U4 not containingu;

(ii) K(2) be the stabilizer in L[1](u) of an element from P8;(iii) K(3) be the stabilizer in L[1](u) of a ordered pair of disjoint maximal

totally singular subspaces in V +6 .

Then K(1) ∼= K(2) ∼= K(3) ∼= L3(2), furthermore K(1) is conjugate in L[1](u)to K(3) but not to K(2).

Proof Since 〈v ∧ w|v, w ∈ V 〉 and 〈u ∧ v|v ∈ V 〉 are disjoint maximal totallysingular subspaces in V +

6 stabilized by K(1), the latter is a conjugate of K(3).On the other hand by (1.10.1) the subgroup K(3) acts doubly transitively on P8as PSL2(7) acts on the projective line over GF (7). Hence K(2) and K(3) cannotpossibly be conjugate in L[1](u).

Notice that the actions of K(1) on U4 and V +6 are semisimple and the action

of K(1) on P8 is doubly transitive, while the actions of K(2) on U4 and V +6 are

indecomposable while on P8 it is intransitive.

Lemma 3.3.5 Assume that K(2) is the stabilizer in L[1](u) of a point p ∈ P8 andthat d1 ∈ Z [1]u corresponds to the partition of P8 into p and its complement.Then d1 centralizes K(2) but not K(1) and it maps O2(L[1](u)) onto the uniqueK(2)-invariant complement to Z [1] in Z [1]O2(L[1](u)) distinct from O2(L[1](u)).

Proof By (3.3.1) CL[1](d1) = L[1](p) ∼= Alt7 intersects L[1](u) in K(2). Since (asa K(2)-module) the quotient of Z [1]O2(L[1](u)) over its centre is the direct sumof two copies of O2(L[1](u)) (one of them being in Z [1]), we have the uniquenessstated in the lemma.

Let W be a 2-dimensional subspace in U4. Then

L[1](W ) ∼= 24 : (Sym3 × Sym3)

and in terms of the permutation action on P8 it is the stabilizer of a partition ofP8 into two equal parts, say A and B, and

L[1](W ) ∼= (Sym4 ⊗ Sym2)+.

Let T be a Sylow 3-subgroup in L[1](W ), and let T (1) and T (2) be the subgroupsof order 3 in T normal in NL[1](W )(T ) ∼= Sym3 × Sym3. Then in the linear termsNL[1](W )(T ) is a Levi complement in L[1](W ) which is the stabilizer of a directsum decomposition

U4 = W (1) ⊕ W (2),

where W (1) = W . We assume that T (i) is the kernel of T on W (3−i) for i = 1, 2.Let u and v be non-zero vectors from W (1) and W (2), respectively and let p(u, v)


denote the transvection with centre u and axis 〈W (1), v〉. Then the elementsp(u, v) taking for all the nine choices for the pair (u, v) generate O2(L[1](W )).

In the permutation terms NL[1](W )(T ) is the stabilizer of a partition

P8 = a, b ∪ (A \ a) ∪ (B ∪ b)

for some a ∈ A and b ∈ B. The subgroup T (i) is generated by an element ofcycle type 32, a Sylow 2-subgroup R of NL[1](W )(T ) contains three involutions,say r(1), r(2) and r(12), such that r(1) and r(2) are of cycle type 24 and r(12) is ofcycle type 2214. We assume r(i) centralizes T (3−i) and inverts T (i) for i = 1, 2 sothat r(12) = r(1)r(2) acts on T fixed-point freely. Let d be a deck automorphism ofZ [1]L[1] which centralizes T . Then there are four choices for d, but independentlyof the choice we have the following.

Lemma 3.3.6 In the above terms the following assertions hold:

(i) CL[1](d) ∩ O2(L[1](W )) ∼= 22;(ii) CL[1](d) ∩ R = 〈r(12)〉;(iii) [r(1), d] = [r(2), d] = δ, where δ is the element from Z [1] corresponding to

the partition of P8 into a, b and its complement.

3.4 Automorphism group of H[01]

As above let H = H [0], H [1] be the amalgam formed by the stabilizers in H ∼=O+

10(2) of a vertex x and an edge x, y of the dual polar graph Ω = D+(10, 2).In terms of the orthogonal space V +

10 , the vertex x is a maximal totally singularsubspace U5, while x, y is a hyperplane U4 in U5. The amalgam H can belooked at in the following way:

(H0) H [0] is the semidirect product of Q[0] ∼=∧2

U5 and L[0] ∼= L5(2) actingnaturally on Q[0];

(H01) H [01] is the semidirect product of Q[0] and the maximal parabolic sub-group L[0](U4) ∼= 24 : L4(2) in L[0], which is the stabilizer of thehyperplane U4 in U5;

(H1) H [1] = 〈H [01], t0〉, where t0 is an element of order 2 which induces anautomorphism τ0 of H [01] which does not normalize Q[0].

We are going to modify H by keeping (H0) with (H01) and changing theelement t0 in (H1) by an element t1 inducing a different automorphism of H [01]

still not normalizing Q[0]. Towards this end we calculate the automorphism groupof H [01].

First let us recall and refine the geometrical setting from Section 2.1. LetW2 be a complement to U4 in U⊥

4 (so that W2 ∼= V +2 ) and let T4 be a maximal

totally singular subspace in W⊥2

∼= V +8 , disjoint from U4. Let u and v denote

the non-zero singular vectors in W2. We assume without loss that 〈U4, u〉 = U5,while 〈U4, v〉 is the only other maximal totally singular subspace in V ∼= V +

10

Automorphism group of H [01] 41

which contains U4 (and corresponds to the vertex y of Ω). Then

C [1] = L[1] × R[1] ∼= L4(2) × 2

is the stabilizer of T4 in H [1] (C [1] also stabilizes W2 = 〈U4, T4〉⊥). The kernel Q[1]

of the action of H [1] on U⊥4 is a special 2-group of order 214. The centre Z [1] of

Q[1] is generated by the Siegel transformations sT taken for all the 2-dimensionalsubspaces T in U4 so that by (1.3.5) Z [1] and

∧2U4 are isomorphic as L[1]-

modules. Let A[1] and B[1] be the subgroups in Q[1] generated by the Siegeltransformations s〈u,w〉 and s〈v,w〉 taken for all w ∈ U#

4 . Then by the proof of(1.3.5) we have the following commutator relations

[s〈u,w1〉, s〈v,w2〉] =

1 if w1 = w2;s〈w1,w2〉 otherwise.

Therefore the following lemma holds.

Lemma 3.4.1 Let α :U4 → A[1], β :U4 → B[1] and ζ :∧2

U4 → Z [1] be theisomorphism commuting with the action of L[1]. Then the following key com-mutator relation holds

[α(w1), β(w2)] = ζ(w1 ∧ w2)

for all w1, w2 ∈ U4.

By (2.1.2) Q[1] = Z [1]A[1]B[1], Q[0] = Z [1]A[1], and H [01] = Q[1]L[1]. Finally,R[1] is of order 2 generated by the element t0 which is the orthogonal transvectionwith respect to the unique non-singular vector u + v of W2. Then t0 centralizesboth Z [1] and L[1], and conjugates s〈u,w〉 onto s〈v,w〉 for every w ∈ U#

4 . Thereforet0 conjugates A[1] onto B[1]. In particular it does not normalize Q[0] = Z [1]A[1].

Put

F [1] = 〈Z [1], L[1]〉 ∼= 26 : L4(2).

Then by (3.2.3 (iii)) H1(L[1], Z [1]) is of order 2, so that by (3.1.6) there exists acocycle γ : L[1] → Z [1] such that

L[1]γ = lγ(l) | l ∈ L[1]

is a complement to Z [1] in F [1] not conjugate to L[1]. By (3.1.2) the mapping

dγ : lz → lγ(l)z

(where l ∈ L[1] and z ∈ Z [1]) is an outer deck automorphism of F [1] which mapsL[1] onto L

[1]γ . Since Z [1] is the centre of Q[1], by (3.1.9) the mapping

σγ : lq → lγ(l)q

(where l ∈ L[1], q ∈ Q[1]) is an outer deck automorphism of

H [01] = Q[1] : L[1].


Proposition 3.4.2 The following assertions hold:

(i) Q[1] = O2(H [01]);(ii) Z [1] = Z(O2(H [01]));(iii) if F [1] is a normal subgroup of H [01] of order 2a for a ≥ 10 and F [1] < Q[1]

then a = 10, F [1] is elementary abelian and coincides with one of thefollowing A[1]Z [1], B[1]Z [1] and

D[1] = 〈s〈u,w〉s〈v,w〉 | w ∈ U#4 ;

(iv) H [01] contains exactly two conjugacy classes of L4(2)-complements to Q[1]

with representatives L[1] and L[1]γ ;

(v) if L is a complement to Q[1] in H [01] then A[1]Z [1] and B[1]Z [1] aresemisimple while D[1] is indecomposable (as L-modules);

(vi) the outer automorphism group of H [01] is elementary abelian of order 4generated by the image of the automorphism τ0 induced by t0 and by theimage of the deck automorphism σγ .

Proof Assertions (i) and (ii) are (2.1.2 (i), (ii)) restated. By (2.1.2 (vii))Q[1]/Z [1] as a module for H [01]/Q[1] ∼= L4(2) is isomorphic to the direct sumof two copies of U4; therefore there are exactly three proper submodules, whichare A[1]Z [1]/Z [1], B[1]Z [1]/Z [1] and the diagonal one denoted by D[1]. The com-mutator relation in (3.4.1) shows that D[1] is elementary abelian and containsZ [1], so that (iii) follows. By (3.2.5) every L4(2)-complement in

H [01]/Z [1] ∼= (24 × 24) : L4(2)

is a conjugate of L[1]Z [1]/Z [1]. The preimage of the latter in H [01] is Z [1] :L[1] ∼= 26 : L4(2). By (3.2.2 (iii)) this preimage contains exactly two classes ofcomplements and (iv) follows. Therefore in (v) we can assume without loss thatL = L[1]; then the assertion is immediate by (3.4.1).

Let δ be an automorphism of H [01]. In view of (iv), multiplying δ by innerautomorphisms and/or by σγ , if necessary, we can assume that δ normalizes L[1].It is known that

Aut L[1] ∼= L4(2) : 〈τ〉,

where τ is the contragredient automorphism. Since Q[1]/Z [1] involves two iso-morphic copies of U4 (and does not involve the dual of U4), δ cannot induce onL[1] a conjugate of the contragredient automorphism. Therefore, adjusting δ byinner automorphisms of L[1], we can assume that δ centralizes L[1]. By (v), δeither normalizes each of A[1]Z [1] and B[1]Z [1], or permutes them. Multiplyingδ by the automorphism τ0, if necessary, we assume that the former possibilityholds. Since A[1]Z [1] is the direct sum of two non-isomorphic absolutely irreduc-ible L[1]-modules A[1] and Z [1] (isomorphic to U4 and

∧2U4, respectively) and

δ commutes with the action of L[1] on these modules, we conclude that δ cent-ralizes both A[1] and Z [1]. Similarly δ centralizes B[1] and (vi) follows, therebycompleting the proof.

Amalgam G = G[0], G[1] 43

Notice that in terms on (3.2.4), the module D[1] for L[1] ∼= L4(2) is isomorphicto the section

Pe15/R1 ∼= 26+4

of the permutation module on the non-zero vectors of U#4 .

3.5 Amalgam G = G[0], G[1]In this section we modify the amalgam H = H [0], H [1] (taken from O+

10(2)) toobtain an amalgam G = G[0], G[1] which will eventually lead us to the group J4.

We preserve H [0] and H [01], so that we assume that G[0] = H [0] andG[01] = H [01], while G[1] is the semidirect product of G[01] and a group of order 2generated by an element t1 such that the automorphism τ1 of G[01] induced byt1 is the product of τ0 and the automorphism σγ defined before (3.4.2).


(i) the automorphism τ1 is of order 2, so that G = G[0], G[1] is correctlydefined;

(ii) the above definition specifies G up to isomorphism.

Proof The automorphism τ0 centralizes Z [1]L[1] and induces a permutation oforder 2 on A[1] ∪ B[1] while σγ centralizes Q[1] and induces an automorphismof order 2 of the subgroup Z [1]L[1]. Therefore τ1 is of order 2 and (i) follows.The isomorphism classes of rank two amalgams are controlled by Goldschmidt’stheorem (cf. Goldschmidt (1980) or proposition 8.3.2 in Ivanov and Schpectorov(2002)). The number of such classes is the number of double cosets in O :=Out G[01] of the subgroups O[0] and O[1], where O[i] is the natural image in O ofNAut G[i](G[01]) for i = 0 and 1. By (3.4.2 (vi)) O is elementary abelian of order4 and O[1] is the whole of O, which gives (ii).

The next result shows that in a certain sense G is the only possiblemodification of the amalgam H.

Proposition 3.5.2 Let X = X [0], X [1] be an amalgam of rank 2 satisfying thefollowing conditions (i) to (iii), where X [01] = X [0] ∩ X [1]:

(i) there is an isomorphism α : H [0] → X [0] such that X [01] = α(H [01]);(ii) [X [1] : X [01]] = 2;(iii) no proper subgroup from X [01] is normal in both X [0] and X [1].

Then X is isomorphic to either H or G.

Proof The restriction of α to H [01] induces an isomorphism of Aut H [01] ontoAut X [01]. Abusing the notation, we denote this isomorphism by the same let-ter α. Let t ∈ X [1] \ X [01]. Since X [01], being a subgroup of index 2, is normal inX [1], t induces an automorphism τ of X [01]. The only proper normal subgroup of


H [0] contained in H [01] is O2(H [0]). Therefore condition (iii) states that α−1(τ)does not normalize O2(H [0]) = Q[0] = Z [1]A[1], while by (3.4.2 (vi))

NAut H[01](Q[0]) = 〈Inn H [01], σγ〉.

By (3.4.2 (vi)) we can adjust τ by inner automorphisms of X [01] so that either

α−1(τ) = τ0, or α−1(τ) = τ1 = τ0σγ .

We claim that whenever t is chosen in this way, it is an involution. In fact,t2 ∈ X [01] and hence t2 induces an inner automorphism of X [01]. On the otherhand we have seen in the proof of (3.5.1) that both τ0 and τ1 are involutoryautomorphisms. Therefor t2 is in the centre of X [01] (compare (3.4.2) and itsproof). Hence the claim follows. Thus X [1] is the semidirect product of X [01]

and a group of order 2 generated by t and the latter induces the automorphismα(τi) for i = 0 or 1. Therefore there are two possibilities for the isomorphism typeof X [1] and arguing as in the proof of (3.5.1) (applying Goldschmidt’s theoremin a rather trivial way), we conclude that the isomorphism of X is uniquelydetermined by making the choice of τ between α(τ0) and α(τ1)).

3.6 Vectors and hyperplanes in Q[0]

From now on we require a better understanding of the structure of Q[0] =O2(G[0]), which we know is the exterior square of the 5-dimensional GF (2)-space U5. We describe the set P of hyperplanes (subgroups of index 2) in Q[0].By (1.4.4) P is in a natural bijection with the set of non-zero vectors in

∧2U∗

5and by (1.4.3) these vectors in turn correspond to the non-zero symplectic formson U5. The group

L[0] ∼= G[0]/Q[0] ∼= L5(2)

has two orbits on the set of non-zero symplectic forms on U5, consisting of theforms of rank 2 and 4, respectively. There are 155 and 15 subspaces in U5 ofdimension 2 and 4 respectively. Since there is a unique non-singular form ina 2-dimensional space and exactly 28 =

(82)

non-singular forms in a space ofdimension 4 (compare (3.3.1)) we obtain the following.

Lemma 3.6.1 The group L[0] ∼= L5(2) acting on the set P of hyperplanes inQ[0] has two orbits P1 and P2 with lengths 155 and 868, respectively.

The hyperplanes in P1 correspond to the rank 2 forms on U5 which areuniquely determined by their 3-dimensional kernels. Let V3 be a 3-subspace inU5 and let P (V3) be the corresponding hyperplane from P1. Then the Siegeltransformation sT for a 2-subspace T in U5 is contained in P (V3) if and onlyif V3 ∩ T = 0. The stabilizer of P (V3) in G[0] coincides with that of V3 and ifV3 = U3 the stabilizer coincides with G[02]. Recall that

G[02]/Q[0] ∼= 22 : (L3(2) × Sym3).

Structure of G[1] 45

The hyperplanes in P2 correspond to the rank 4 symplectic forms on U5.By (1.4.1 (iii)) such a form f is determined by its radical V1 (which is a1-dimensional subspace in U5) and by the non-singular symplectic form f onU5/V1 induced by f . The hyperplane determined by f will be denoted by P (V1, f)(although V1 is determined by f , we prefer to keep the explicit reference to theradical). If V1 = U1 then the stabilizer S of P (V1, f) is contained in G[04] withindex 28 (which is the number of non-singular symplectic forms on U5/U1).Furthermore, S = Q[0] : L[0](f) and by (1.5.1)

L[0](f) = Sf∼= 24 : S4(2) ∼= 24 : Sym6.

We also have the following ‘contragredient’ version of (3.6.1).

Lemma 3.6.2 The group L[0] ∼= L5(2) acting on the set of non-zero vectorsof Q[0] (considered as a GF (2)-module) has two orbits Q1 and Q2 with lengths155 and 868, respectively. The vectors in Q1 are indexed by the 2-dimensionalsubspace in U5 while the vectors in Q2 are indexed by the pairs (V4, g), whereV4 is a hyperplane in U5 and g is a non-singular symplectic form on V4.

Finally (3.6.1) and (3.6.2) are combined in the following.


(i) a hyperplane from P1 contains 91 vectors from Q1 and 420 vectorsfrom Q2;

(ii) a hyperplane from P2 contains 75 vectors from Q1 and 436 vectorsfrom Q2;

(iii) a vector from Q1 corresponding to a 2-dimensional subspace V2 of U5 iscontained in the hyperplane from P1 corresponding to a 3-dimensionalsubspace V3 of U5 if and only if V2 ∩ V3 is non-zero;

(iv) a vector from Q1 corresponding to a 2-dimensional subspace V2 is con-tained in the hyperplane from P2 corresponding to a pair (V1, f) (whereV1 is a 1-dimensional subspace in U5 and f is a non-singular symplecticform on U5/V1) if and only if either V1 ⊆ V2 or the image of V2 in U5/V1is a maximal totally isotropic subspace in (U5/V1, f).

3.7 Structure of G[1]

By the definition from Section 3.5, G[1] is a semidirect product of G[01] and agroup of order 2 generated by an element t1. Furthermore, G[01] (through itsidentification with H [01]) can be written as

G[01] = Z [1]A[1]B[1]L[1],

where L[1] ∼= L4(2) with the natural module U4, Z [1], A[1] and B[1] are elementaryabelian 2-subgroups normalized by L[1] with

Z [1] ∼=∧

2U4, A[1] ∼= B[1] ∼= U4


(as L[1]-modules). Therefore there are L[1]-isomorphisms

α : U4 → A[1], β : U4 → B[1] and ζ :∧

2U4 → Z [1].

If W2 is a complement to U4 in U⊥4 with singular vectors u, v and the only non-

singular vector u + v, then (with G[01] identified with H [01]) the mappings α, β,and ζ are defined in terms of Siegel transformations as follows:

α : w → s〈w,u〉, β : w → s〈w,v〉, and ζ : w1 ∧ w2 → s〈w1,w2〉,

where w, w1, w2 are distinct non-zero vectors in U4. Then the key commutatorrelation is

[α(w1), β(w2)] = ζ(w1 ∧ w2).

Let D[1] be the subgroup in G[01] generated by the elements α(w)β(w) takenfor all w ∈ U#

4 . Then D[1] is elementary abelian and as a module for L[1] it isan indecomposable extension of

∧ 2 U4 by U4, isomorphic to a quotient of theGF (2)-permutation module of L[1] acting on the set of non-zero vectors in U#

4(compare with the last paragraph of Section 3.4).

The element t0 which extends G[01] = H [01] to H [1] can be taken to be theorthogonal transvection associated with u + v. Then t0 centralizes Z [1]L[1] andpermutes α(w) and β(w) for every w ∈ U#

4 , so that

CG[01](t0) = D[1]L[1] ∼= 26+4 : L4(2).

The element t1 which extends G[01] to G[1] can still be taken to be of order 2and to induce the automorphism which is the product of the one induced byt0 and the deck automorphism σγ as in (3.4.2 (vi)). This automorphism stillnormalizes D[1]L[1] centralizing D[1] but no longer normalizes any one of theL4(2)-complements. Put Q[m] = 〈D[1], t1〉.

Lemma 3.7.1 The subgroup

Q[m] := 〈t1, α(w)β(w) | w ∈ U4〉

is elementary abelian of order 211. Moreover, it is the only subgroup of order 211

normal in G[1].

Proof Since t1 is an involution commuting with D[1], the subgroup Q[m] iselementary abelian. Let F [1] be a normal subgroup of order 211 in G[1]. ThenF [1] ∩ G[01] is a normal subgroup of order 2a in G[01] for a ∈ 10, 11. By (3.4.2(iii)), a = 10 and F [1]∩G[01] is one of the following: Z [1]A[1], Z [1]B[1], or D[1]. Bythe order consideration F [1]G[01] = G[1] and hence F [1] ∩ G[01] must be normalin G[1]. Therefore F [1] = CG[1](D[1]) = Q[m] and the result follows.

By the above lemma we can treat G[1] as a semidirect product of Q[m] ∼= 211

and A[1]L[1] ∼= 24 : L4(2).

A shade of a Mathieu group 47

3.8 A shade of a Mathieu group

Already at this stage we are in a position to relate G[1] and the semidirectproduct C11 : M24 where M24 is the Mathieu group of degree 24 and C11 is theirreducible Todd module (cf. Section 11).

Lemma 3.8.1 Let X be the semidirect product of the Todd module C12 and thestabilizer Mb of an octad in M24 (the semidirect product is with respect to thenatural action of M24 restricted to Mb). Then

(i) the commutator subgroup X ′ of X is isomorphic to H [01] = G[01];(ii) X ∼= Aut G[01];(iii) the subgroup C11 : Mb of X is isomorphic to G[1].

Proof Let Qb = O2(Mb). By (11.3.1) X ′ is the semidirect product E10 : Mb,where E10 = [C11, Qb] is the only hyperplane in C11 stabilized by Mb. If B isthe octad stabilized by Mb then E10 consists of the 2-element subsets of P24and the sextets which intersect B evenly (for a sextet this means that everytetrad intersects B in an even number of elements). By (11.3.1) the only properMb-submodule in E10 is

CE10(Qb) ∼=

∧2Qb

and the corresponding quotient is isomorphic to Qb.We claim that E10, as a module for Kb

∼= L4(2), is an indecomposable10-dimensional quotient of the permutation module on the non-identity elementsof Qb (compare the last paragraph from Section 3.4). Let p ∈ P24\B be theelement stabilized by Kb, put

S = p, r | r ∈ P24\(B ∪ p)and define a mapping ω : Qb → S by ω : q → p, pq. Then clearly ω is abijection which commutes with the action of Kb. Now it only remains to showthat S (considered as a subset of E10) generates the whole of E10. But since every2-element subset of P24\B is either in S already or is the symmetric differenceof a pair of subsets from S, this is quite clear. Now take a look at

G[01] = Z [1]A[1]B[1]L[1].

Let λ be an isomorphism of L[1] onto Kb which induces an isomorphism µ of U4onto Qb. Then we can uniquely define isomorphisms of A[1] onto Qb and of D[1]

onto E10 (the latter maps α(w)β(w) onto p, pµ(w) for every w ∈ U4) commutingwith λ. It is straightforward to check using the uniqueness property of the keycommutator relations in (3.4.1) that in this way we get an isomorphism of G[01]

onto E10 : Mb establishing (i). By (3.4.2 (vi)) Out (G[01]) has order 4. Since[X : X ′] = 4 and the centralizer of X ′ in X is trivial, (i) implies (ii).

By (3.4.2 (vi)) the outer automorphism group of G[01] is elementary abelianof order 4 and its non-identity elements are the images of the automorphismsτ0, τ1 and τ0τ1 = σγ . Furthermore, τ0 commutes with L[1], τ0τ1 commutes


with A[1] while τ1 commutes with none of those two subgroups. On the otherhand, let r ∈ B. Then p, p, r, and r (considered as vectors in C12) gen-erate a 2-dimensional subspace which complements E10. The automorphism ofX ′ induced by p commutes with Kp while the automorphism induced by rcommutes with Qb. Since p, r ∈ C11, (iii) follows.

Lemma 3.8.2 The group A[1]L[1] ∼= 24 : L4(2) acting on the non-identityelements of the subgroup Q[m] ∼= 211 as in (3.7.1) has exactly six orbits withlengths 28, 35, 120, 840, 128, and 896 with stabilizers isomorphic to 24 : S4(2),24 : 24 : (Sym3 ×Sym3), 24 : L3(2), Sym4 ×2, Alt7, (Sym3 ×Sym5). The formertwo orbits are in Z [1] while the former four are in D[1].

Proof Because of the isomorphism established in (3.8.1) in order to determinethe orbits in question it is sufficient to calculate the orbits of Mb on the 2-elementsubsets of P24 and on the sextets. An orbit on 2-element subsets is determinedby the size of the intersections with B. These sizes are 2, 1, or 0 while thecorresponding orbit lengths are 28, 128, and 120. The orbits on the sextets canbe read from the diagram Db(M24) in Section 11.5, their lengths are 35, 840,and 896. For the structure of the stabilizers we refer to Lemma 3.7.2 in Ivanov(1999).

It is worth noticing that the orbit lengths of the actions of A[1]L[1] on D[1]

and Z [1]B[1] are the same, although the actions are not isomorphic.

Exercises

1. Prove (3.1.10) and (3.6.3).2. Calculate directly the orbit lengths of A[1]L[1] on Q[m].3. Define the Cayley graph Ξ on Q[m] with respect to the union of the A[1]L[1]-

orbits of lengths 28, 120, and 128 (so that Ξ is regular of valency 276). Provethat the automorphism group of Ξ is the semidirect product C11 : M24.

4

PENTAD GROUP 23+12 · (L3(2) × Sym5)

The geometric subgroup at level 2 of the modified amalgam G is known as thepentad group and it has a rather complicated structure. In a very beautifulway the structure can be described in terms of the non-singular 4-dimensionalsymplectic space over GF (2) and the associated orthogonal spaces. This makesthe pentad group an example of a class of so-called tri-extraspecial groups studiedby S. V. Shpectorov and the author in Ivanov and Shpectorov (submitted).

4.1 Geometric subgroups and subgraphs

Let G = G[0], G[1] be the amalgam defined in (3.5.1). We believe it is appro-priate to recall here the abstract definition of G (we continue to assume thatG[0] = H [0] and G[01] = H [01]).

Let U5 be a 5-dimensional GF (2)-vector space, let L[0] ∼= L5(2) be the generallinear group of U5, and let Q[0] =

∧2U5 be the exterior square of U5. Then G[0]

is the semidirect product of Q[0] and L[0] with respect to the natural action. Weassume that G[0] acts on U5 so that Q[0] is the kernel and G[0]/Q[0] ∼= L5(2) actsin the natural way. Let U4 be a hyperplane in U5. Then G[01] = G[0] ∩ G[1] isthe stabilizer of U4 in G[0], so that G[01] is the semidirect product of Q[0] andL[0](U4) ∼= 24 : L4(2). The structure of G[01] can be described as follows. Let

d :[U5

2

]→ Q[0]

be the injection which commutes with the action of L[0] (cf. (1.3.4 (i))). Let Z [1]

be the subgroup in Q[0] generated by the images under d of the 2-dimensionalsubspaces contained in U4. Then Z [1] is normal in G[01]. Furthermore, restrictingd to

[U42

]and applying (1.3.4) we obtain an isomorphism

ζ :∧

2U4 → Z [1],

which commutes with the action of G[01]. Let u ∈ U5\U4, so that

L[1] := L[0](U4, u) ∼= L4(2).

For w ∈ U4 let α(w) = d(〈w, u〉), let β(w) be the transvection of U5 with centrew and axis U4 (this transvection is an element of L[0]). Put

A[1] = 〈α(w) | w ∈ U4〉, B[1] = 〈β(w) | w ∈ U4〉.

Then A[1] is a complement to Z [1] in Q[0], B[1] = O2(L[0](U4)), the above definedmappings α : U4 → A[1] and β : U4 → B[1] are isomorphisms of L[1]-modules

49

50 Pentad group 23+12 · (L3(2) × Sym5)

and the key commutator relation

[α(w1), β(w2)] = ζ(w1 ∧ w2)

(which holds for all w1, w2 ∈ U4) describes Q[1] := Z [1]A[1]B[1] = O2(G[01]) up toisomorphism. Then G[1] is a semidirect product of G[01] and a group of order 2with generator t1, such that t1 (acting by conjugation) permutes α(w) and β(w)for every w ∈ U4 and t1 induces a non-trivial deck automorphism of Z [1]L[1] ∼=26 : L4(2). This means that t1 centralizes both Z [1] and Z [1]L[1]/Z [1] ∼= L4(2).

The subgroup P := 〈Z [1]L[1], t1〉 is isomorphic to the semidirect productwith respect to the natural action of P8/Pc

8∼= 27 and L[1], where P8 is the

GF (2)-permutation module of the natural action of L[1] ∼= Alt8 of degree 8. Thegroup G[1] can be described as a partial semidirect product of Q[1] and P (forthe definition of the partial semidirect product see p. 28 in Gorenstein (1968)).By (3.8.1) and (3.8.2) G[1] can also be described as a semidirect product ofQ[m] = 〈t1, α(w)β(w) | w ∈ U4〉 ∼= 211 and A[1]L[1] ∼= 24 : L4(2).

Let

ϕ : G → G

be a faithful generating completion of G (so far we have not got any partic-ular completion to think about, but we can always think about the universalcompletion).

As in Section 2.3 we associate with the triple (G, ϕ, G) a graph

Γ = Λ(G, ϕ, G)

whose vertices and edges are the cosets of ϕ(G[0]) and ϕ(G[1]) in G and the incid-ence is via non-emptiness of intersections. Notice that Γ is the incidence graph ofthe cosets geometry F(G, G). The vertex and the edge of Γ corresponding to thecosets of G[0] and G[1] containing the identity will be denoted by x and x, y,respectively. Hopefully it will always be clear from the context which graph isunder consideration.


(i) Γ is an undirected connected graph of valency 31 without multiple edges;(ii) G acts faithfully on Γ and this action is transitive on the set of incident

vertex–edge pairs;(iii) G(x) = G[0], Gx, y = G[1];(iv) there is a unique projective space structure Πx on Γ(x) preserved by

G(x) with points indexed by the hyperplanes in U5;(v) G(x)Γ(x) ∼= L5(2) is the full automorphism group of Πx;(vi) G1(x) = Q[0];(vii) G1(y) = Z [1]B[1] induces on Γ(x) an elementary abelian group of order

24 whose non-identity elements are the transvections of Πx with centre y;(viii) Gx, y stabilizes a unique bijection between Lx(y) and Ly(x) (where

Lu(v) is the set of lines in Πu passing through v ∈ Γ(u)).

Kernels and actions 51

Proof Since the properties of H recorded in the third column of Table 1 inSection 2.3 are inherited by the amalgam G, all the properties in the secondcolumn hold with (G, Γ) in place of (H, Ω) which gives (i) to (vi). The elementt1 maps x onto y and conjugates G1(x) = Z [1]A[1] onto G1(y) = Z [1]B[1]. Onthe other hand, B[1] is of order 24 trivially intersecting Q[0], which gives (vii).Since t1 commutes with G(x, y)/(G1(x)G1(y)) ∼= L4(2), (viii) follows.

We can define geometric subgroups in G in the following way. For 2 ≤ i ≤ 4set G[0i] = H [0i], G[01i] = H [01i], G[1i] = 〈G[01i], t1〉 = NG[1](G[01i]) and put

G[i] = 〈ϕ(G[0i]), ϕ(G[1i])〉.

The important property of t1 we use here is that it centralizes G[01]/O2(G[01]).

It is clear that there is a maximal flag

0 < U1 < U2 < U3 < U4 < U5

in U5 such that the geometric subgroup H [i] in H ∼= O+10(2) is the stabilizer of

U5−i in H. Since G[0] still acts on U5 we can define G[0i] and G[01i] to be stabilizersof U5−i in G[0] and G[01], respectively (here 2 ≤ i ≤ 5), while the rest of thedefinition stays as it is.

Having the geometric subgroups we can define the geometric subgraphs Γ[i]

for 2 ≤ i ≤ 4 as follows. The vertices and edges of Γ[i] are the images under G[i]

of the vertex x and the edge x, y, respectively. By the definition (x, x, y) isan incident pair and every image of an incident pair is incident as well.

4.2 Kernels and actions

We continue to use the notation introduced in the previous section. Put G[i] =G[0i], G[1i], denote by ϕ[i] the restriction of ϕ to G[i] and consider the cosetsgraph

Λ[i] = Λ(G[i], ϕ[i], G[i]).

Then Λ[i] is of valency 2i − 1 = [G[0i] : G[01i]] and the natural action of G[i] onΛ[i] is locally projective of type (i, 2). We can further define a mapping

µ[i] : Λ[i] → Γ

via gG[0i] → gG[0]. The mapping µ[i] is an injection if and only if G[i] is a propersubgroup in G (cf. Lemma 9.6.4 in Ivanov (1999)) in which case the image of µi

is the geometric subgraph Γ[i].Let N [i] denote the largest subgroup from G[01i] which is normal in both G[0i]

and G[1i] for 2 ≤ i ≤ 4.


(i) the subgroup N [i] is independent of the choice of the completionϕ : G → G;


(ii) N [i] is the kernel of the action of G[i] on Λ[i];(iii) if G[i] is proper in G, then N [i] is the vertexwise stabilizer of Γ[i] in G[i];(iv) whenever N [i] is non-identity, G[i] is proper in G;(v) N [4] ≤ N [3] ≤ N [2].

Proof Since N [i] is defined solely in terms of G[i], (i) is immediate. Since N [i]

is the largest subgroup in G[01i] which is normal in both G[0i] and G[1i], (ii)follows and implies (iii). Since G acts faithfully on Γ, it does not contain normalsubgroups inside G[0]. Therefore N [i], in a non-identity group, cannot possiblybe normal in G and (iv) follows. Suppose that G is the universal completion ofG. Then N [i] is the vertexwise stabilizer in G of the geometric subgraph Γ[i].Since

Γ[2] ⊂ Γ[3] ⊂ Γ[4],

we obtain (v).

Lemma 4.2.2 Let G[i] be the image of G[i] in the outer automorphism group ofN [i] for 2 ≤ i ≤ 4. Then

(i) G[i] = G[i]/(N [i]CG[i](N [i]);(ii) G[i] is generated by the images in Out N [i] of G[0i] and G[1i] (equivalently

of G[0i] and t1);(iii) G[i] is independent of the choice of the completion ϕ : G → G;(iv) |G[i]| ≥ |N [i]| · |G[i]|.

Proof Statement (i) holds by the definition, while (ii) is clear since G[i] =〈G[0i], G[1i]〉. Since, in (ii), the group G[i] is defined in terms of G, (iii) follows.Statement (iv) also follows from (i) and the order formula for products.

Recall that if the completion ϕ : G → G is universal then G is the free productof G[0] and G[1] amalgamated over the common subgroup G[01]. In this case G[i]

is also a free amalgamated product and it is proper in G. Finally each of Γ, Γ[i],and Λ[i] is a tree of a suitable valency.

We will explore the relationship between the N [i] and the subgroups K [i]

associated with the geometric subgroups in H ∼= O+10(2) as in (2.1.3 (vi)). The fol-

lowing data can be read from (2.1.2) and (2.1.3 (vi)). Because of our identificationof G[0] and H [0] we consider the K [i]’s as subgroups in G[0].

Lemma 4.2.3 For 2 ≤ i ≤ 4, let K [i] be the kernel of the action of the geometricsubgroup H [i] in H ∼= O+

10(2) on the geometric subgraph Ω[i] in the dual polargraph Ω = D+(10, 2). Then

(i) K [2] is a semidirect product of a special group Q[2] (of order 215 withcentre Z [2] of order 23) and L[2] ∼= L3(2), every chief factor of K [2] insideQ[2] is either the natural or the dual natural module of K [2]/Q[2] ∼= L3(2);

Kernels and actions 53

(ii) K [3] is a semidirect product of an extraspecial group Q[3] ∼= 21+12+ and a

group L[3] ∼= L2(2) ∼= Sym3, a Sylow 3-subgroup of L[3] acts fixed-pointfreely on Q[3]/Z [3] ∼= 212 where Z [3] is the centre of Q[3];

(iii) K [4] = Q[4] is elementary abelian of order 28.

In order to get a lower bound for the order of the N [i] we present a restrictionon the possible composition factors of G[0i]/N [i] which comes from a well-knownproperty of graphs with a locally projective action.

Lemma 4.2.4 Let Ψ be a connected graph of valency 2k − 1 for k ≥ 2 and F bea group of automorphisms of Ψ such that

(i) F (a) is finite for every vertex a of Ψ;(ii) F acts transitively on the set of incident vertex–edge pairs in Ψ;(iii) for every vertex a in Ψ the group F (a) induces on the set Ψ(a) of

neighbours of a in Ψ the natural doubly transitive action of Lk(2);(iv) if a, b is an edge of Ψ then there is a bijection ψab of La(b) onto Lb(a)

commuting with the action of the stabilizer of a, b in F (where La(b)is the set of lines passing through b in the unique projective space Πa onΨ(a) preserved by F (a) and similarly for Lb(a)).

Then F1(a) (which is the vertexwise stabilizer of Ψ(a) in F (a)) is a 2-group, sothat F (a)/O2(F (a)) ∼= Lk(2).

Proof We claim that F1(a) induces a 2-group on Ψ(b) for every b ∈ Ψ(a). Infact F1(a) fixes every line La(b) pointwise and by (iv) it stabilizes every linein l ∈ Lb(a) as a set. Since l = a, c, d for some c, d ∈ Ψ(b)\a and a isfixed by F1(a), the latter can only permute the vertices c and d thus inducingon l an action of order at most 2. Since every vertex from Ψ(b) is on a linefrom Lb(a), the claim follows. Thus O2(F1(a)) acts trivially on Ψ(b) and henceO2(F1(a)) ≤ F1(b). Now it is easy to apply induction and to use the connectivityof Ψ to conclude that O2(F1(a)) fixes all the vertices of Ψ. This gives the resultsince by definition the action of F on Ψ is faithful.

Notice that the existence and uniqueness of the projective space structure Πa

in (4.2.4 (iv)) can be established as in the proof of (2.2.1).


(i) N [2] contains K [2];(ii) N [3] contains the commutator subgroup of K [3], which is of index 2 in K [3].

Proof By (4.2.4) we know that the only chief factor of G[0i]/N [i] of order greaterthan 2 is isomorphic to Li(2) if i = 3 or 4 and to the cyclic group of order 3 if i =2. On the other hand it is immediate from the structure of the maximal parabolic


subgroups in G[0]/O2(G[0]) ∼= L5(2) that G[0i]/O2(G[0i]) is isomorphic to

Sym3 × L3(2), L3(2) × Sym3, and L4(2)

for i = 2, 3, and 4, respectively. Therefore L[2] ∼= L3(2) is contained in N [2]. By(4.2.3 (i)) [L[2], G[02]] = K [2] which gives (i). Similarly, if T is a Sylow 3-subgroupof L[3] ∼= Sym3 then T is in N [3] while by (4.2.3 (ii)) [T, G[03]] = Q[3]T . Bynoticing that Q[3]T is the only index 2 subgroup in K [3] we obtain (ii).

The above discussion says nothing about N [4]. We will see in due course thatin fact this group is trivial.

Lemma 4.2.6 N [2] = K [2].

Proof By (4.2.5 (i)) and (4.2.1 (iv)) we know that G[2] is a proper subgroup inG and hence the geometric subgraph Γ[2] exists. Since G[02]/K [2] ∼= H [02]/K [2] ∼=Sym3 × 2, it is sufficient to show that the action induced by G[02] on Γ[2] hasorder 12 at least. Clearly G[02] induces Sym3 on Γ[2](x). By (4.1.1 (vii)) it is easyto see that G1(x) induces on Γ[2](y) an action of order 2. Hence the result.


(i) O2(G[02]/N [2]) = O2(G[02])N [2]/N [2];(ii) G[12]/N [2] ∼= D8;(iii) A[1]B[1]N [2]/N [2] and Q[m]N [2]/N [2] are the two elementary abelian

subgroups of order 4 in G[12]/N [2].

Proof A comparison of the proofs (4.2.5 (i)) and (4.2.6) gives (i). SinceG[012]/N [2] ∼= 22 and since G[12]\G[012] contains the involution t1, (ii) follows.Since D8 contains only two elementary abelian subgroups of order 4, we alsoobtain (iii).

Lemma 4.2.8 [K [3] : N [3]] = 2.

Proof We follow the notation in the proof of (4.2.5). So that T is a Sylow3-subgroup of L[3] ∼= Sym3 which is also a Sylow 3-subgroup of O2,3(K [3]). By(4.2.3) Q[3]T is normal in both G[03] and G[13] and we have to show that K [3] isnot normal in at least one of these groups. It is clear that K [3] is normal in bothG[03] = H [03] and in H [13]. In addition

H [13] = 〈H [013], t0〉, G[13] = 〈H [013], t1〉,

where the automorphism of G[01] = H [01] induced by t1 is that induced byt0 multiplied by the deck automorphism σγ (compare Section 3.5). Therefore inorder to show that K [3] is not normal in G[13], it is sufficient to take an involutions ∈ L[3] and show that [s, σγ ] ∈ Q[3]T .

By the definition L[3] ∼= L2(2) ∼= Sym3 is the vectorwise stabilizer in G[03] =H [03] of W6 and also stabilizes T2. Here as in Section 2.1 W6 is a complementto U2 in U⊥

2 and T2 is a maximal totally singular subspace in W⊥6

∼= V +4 disjoint

Inspecting N [2] 55

from U2. By our notation convention in Section 2.1, U2 < U4 and U4 ∩ W6is a complement to U2 in U4. Therefore L[3] < L[1] and [L[3], σγ ] ≤ Z [1]L[1]

and we only have to deal with the restriction of σγ to Z [1]L[1] which is a deckautomorphism d as in Section 3.3. Now it only remains to relate our currentnotation with that in the paragraph after the proof of (3.3.5). We have U = U4,W = W (1) = U2 and W (2) = U4 ∩ W6, T (2)〈r(1)〉 = L[3] = T 〈s〉. Then by(3.3.6 (iii)) [s, d] = δ, where δ corresponds to a partition of P8 into 2- and 6-element subsets (when Z [1] is treated as the heart of the GF (2)-permutationmodule). Suppose that δ ∈ Q[3]T . Since δ is of order 2, we must have δ ∈ Q[3].Since T acts fixed-point freely on Q[3]/Z [3], while T centralizes δ, we furtherget δ ∈ Z [3]. On the other hand Z [3] is of order 2, generated by the Siegeltransformation sU2 corresponding to U2. This is a contradiction since sU2 is asingular vector while δ is a non-singular vector with respect to the quadraticform on Z [1] ∼= V +

6 preserved by L[1] ∼= Ω+6 (2). In other terms sU2 belongs to

the 35-orbit while δ belongs to the 28-orbit of L[1] on the set of non-identityelements of Z [1].

4.3 Inspecting N [2]

The ultimate goal of this chapter is to show that G[2] is isomorphic to Sym5. Forthis we first analyse the structure of N [2] and calculate its automorphism group.Since N [2] = K [2] is contained in G[0], now identified with H [0], we can carryout calculations inside H, being careful not to miss the difference between theabstract properties of N [2] and those associated with its embedding into H.

Let us recall the geometric setting from Section 2.1. Let U3 be the3-dimensional totally singular subspace in

(V, f, q) ∼= V +10

such that H [2] is the stabilizer of U3 in H ∼= O+10(2). Let W4 be a complement to

U3 in U⊥3 , so that W4 ∼= V +

4 and let T3 be a maximal totally singular subspacein W⊥

4∼= V +

6 disjoint from U3. The following lemma makes a modest refinementof (2.1.2) and (2.1.3 (vi)).

Lemma 4.3.1 The group N [2] = K [2] is the semidirect product of Q[2] andL[2], where

(i) L[2] ∼= L3(2) is generated by the Siegel transformations s〈u,t〉 taken forall the orthogonal pairs (u, t), u ∈ U#

3 , t ∈ T#3 ;

(ii) Q[2] is a special group of order 215 whose centre Z [2] is elementary abelianof order 23 generated by the Siegel transformations s〈u,v〉 taken for alldistinct pairs u, v ∈ U#

3 ;(iii) if w is a singular vector in W4, then the Siegel transformations s〈v,w〉

taken for all v ∈ U#3 generate in Q[2] an elementary abelian subgroup

E(w) of order 23 disjoint from Z [2] and normalized by L[2].


By (2.1.2 (vii)) Q[2]/Z [2], as a module for H [2]/Q[2] ∼= L3(2) × O+4 (2), is

isomorphic to the tensor product of U3 and W4. Thus, as a module for L[2]

it is isomorphic to the direct sum of four copies of U3. Therefore Q[2] containsexactly 15 = 24−1 subgroups of order 26 containing Z [2] and normal in N [2]. Fora singular vector w ∈ W4 put D(w) = Z [2]E(w), where E(w) is as in (4.3.1 (iii)).For a non-singular vector t ∈ W4 write t = a + b where a and b are singularvectors in W4 and put

D(t) = 〈s〈a,u〉s〈b,u〉 | u ∈ U#3 〉.

Lemma 4.3.2 In the above notation the following assertions hold:

(i) for every v ∈ W#4 the subgroup D(v) is elementary abelian of order 26,

contains Z [2], and is normalized by L[2];(ii) the subgroup D(v) for non-singular v is independent of the choice of a

and b.

Proof If v is singular then D(v) contains Z [2] by definition and, by (4.3.1 (iii)),it is elementary abelian normalized by L[2]. If v is non-singular the result followsfrom the commutator relations in (3.4.1). Since there are exactly 15 subgroupsof order 26 in Q[2] containing Z [2] and normal in N [2] they all are of the formD(v) for some v ∈ W#

4 . In particular (ii) holds.

By (4.3.2),

E = D(v) | v ∈ W#4

is the set of all subgroups of order 26 in Q[2] containing Z [2] and normal in N [2].The subgroups from E will be called dents. Define a binary operation on E in thefollowing way. For distinct D1, D2 ∈ E put F = 〈D1, D2〉. Then F/Z [2] is thedirect sum of D1/Z

[2] and D2/Z[2], and therefore it contains just one further

3-dimensional N [2]-submodule whose preimage is a dent from E which we denoteby D1 + D2.

Lemma 4.3.3 In the above terms D(v) + D(w) = D(v + w).

Proof The equality is immediate from the fact that Q[2]/Z [2] carries the tensorproduct structure U3 ⊗ W4.

By (4.3.3) we can recover the vector space structure of W4 by adjoining zeroto the set E of dents and defining the addition as in the paragraph before (4.3.3)together with the obvious rule for the zero element. Next we show that thesymplectic space structure can also be recovered. For D1, D2 ∈ E put

h(D1, D2) =

0 if 〈D1, D2〉 is abelian;1 otherwise.

Cohomology of L3(2) 57

Lemma 4.3.4 h(D(v), D(w)) = 0 if and only if v and w are perpendicularin W4.

Proof By (4.3.3) 〈D(v), D(w)〉 = 〈D(v), D(v + w)〉. Hence without loss wecan assume that v and w are either both singular or both non-singular. In theformer case, if v and w are perpendicular, then 〈U3, v, w〉 is totally singularand [E(w), E(v)] = 1 by (1.3.5). If v and w are singular but non-perpendicular,then v + w is non-singular and the commutator relations in (3.4.1) imply that[E(v), E(w)] = Z [2].

Suppose now that v and w are non-singular and perpendicular. Then we canfind perpendicular hyperbolic pairs (a1, a2) and (b1, b2) such that v = a1 + a2and w = b1 + b2. Since 〈U3, ai, bj〉 is totally singular for 1 ≤ i, j ≤ 2, everygenerator of D(w) (as defined before (4.3.2)) commutes with every generatorof D(w) and hence [D(v), D(w)] = 1. Finally, if v and w are non-singular andnon-perpendicular, the result is easy to deduce from the commutator relationsin (3.4.1).

By (4.3.4) the mapping v → D(v) extends to an isomorphism of thesymplectic spaces (W4, f |W4) and (E ∪ 0, h).

Define qL[2]to be a GF (2)-valued function on E such that qL[2]

(D) = 0 if D

is a semisimple module for L[2] and qL[2](D) = 1 if D is indecomposable.

Lemma 4.3.5 qL[2](D(v)) = 0 if and only if v is singular.

Proof If v is singular then D(v) is the direct sum of L[2]-submodules Z [2] andE(v) (where the latter is defined in (4.3.1 (iii)). If v is non-singular then thegenerators of D(v) (as defined before (4.3.2)) form the only L[2]-orbit of length7 on D(v)\Z [2]. The commutator relations in (3.4.1) now show that D(v) isindecomposable.

By (4.3.5) the mapping v → D(v) extends to an isomorphism

(W4, f |W4 , q|W4) ∼= (E ∪ 0, h, qL[2])

of quadratic spaces. Thus it appears that the quadratic space structure of W4 canbe recovered on E ∪ 0. But actually the recovery of qL[2]

has not been done inpurely abstract terms, but rather in terms of a particular complement L[2] to Q[2]

in N [2]. This complement is distinguished in H ∼= O+10(2) since it is a so-called

Levi complement, but abstractly it is just one of the complements. In fact wewill show that N [2] contains quite a few conjugacy classes of L3(2)-complementswhich lead to different quadratic forms on E ∪ 0.

4.4 Cohomology of L3(2)

Let L ∼= L3(2), let U3 be the natural module of L and let U∗3 be the dual

of U3, which is also isomorphic to the exterior square of U3. By (1.10.1) weknow that L3(2) ∼= L2(7) and Aut L3(2) ∼= PGL2(7) ∼= L3(2) : 〈τ〉, where τ is


a contragredient automorphism which permutes U3 and U∗3 . By (3.2.3 (ii)) and

(3.1.8) we have the following.

Lemma 4.4.1 Let W be U3 or U∗3 . Then

(i) H1(L, W ) is of order 2;(ii) the semidirect product W : L with respect to the natural action contains

exactly two classes of complements;(iii) if L(1) and L(2) are representatives of the classes of L3(2)-complements

in W : L, then L(1) ∩ L(2) ∼= F 37 (the Frobenius group of order 21);

(iv) there exists a unique indecomposable extension Wu of W by thetrivial 1-dimensional module and a unique extension W d of the trivial1-dimensional module by W (here ‘u’ is for up and ‘d’ is for down);

(v) (Uu3 )∗ ∼= (U∗

3 )d;(vi) Wu : L is isomorphic to the automorphism group of W : L.

Let P7 be the set of non-zero vectors in U3. Then L acts doubly transitivelyon P7 with stabilizer L(p) ∼= Sym4 for p ∈ P7. Let A ∼= Alt4 be the only index2 subgroup in L(p) and let P14 be the set of cosets of A in L on which L acts inthe natural way. Then there is a unique surjective map

π : P14 → P7

which commutes with the action of L. Let P7 and P14 be the GF (2)-permutationmodules of L on P7 and P14, respectively.

Lemma 4.4.2 (i) P7 = Pc7 ⊕ Pe

7 and Pe7 is an indecomposable extension of

U∗3 by U3;

(ii) Ud3 is a quotient of P14 but not a quotient of P7;

(iii) if C is the submodule in P14 formed by the functions constant on π−1(p)for every p ∈ P7, then C ∼= P14/C ∼= P7;

(iv) if Ce is the unique codimension 1 submodule in C, then Pe14/Ce is an

indecomposable extension of the trivial 1-dimensional module by Pe7 .

Proof Statement (i) is well known and easy to check (cf. proposition 3.3.5 inIvanov and Shpectorov (2002)). Notice that if P7 is identified with the pointsof the Fano plane, then U∗

3 is generated by the vectors whose support is thecomplement of a line. Let χ : Ud

3 → U3 be the natural homomorphism. We claimthat L(p) permutes transitively the two vectors in χ−1(p). In fact, otherwise L(p)would fix these vectors and Ud

3 would be a quotient of P7 which contradicts (i).Thus L(p) acts on the vectors in χ−1(p) as on the cosets of A and (ii) follows. (iii)is easy and of a rather general nature. Let α : P14 → Ud

3 be the homomorphismwhose existence follows from (ii) and let K be the kernel of α. If K would containC, Ud

3 would be a quotient of P7, which contradicts (i). On the other hand,since Ud

3 possesses the unique proper quotient which is isomorphic to U3, it iseasy to see that K contains the unique codimension 1 submodule in C and thatKPe

14 = P14. Thus (iv) follows.

Cohomology of L3(2) 59

Lemma 4.4.3 H1(L,Pe7) is of order 2.

Proof Since Pe7 is self-dual, H1(L,Pe

7) is of order at least 2 by (3.1.8). Supposethe order of H1(L,Pe

7) is greater than 2. Then there exists an indecomposableextension Y of the trivial 1-dimensional module by Pe

14/Ce. The latter module isa quotient of P14, since it contains a generating set of 14 vectors on which L actsas on the cosets of A. Since A has no subgroups of index two, this orbit lifts toa similar orbit in Y . Hence Y also must be a quotient of P14, which contradicts(4.4.2 (i), (iii)).

In order to simplify the notation in what follows we denote Pe7 by D (where

‘D’ is for dent). Let Z be the unique L-submodule in D. Then Z ∼= U∗3 and

D/Z ∼= U3. In fact D is the only indecomposable extension of U∗3 by U3 (cf. Bell

(1978)). Let Y = D : L be the semidirect product with respect to the naturalaction and let Q = ZL. Then Q ∼= 23 : L3(2) is as in (4.4.1 (ii)) and it containstwo classes of L3(2)-complements with representatives L and LZ , where LZ isthe image of L under an outer automorphism of Q which acts trivially both on Zand Q/Z. It is easy to see that L and LZ are not conjugate in Y and by (4.4.3)any complement to D in Y is conjugate either to L or to LZ . On the other hand,Y/Z ∼= 23 : L3(2) is also as in (4.4.1 (i)) and Q/Z ∼= L3(2) is a complement toD/Z in Y/Z. Let QD be the preimage in Y of an L3(2)-complement to D/Z inY/Z which is not conjugate to Q/Z in Y/Z. Then QD is an extension of Z ∼= U∗

3by L3(2) and the extension cannot split, since we have already accounted all thecomplements to D in Y . Therefore we have the following.

Lemma 4.4.4 The subgroup QD is a non-split extension 23 · L3(2) of U∗3

by L3(2).

It is well-known (Bell 1978) that the second cohomology group H2(L, U∗3 ) is

of order 2 and therefore there is a unique non-split extension as in (4.4.4).

Lemma 4.4.5 The outer automorphism group of QD ∼= 23 · L3(2) is oforder 2.

Proof First we show that QD possess an outer automorphism. Let σ be anon-trivial automorphism of Y = D : L which acts trivially both on D andon Y/D (such an automorphism exists by (3.1.6) and (4.4.3). We claim thatσ normalizes QD and induces on it an outer automorphism. Consider Y/Z ∼=23 : L3(2). Since σ is trivial on D/Z and on Y/D, it either centralizes Y/Z orpermutes two classes of L3(2)-complements in Y/D. We know that Q/Z andQD/Z are representatives of the classes of complements. Since Q splits over Zwhile QD does not split, these complements cannot be permuted by σ. Henceσ normalizes QD. On the other hand σ cannot centralize QD since it alreadycentralizes D and since Y = DQD.

Next we claim that the order of Out QD is at most 2. Let δ be an automorph-ism of QD and let F ∼= F 3

7 be the normalizer of a Sylow 7-subgroup in QD. By a


Frattini argument (working modulo inner automorphisms) we can assume thatδ normalizes F . By 3-subgroup lemma in this case δ centralizes F . Then δ nor-malizes a Sylow 2-subgroup S in the normalizer in QD of a Sylow 3-subgroupin F . It is easy to see that S contains three involutions, say z, x, and y, wherez ∈ Z. Since QD does not split

〈F, x〉 = 〈F, y〉 = QD

and the automorphism δ is uniquely determined by its action on x, y. If theaction is trivial, it is the identity automorphism. If δ permutes x and y then itis an automorphism from the previous paragraph.

Let S3 be a Sylow 3-subgroup of L (which is also a Sylow 3-subgroup ofY ) and let I = I(Y ) be a Sylow 2-subgroup in NY (S3). Since NL(S3) ∼= Sym3and the centralizer in U3 of S3 and NU3(S3) coincide and have dimension 1,we conclude that I is elementary abelian of order 23. By a Frattini argument wemay assume that the non-identity automorphism σ of Y which centralizes bothD and Y/D also normalizes I. Then it is easy to see that σ acts on I as thetransvection whose axis is I ∩ D and whose centre is the non-identity elementz ∈ Z ∩ I.

4.5 Trident group

In this section U is a 4-dimensional GF (2)-space, M ∼= L4(2) is the linear groupof U and W is the exterior square of U (which is also the exterior square of thedual of U). If we treat M as Ω+

6 (2), then W ∼= V +6 and if we treat M as Alt8,

then W is the heart H8 of the GF (2)-permutation module of the natural actionon a set P8 of size 8.

Let u ∈ U#, M(u) ∼= 23 : L3(2) be the stabilizer of u in M and let T = W :M(u) be the semidirect product with respect to the natural action. We call Tthe trident group. In this section we analyse the structure of T and calculate itsautomorphism group.

First observe that M(u) is the semidirect product of B ∼= 23 and L ∼= L3(2),where the non-identity elements of B are the transvections with centre u and L isthe stabilizer in M(u) of a hyperplane which does not contain u. We identify thishyperplane with the dual natural module of L and denote it by U∗

3 . The subgroupM(u) stabilizes in W a unique proper submodule Z which is 3-dimensional,generated by the elements u ∧ v taken for all v ∈ U3\u; this submodule iscentralized by B, while L acts on it as on U∗

3 . Although W is indecomposableas an M(u)-module, L stabilizes a complement A to Z, which is generatedby the elements w ∧ v taken for all w, v ∈ U∗

3 . This gives us the followingdescription of T .

Lemma 4.5.1 We have T = ZABL, where L ∼= L3(2), while Z, A and B areelementary abelian of order 23. There are L-module isomorphisms α : U3 → A,β : U3 → B and ζ : U∗

3 → Z and the only non-trivial commutator relations

Trident group 61

among the elements of ZAB is

[α(v), β(w)] = ζ(〈v, w〉),

where v and w are distinct non-zero elements of U3.

Proof Almost everything follows from the paragraph before the lemma. It isclear that ZAB is non-abelian with centre Z and the unique non-trivial bilinearL-invariant mapping from A × B into Z is the one described by the abovecommutator relation.

For an L3(2)-complement K to ZAB in T let KZ denote the image of Lunder an outer deck automorphism of ZK ∼= 23 : L3(2).

Suppose that E is a subgroup of order 23 in ZAB disjoint from Z and normal-ized by K. Then KE denotes the image of K under an outer deck automorphismof EK ∼= 23 : L3(2). Put

C = 〈α(v)β(v) | v ∈ U3〉.


(i) ZAB contains exactly three subgroups of order 26 normal in T which areAZ, BZ and C;

(ii) as L-modules AZ and BZ are semisimple while C is indecomposable;(iii) T contains six classes of L3(2)-complements and

L = L, LZ , LA, LAZ , LB , LB

Z

is a set of representatives for these classes.

Proof (i) and (ii) are similar to (3.4.2 (v)). Since ZAB/Z is the direct productof two copies of U3, T/Z contains exactly four classes of complements withrepresentatives

LZ/Z, LAZ/Z, LBZ/Z, (LZ)C/Z.

Here (LZ)C can be defined as follows. The group LC/Z ∼= 23 : L3(2) containsLZ/Z ∼= L3(2) and possesses a non-trivial deck automorphism d. Then (LZ)C isthe preimage in T of the image of LZ/Z under d. By (ii) and (4.4.4) the preimageof (LZ)C/Z is the non-split extensions 23 · L3(2), while the preimages of theformer three complements split over Z. Since ZL has two classes of complementswith representatives L and LZ (iii) follows.

A normal subgroup of order 26 in T will be called a dent. By (4.5.2)

ET = AZ, BZ, C

is the set of dents. The name trident that we gave to T is explained by the factthat it contains exactly three dents. A subgroup Q in T will be called a quasi-complement if Q ∩ ZAB = Z and QAB = T , so that Q is an extension of Z by


Table 2. Dents under variousquasi-complements

LZ LAZ LBZ (LZ)C

AZ 0 0 1 1BZ 0 1 0 1C 1 0 0 1

L3(2) (which might or might not split). By the proof of (4.5.2)

Q = LZ, LAZ, LBZ, (LZ)C

is the set of quasi-complements. The following table shows which dents aresemisimple ‘0’ and which are indecomposable ‘1’ under various quasi-complements.

We are going to introduce some automorphisms of T . Let λ be the automorph-ism which centralizes ZAB, and maps L onto LZ (its existence is guaranteedby (3.1.9)). Thus λ is a deck automorphism of T , when T is regarded as a semi-direct product of ZAB and L. Let µ be the automorphism of T which centralizesZ and L and which transposes α(v) and β(v) for every v ∈ U3. It is immedi-ate from (4.5.2) that µ is indeed an automorphism of T . Finally, let ν be therestriction to T of an outer deck automorphism of W : M ∼= 26 : L4(2). We willassume without loss that the complements in L contain a common normalizer ofa Sylow 7-subgroup and that this normalizer is centralized by the automorphismsλ, µ, and ν.

Lemma 4.5.3 The automorphisms λ, µ and ν induce on L the followingpermutations:

λ : (L, LZ)(LA, LAZ)(LB , LB

Z );

µ : (L)(LZ)(LA, LB)(LAZ , LB

Z );

ν : (L, LA)(LZ , LAZ)(LB)(LB

Z ).

In particular Aut T permutes transitively the classes of L3(2)-complements.

Proof The actions of λ and µ are immediate, while the action of ν follows from(3.3.5) and Table 2.


(i) the automorphism group of T permutes transitively the classes of L3(2)-complements;

(ii) µ is the only non-identity automorphism of T which centralizes L;(iii) Out T ∼= Sym3 × 2;

Trident group 63

(iv) Out T induces Sym3 on the set of quasi-complements, the kernel of theaction is generated by the image of λ.

Proof (i) follows directly from (4.5.3) while (ii) is easy to deduce from (4.5.2)and (4.5.3). Considering the actions of λ, µ, and ν on L we obtain (iii) and (iv).

Notice that if we treat ET as the set of non-zero vectors of a non-singular sym-plectic 2-dimensional GF (2)-space (ET ∪0, f) and consider columns of Table 2as GF (2)-valued functions, then (assuming that zero always maps to zero) weobtain all the orthogonal forms associated with f . This can be formulated asfollows.

Lemma 4.5.5 For a quasi-complement Q define pQ : (ET ∪ 0) → GF (2) asfollows: pQ(0) = 0 and for D ∈ E we have pQ(D) = 0 if and only if Q actssemisimply on D. Then

(i) (ET ∪ 0, f, pQ) is an orthogonal space for every Q ∈ Q;(ii) Q → (ET ∪ 0, f, pQ) is a bijection between Q and the set of orthogonal

spaces associated with (V, f);(iii) pQ is of plus type if Q splits over Z and of minus type if Q does not split.

We conclude this section with the following lemma which describes how toswitch a quasi-complement over a dent.

Lemma 4.5.6 Let Q ∈ Q and D ∈ ET . Then QD contains exactly two classesof quasi-complements with representatives Q and QD. Furthermore, in termsof (4.5.5) we have pQ(D) = pQD

(D) while pQ(D1) = pQD

(D1) for every D1 ∈ET \ D.

Proof Since QD/Z ∼= 23 : L3(2) and Q/Z is an L3(2)-complement in this group,by (4.4.1) we have two classes of quasi-complements in QD. The second claim isimmediate from Table 2.

Let S3 be a Sylow 3-subgroup of L (which is also a Sylow 3-subgroup ofT ) and let I = I(T ) be a Sylow 2-subgroup of NT (S3). Let z, a, b and l bethe non-identity elements in the intersection of I with Z, A, B and L, respect-ively. These four elements are pairwise commuting involutions which generate I.Since they are also independent, I ∼= 24. Without loss of generality we canassume that the automorphisms λ, µ and ν in (4.5.3) normalize I. It is obviousfor instance that λ induces the transvection with axis 〈z, a, b〉 and centre z. Theaction of µ to be used later is recorded in the following lemma which is immediatefrom Table 2 and the action of µ on the classes of L3(2)-complements.

Lemma 4.5.7 The automorphism µ (which is the restriction to T of a deck auto-morphism of 26 : L4(2)) acts on I = I(T ) as the transvection with axis 〈z, a, lb〉and centre a. In particular µ centralizes the non-split quasi-complement (LZ)C .


4.6 Automorphism group of N [2]

In this section we apply the isomorphism

(W4, f |W4 , q|W4) ∼= (E ∪ 0, h, qL[2]),

established in Section 4.3 to obtain an abstract description of N [2]. Next weapply results from Section 4.4, to classify all the L3(2)-complements and allthe quasi-complements in N [2]. Then, using the properties of the trident groupfrom Section 4.5 we observe that all the L3(2)-complements lead to equivalentabstract descriptions of N [2]. This will allow us to conclude that Aut N [2] actstransitively on the classes of L3(2)-complements.

Thus N [2] is a semidirect product of L[2] and Q[2], where L[2] ∼=L3(2) and Q[2] is special of order 215 with centre Z [2] of order 23. Let U3and U∗

3 be the natural and dual natural modules of L[2]. Then there is anL-isomorphism

ζ : U∗3 → Z [2].

The group Q[2] contains a set E of 15 dents which are the elementary abeliansubgroups of order 26 containing Z [2] and normal in N [2]. The dents gen-erate the whole Q[2] while every dent is normalized by L[2]. Furthermore adent D is semisimple (as an L[2]-module) if it is singular with respect to theform qL[2]

, and it is an indecomposable extension of U∗3 by U3 if D is non-

singular. We have seen in Section 4.4 that the indecomposable extension isthe even half Pe

7 of the permutation module of L[2] on the set P7 of non-zerovectors of U3.

Lemma 4.6.1 Let D be a dent from E. Then there is a unique L[2]-invariantinjection

δD : U#3 → D.

The image of δD generates a complement to Z [2] if D is singular with respect toqL[2]

and the whole of D if it is non-singular. If u, v, w is the set of non-zerovectors of a 2-subspace V in U3, then

δD(u)δD(v)δD(w) =

1 if qL[2]

(D) = 0;ζ(V ) otherwise.

Proof The uniqueness of δD can be seen as follows. If there were two distinctsuch injections δ1 and δ2, then

u → δ1(u)δ2(u)

would be a non-trivial L-invariant mapping of U3 into U∗3 , which is impossible.

The existence of δ follows from the paragraph before the lemma. The 2-subspacecondition is quite clear.

The vector space structure on E ∪ 0 is reflected in the following.

Automorphism group of N [2] 65

Lemma 4.6.2 Let D1 and D2 be distinct dents. Then the mapping of U#3 into

D1 + D2 as in (4.6.1) is the following:

u → δD1(u)δD2(u)

for u ∈ U#3 .

Finally the symplectic form h on E ∪ 0 is reflected in the following (recallthat D1 and D2 commute whenever h(D1, D2) = 0.)

Lemma 4.6.3 Suppose that h(D1, D2) = 1 and u, v ∈ U#3 . Then

[δD1(u), δD2(v)] =

1 if u = v;ζ(〈u, v〉) otherwise.

Proof The subgroup L[2](u) does not stabilize non-zero vectors in U∗3 while

L[2](u) ∩ L[2](v) stabilizes a unique such vector, namely the one correspondingto 〈u, v〉.

Let us turn to the classes of L3(2)-complements in N [2]. First of all

Z [2]L[2] ∼= 23 : L3(2)

contains two classes of complements whose representatives are L[2] and the imageL

[2]Z of L[2] under an outer deck automorphism of Z [2]L[2]. Since Z [2] is the centre

of Q[2], the complements L[2] and L[2]Z have the same image in Aut Q[2].

Lemma 4.6.4 The form defined with respect to L[2]Z as in the paragraph before

(4.3.5) coincides with qL[2]. There is a deck automorphism ξ of N [2] which

centralizes Q[2] and maps L[2] onto L[2]Z .

Proof The existence of ξ follows from (3.1.9).

An L3(2)-complement in N [2] and its image under ξ act on Q[2] in exactlythe same way. Therefore it is convenient for us to deal with quasi-complementswhich are subgroups R in N [2] such that R ∩ Q[2] = Z [2] and Q[2]R = N [2]. Forinstance

Q := L[2]Z [2] = L[2]Z Z [2] = L[2]L

[2]Z

is a quasi-complement.The quasi-complements are bijective with the complements to Q[2]/Z [2] in

N [2]/Z [2]. Since the former (as a module for N [2]/Q[2] ∼= L3(2)) is isomorphic tothe direct sum of four copies of U3, it follows from (4.4.1) that there are exactly24 classes of quasi-complements.


Lemma 4.6.5 Let R be a quasi-complement in N [2]. Then after conjugation ofR by a suitable element of N [2] one of the following holds:

(i) R = Q = L[2]Z [2];(ii) there is a unique dent D ∈ E such that Q and R are representatives of

two classes of quasi-complements in DQ.

Proof Since both Q/Z [2] and R/Z [2] are L3(2)-complements in N [2]/Z [2] andR is chosen up to conjugation in N [2], we can assume that there is an invariantsubmodule D in Q[2]/Z [2] such that

(Q/Z [2])D = (R/Z [2])D.

Since D = D/Z [2] for a dent D, the result follows.

The above lemma implies that once we have a quasi-complement R, we canobtain the representatives of the remaining 15 classes of quasi-complements byswitching R over the 15 dents in E . Therefore

Q = Q ∪ QD | D ∈ E

is a set of representatives of the classes of quasi-complements in N [2]. Some ofthese quasi-complements split, some do not. We know that Q splits. In order tospecify the choice of representatives we will assume that the quasi-complementsin Q share the normalizer of a Sylow 7-subgroup (which is the Frobenius groupof order 21).

Lemma 4.6.6 The quasi-complement QD splits if and only if D is singular withrespect to qL[2]

.

Proof If D is singular then there is an L[2]-invariant complement E to Z [2] in D.If LE is the image of L[2] under an outer deck automorphism of EL[2] ∼= 23 : L3(2)then QD = LEZ [2]. If D is non-singular, then it is indecomposable with respectto L[2] and the result follows from (4.4.4).

For a quasi-complement R ∈ Q define, qR to be the GF (2)-valued functionon E ∪ 0 such that qR(0) = 0 and

qR(D) =

0 if R acts on D semisimply;1 if R acts on D indecomposably.

Comparing this definition with the paragraph before (4.3.5), we observe thatqL[2]

= qQ is a quadratic form of plus type on E ∪ 0 associated with thesymplectic form h (recall that Q = L[2]Z [2]). The following crucial lemma showshow the function qR changes when we switch the quasi-complement R over adent (recall that the quasi-complements in Q share a Sylow 7-normalizer).

Automorphism group of N [2] 67

Lemma 4.6.7 Let R ∈ Q be a quasi-complement, let D ∈ E be a dent andlet R1 = RD ∈ Q be R switched over D. Then for a dent D1 ∈ E we haveqR1(D1) = qR(D1) if and only if h(D, D1) = 0.

Proof Suppose that h(D, D1) = 0. Then 〈D, D1〉 is abelian which means thatthe actions of R and RD on D1 coincide. If h(D, D1) = 1 and R splits over Z [2]

then DD1R is isomorphic to the trident group from the previous section and theresult follows from (4.5.6). But in fact one can observe that whether or not Rsplits is irrelevant, since this does not effect the action of R on DD1.

In terms of Section 1.7 the above lemma can be reformulated as follows.

Corollary 4.6.8 For D ∈ E let lD be the linear function on E ∪ 0 defined bylD : D1 → h(D, D1). Then qRD

= qR + lD.

Since qQ is a quadratic form of plus type associated with h, combining (1.7.2),(4.6.6), and (4.6.8) we obtain the following.

Proposition 4.6.9 The set qR | R ∈ Q consists of all the quadratic forms onE ∪ 0 associated with the symplectic form h. Furthermore, for R = QD, wehave the following:

(i) if D is semisimple under Q then qR is of plus type and R splits over Z [2];(ii) if D is indecomposable under Q then qR is of minus type and R does not

split over Z [2].

By (1.2.4) we observe that N [2] contains 10 classes of split and 6 classesof non-split quasi-complements and, particularly there are exactly 20 classes ofL3(2)-complements.

Lemma 4.6.10 The automorphism group of N [2] permutes transitively theclasses of L3(2)-complements in N [2].

Proof Let L ∼= L3(2) be a complement and let R = Z [2]L be the correspondingquasi-complement. Let α be an isomorphism of L onto L[2] such that l−1α(l) ∈Q[2] for every l ∈ L. Let β be an isomorphism

β : (E ∪ 0, h, qR) → (E ∪ 0, h, qL[2])

of orthogonal spaces whose existence follows from (4.6.9) and (1.1.8). We claimthat the pair (α, β) can be extended to an automorphism of N [2]. In fact, using(4.6.1), (4.6.2), and (4.6.3) we can obtain a description of N [2] using L insteadof L[2]. In particular for every D ∈ E there is a unique L-invariant bijection

σD : U#3 → D.

We refine β to

γ : σD(u) → δβ(D)(u),


where u ∈ U#3 . Since qR(D) = qQ(β(D)) for every D ∈ E , the relations for

σD(u) and δβ(D)(u) coming from (4.6.2) and (4.6.3) are identical. Finally, by thechoice of α, the actions of l and α(l) on U3 are also identical. Therefore the map-ping lσD(u) → α(l)δβ(D)(u) extends to an automorphism of N [2] which maps L

onto L[2].

Proposition 4.6.11 The following assertions hold;

(i) Aut N [2] induces S4(2) ∼= Sym6 on the dent space E ∪ 0 as well as onthe set Q of quasi-complements;

(ii) the kernel of the action of Out N [2] on the set of quasi-complements isof order 2 and is generated by the image of the deck automorphism ξ asin (4.6.4);

(iii) Out N [2] ∼= Sym6 × 2.

Proof Since the symplectic space structure (E ∪ 0, h) is described inSection 4.3 in abstract terms of N [2], the action of Aut N [2] on the set ofdents is a subgroup of S4(2) ∼= Sym6. On the other hand, H [2] stabilizes aclass of L3(2)-complements and induces O+

4 (2) on the set of dents. By (4.6.9)and (4.6.10), the action of Aut N [2] on the set of quadratic forms of plus typeassociated with h is transitive. This implies (i). By (4.6.5) an automorphismα of N [2] which normalizes every quasi-complement also normalizes every dent.Adjusting α by an outer automorphism of Q if necessary we can assume that α

induces a deck automorphism of Q transposing L[2] and L[2]Z . Since the actions

of these two complements are identical on every dent D, α must commute withthis action, forcing α to centralize D and hence to centralize every dent. There-fore α is a non-trivial deck automorphism of N [2], and hence coincides with ξ,which gives (ii). Thus O = Out N [2] is an extension of a group Y of order 2(generated by the image of ξ as in (4.6.4)) by O/Y ∼= Sym6 and to establish(iii) we have to show that O splits over Y . Notice that the image of H [2] in O isisomorphic to O+

4 (2) and intersects Y trivially (since H [2] stabilizes the class ofL3(2)-complements containing L[2]). Let A be the preimage of the subgroup Alt6in O/Y . We claim that A splits over Y . If not, Y would be the unique non-splitextension SL2(9) of a group of order 2 by Alt6. But the group SL2(9) containsno involutions except the one in the centre while, on the other hand

Ω+4 (2) = O+

4 (2) ∩ Alt6

(where O+4 (2) is the image of H [2] in O) contains more than one involution.

Hence A splits over Y . Now it only remains to show that O contains involutionsoutside A. Let D1 and D2 be non-commuting dents which are semisimple underL[2] (equivalently singular with respect to qL[2]

). Then (D1, D2) is a hyperbolicpair and hence

〈D1, D2〉 ∼= V +2 .

D12, D8-amalgams in Sym6 69

The elementwise stabilizer S of the perp of 〈D1, D2〉 in the symplectic group ofthe dent space is S2(2) ∼= Sym3. Using the model of the dent space in terms ofa 6-element set, or otherwise, we observe that S contains involutions which maponto odd elements of O/Y ∼= Sym6. On the other hand T = D1D2L

[2] is thetrident group from Section 4.5. By (4.5.4 (iii)) Out T ∼= Sym3 × 2 (which is asplit extension of Y by S) supplies us with the required involutions.

By (4.6.10) and the remark before it the group N [2] contains exactly 20 classesof L3(2)-complements transitively permuted by Aut N [2] ∼= Sym6 × 2. On theother hand, H [2] stabilizes the class of such complements containing L[2]. Sincethe centralizer of N [2] in H [2] is trivial, the latter maps isomorphically onto itsnatural image in Aut N [2]. Therefore, by a consideration of group orders weobtain the following.

Lemma 4.6.12 The image of H [2] in Aut N [2] coincides with the stabilizer ofthe conjugacy class of L3(2)-complements containing L[2].

4.7 D12, D8-amalgams in Sym6

Recall that the goal of this chapter is to show that

G[2] ∼= Sym5,

where G[2] is the image of G[2] in O = Out N [2]. By (4.2.6), N [2] = K [2]. Weknow that the image H [2] of H [2] in O is O+

4 (2). On the other hand, in view of(4.2.7) both H [2] and G[2] are generated by amalgams consisting of

H [02]/N [2] = G[02]/N [2] ∼= Sym3 × 2 ∼= D12; and H [12]/N [2] ∼= G[12]/N [2] ∼= D8

intersecting in a subgroup of order 4 (recall that we have identified G[0] andH [0]). By (4.6.11 (iii)) we know that Out N [2] ∼= Sym6 × 2 and that it inducesSym6 on the dent space. In this section we are making a further step towardsthe identification of G[2] by classifying the subgroups in Sym6 generated byD12, D8-subamalgams.

We start with the following easy lemma whose proof we leave as an exercise.

Lemma 4.7.1 Up to isomorphism there exists a unique amalgam E = D[0],D[1] such that D[0] ∼= D12 ∼= Sym3×Sym2, D[1] ∼= D8, D[01] := D[0]∩D[1] ∼= 22,no non-identity subgroup of D[01] is normal in both D[0] and D[1].

Let (V, f) be a non-singular 4-dimensional symplectic space and let(V, f, qε) ∼= V ε

4 be an associated orthogonal space, where as usual ε is + or −.By (1.2.4 (iii)) the number of non-singular vectors in V +

4 is 6 while in V −4 there

are 10 such vectors.Define Σε to be the graph on the set of non-singular vectors in V ε

4 in whichtwo such vectors are adjacent if they are perpendicular (with respect to f). Thenusing the combinatorial model V (P6) or otherwise, we observe that Σ+ is the


complete bipartite graph K3,3 while Σ− is the Petersen graph (cf. Lemma Cin Preface for the pictures of these graphs). Consider the action on Σε of theorthogonal group Xε = Oε

4(2) which is the stabilizer of qε in the symplectic groupassociated with (V, f). Let X

[0]ε and X

[1]ε be the stabilizers in Xε of a vertex in

Σε and an edge containing that vertex. The following result is immediate.

Lemma 4.7.2 Independently on whether ε is + or − the amalgam X[0]ε , X

[1]ε

is isomorphic to the amalgam E from (4.7.1).

We need a description of the subamalgams in Sym6 isomorphic to theamalgam E from (4.7.1).

Lemma 4.7.3 Up to conjugation the group Sym6 contains exactly four subamal-gams E as in (4.7.1). The representatives

Dj = D[0]j , D

[1]j | 1 ≤ j ≤ 4

of these subamalgams can be chosen in such a way that

(i) 〈D1〉 ∼= 〈D4〉 ∼= O−4 (2) ∼= Sym5 and there is an outer automorphism of

Sym6 which permutes D1 and D4;(ii) 〈D2〉 = 〈D3〉 ∼= O+

4 (2) ∼= Sym3 Sym2 and there is an outer auto-morphism of Sym6 which permutes D2 and D3 (and hence normalizesthe subgroup generated by the subamalgam);

(iii) D[0]i = D

[0]j if and only if i, j = 1, 2 or i, j = 3, 4.

Proof Let D = D[0], D[1] be a subamalgam in X ∼= Sym6 and let P6 bea set of size 6 on which X induces the symmetric group. Then (taking intoaccount the outer automorphisms of X) we assume without loss of generality thatT = O3(D[0]) is generated by a 3-cycle on P6. Since N := NX(T )/T ∼= Sym3 ×Sym2, D[0]/T is a Sylow 2-subgroup in N and hence is uniquely determined upto conjugation. Furthermore NX(D[01])/D[01] ∼= 22 while CX(D[01])/D[01] ∼= 2and hence there are exactly two ways to extend D[01] to D[1], and the resultfollows.

Consider X ∼= Sym6 as an abstract group and let us follow the notation fromSection 1.8. Then P6 and R6 are two sets of size 6 on which X acts in twoinequivalent ways (permuted by an outer automorphism). Let s(P6) and s(R6)be the symplectic spaces associated with these two actions. Let Fa = X(a) fora ∈ P6 and Fb = X(b) for b ∈ R6, so that Fa

∼= Fb∼= Sym5. Let A, C be

a partition of P6 and let B, D be a partition of R6 such that |A| = |B| =|C| = |D| = 3, a ∈ A, b ∈ B. Finally let E ∼= Sym3 Sym2 be the stabilizerof both the partitions (compare the paragraph before (1.8.5)). Then in terms of(4.7.2), we can assume that 〈D1〉 = Fa, 〈D2〉 = 〈D3〉 = E and that 〈D4〉 = Fb.The subgroup D

[0]1 = D

[0]2 = E(a) is the stabilizer in E of a non-singular vector

A \ a, C ∪ a of the orthogonal space Q(P6, A, C) and also it is the

The last inch 71

stabilizer in E of a maximal totally singular subspace in the orthogonal spaceQ(R6, B, D) (recall that U5 ∩ W4 is assumed to be 2-dimensional).

Let us now relate this to N [2]. Without loss we identify the dent space D∪0with s(R6) via the isomorphism O/Y ∼= X where O = Out N [2] and Y is thekernel of the action of O on the dent space. Then H [02]/N [2] = G[02]/N [2] isthe stabilizer in H [2]/N [2] ∼= O+

4 (2) of a maximal totally singular subspace in(E ∪ 0, h, qL[2]

). Therefore by (4.7.3) we assume without loss that

H [02]/N [2], H [12]/N [2] = D2,

G[02]/N [2], G[12]/N [2] = D1.

Thus by (4.7.3) and in view of (1.8.5) we obtain the following.

Proposition 4.7.4 The group G[2] induces Sym5 on the dent space andpermutes the dents transitively.

In terms of another remarkable isomorphism

Sym5∼= PΣL2(4)

(where the latter is the extension of L2(4) by the non-identity automorphism ofGF (4)), the dent space is the natural (2-dimensional GF (4)-) module for thisgroup.

Now in order to get the precise structure of G[2], we have to decide aboutthe action of G[2] on the 20 classes of complements to Q[2] in N [2]. At this stagewe can say that the action is either Sym5 ×2 or Sym5 and in the latter case thereare still two possibilities. The action on the 20 classes is either transitive or withtwo orbits of length 10 each (notice that Sym6 × 2 contains exactly 4 conjugacyclasses of subgroups isomorphic to Sym5).

4.8 The last inch

In order to get the precise structure of G[2] consider a Sylow 2-subgroup I =I(N [2]) of the normalizer in N [2] of a Sylow 3-subgroup S3 of L[2] (of course S3is also a Sylow 3-subgroup of the whole of N [2]).

Lemma 4.8.1 I = I(N [2]) is elementary abelian of order 26 and CI(S3) is ahyperplane in I.

Proof Let (D1, D2) and (D3, D4) be perpendicular hyperbolic pairs in theorthogonal space (E∪0, h, qL[2]

). Then both T1 = D1D2L[2] and T2 = D3D4L

[2]

are trident groups and I ∩Ti = I(Ti) is elementary abelian of order 24 for i = 1, 2(compare the paragraph before (4.5.7)). Since

I(T1) ∩ I(T2) = I ∩ Z [2]L[2]

and 〈T1, T2〉 = N [2], the result follows.


Let F ∼= F 37 be normalizer of a Sylow 7-subgroup of L[2]. Then F is also

the normalizer of a Sylow 7-subgroup in N [2]. Let C be the centralizer of F inAut N [2]. The subgroup F has trivial centralizer in N [2] and

C Inn N [2] = Aut N [2],

by a Frattini argument. Therefore we conclude that C ∼= Out N [2] ∼= Sym6 × 2(compare (4.6.11)). Without loss of generality we assume that S3 ≤ F and sincethere are three Sylow 2-subgroups in NN [2](S3), we can further assume that Cnormalizes I. Our next goal is to describe I as a module for C.

As in the paragraph before (4.7.4) we consider C as the direct product ofthe symmetric group X ∼= Sym6 of R6 of size 6 and a group Y of order 2, sothat s(R6) is canonically isomorphic to the dent space and the quadratic formsof minus type associated with (E ∪ 0, h) are indexed by the elements of R6.Let R6 be the GF (2)-permutation module of X acting on R6. Then the subsetsof size 1 form a basis B1 of R6 and the subsets of size 5 form another basis B2.Let γ be a linear transformation of R6 which sends B1 onto B2 by mapping a1-element subset onto its 5-element complement in R6. Then γ is of order 2 andit normalizes the natural action of X on R6. Now consider R6 as a module forC ∼= X × Y where X acts naturally and the generator of Y acts as γ (we willdenote this generator also by γ).

Lemma 4.8.2 The action of C on I is isomorphic to the above definedaction on R6.

Proof By (4.6.9) and (4.6.11), there are exactly 6 classes of non-splitquasi-complements in N [2]. The group C acts on these classes as on R6 withY being the kernel. If R is such a quasi-complement containing S3, then I ∩R isof order 4, and contains three involutions, one of them, say z, is inside Z [2] whiletwo others, which are transposed by Y , invert S3. Recall that Y is generated bythe image of the deck automorphism ξ of N [2] and ξ acts on I as the transvectionwith centre z and axis CI(S3). Thus C has an orbit Ω on I of length 12 on whichit acts with imprimitivity classes of size 2 indexed by the elements of R6. Exactlyone of the two subgroups in C isomorphic to Sym6 acts on Ω intransitively withtwo orbits of length 6, and without loss of generality we assume that X is chosento be this subgroup. It is easy to see that each of the two X-orbits of length6 span the whole I. Therefore I, as a module for X is isomorphic to R6. Thegenerator of Y fuses the orbits of length 6 and commutes with the action of X,so its action is also uniquely determined.

Let J = H [02] = G[02], H [12], H [2], G[12]. Since CJ(N [2]) = 1 for everyJ ∈ J , each such J can and will be identified with its natural image in Aut N [2].Therefore, if we put

Jc = J ∩ C

then our goal is just to identify the subgroup in C generated by G[02]c and G

[12]c .

The last inch 73

First we locate in C the subgroup H[2]c and the subamalgam H

[02]c , H

[12]c .

By (4.6.12) H [2] is the stabilizer in Aut N [2] of the conjugacy class ofL3(2)-complements containing L[2]. Therefore H

[2]c is the centralizer of l in C,

where l = I ∩L[2]. The element l is contained in the (unique) orbit of length 20of C on I (this orbit corresponds to the classes of L3(2)-complements in N [2]).In terms of R6 this orbit corresponds to the 3-element subsets of R6. Let A bethe 3-element subset corresponding to l and set B = R6 \ A. The stabilizer inX of the partition A, B is Sym3 Sym2

∼= O+4 (2) and the generator γ of Y

permutes A and B (compare the definition of γ in the paragraph before (4.6.8)).Therefore we have the following lemma.

Lemma 4.8.3 The subgroup H[2]c consists of the elements from X stabilizing

each of the subsets A and B and the elements from X permuting A and B,multiplied by γ.

At this final stage we require some explicit description of the automorphismσγ defined in the paragraph before (3.4.2) and of the restriction of σγ to N [2].We assume without loss that σ ∈ C and as above identify G[0] with H [0] andK [2] with N [2].

Lemma 4.8.4 Let σ be the restriction to N [2] of the automorphism σγ definedbefore (3.4.2). Then

(i) N [2] is the commutator subgroup (of index 4) of the normalizer in H [01]

of Z [2];(ii) Z [1] is a dent in N [2];(iii) T := Z [1]L[1] ∩ N [2] is isomorphic to the trident group;(iv) σ centralizes a non-split quasi-complement in T ;(v) σ acts on the dent space (E ∪ 0, h) as the transvection whose centre is

Z [1] and whose axis is the perp of Z [1];(vi) σ acts on R6 as a transposition and σ ∈ X.

Proof In view of the discussion at the beginning of Section 4.3, H [012] =NH[01](U3) and H [012]/N [2] ∼= 22, since H [012] is the vertexwise stabilizer ofan edge in the action of H [2] on the geometric subgraph Ω[2] of valency 3 inΩ. Since N [2] coincides with its commutator subgroup, (i) follows. Since Z [1]

is elementary abelian of order 26 and is normal in N [2], we obtain (ii). SinceZ [1]L[1] ∼= 26 : L4(2) and L[1] ∩N [2] ∼= 23 : L3(2) is the stabilizer of a hyperplanein the natural module of L[1], (iii) follows directly from the definition of the tri-dent group at the beginning of Section 4.5. The assertion (iv) now follows from(4.5.7). Since σγ acts trivially on Q[1] and the latter contains the centralizer ofZ [1] in N [2], σ centralizes every dent perpendicular to Z [1]. Since σ must alsopreserve the symplectic form h, we obtain (v). Since the dent space is isomorphicto s(R6), symplectic transvections of the dent space act as transpositions on R6.By (v) σ centralizes a non-split quasi-complement in the trident group T . There-fore it centralizes at least two involutions from I contained in the orbit of length


12 of C on I. Therefore σ projects trivially on the direct factor Y of C = X × Yand (vi) follows.

Proposition 4.8.5 Let G[2]c be the subgroup in C generated by G

[02]c and G

[12]c .

Then

(i) G[2]c

∼= Sym5;(ii) G

[2]c acts transitively on the set of 12 involutions in I contained in non-

split quasi-complements;(iii) G[2] ∼= Sym5.

Proof We turn (4.8.3) into explicit permutations. Put R6 = 1, 2, . . . , 6,A = 1, 2, 3, B = 4, 5, 6. Since H [02] does not stabilize quadratic forms ofminus type associated with (E ∪ 0, h) ∼= s(R6), we conclude that O3(H

[02]c )

acts transitively on R6 and without loss we assume that H [02] = 〈y, d1, d2〉,where

y = (1, 2, 3)(4, 5, 6), d1 = (1, 4)(2, 5)(3, 6)γ, d2 = (1, 4)(2, 6)(3, 5)γ.

The permutation of R6 induced by y stabilizes each of A and B, while thoseinduced by d1 and d2 permute A and B. Therefore these two permutations aremultiplied by γ in accordance with (4.8.3). It is easy to deduce from (4.7.3) thatthe element x which extends H

[012]c = 〈d1, d2〉 to H

[12]c can be taken to be

x = (5, 6).

In order to obtain an element t which extends H[012]c to G

[12]c we have to multiply

x by an element s ∈ C acting on R6 as a transposition. On the other hand s isthe action of σ on I. The crucial consequence of (4.8.4 (vi)) is that s is a purepermutation of R6 (i.e. one which does non-project non-trivially on Y ), so that

t = (1, 4)(5, 6).

The action of G[2]c = 〈y, d1, d2, t〉 on R6 is generated by a D12, D8-amalgam and

by (4.7.3) the action is isomorphic to Sym5. Now we observe that the generatorsd1, d2 which induce odd permutations on R6 are multiplied by γ while the evengenerators y, t are not. Therefore G

[2]c is a diagonal subgroup in

Sym5 × Y ≤ X × Y,

which gives (i). Since G[2]c projects onto Y , (ii) follows. Finally (iii) is from the

general setting after the proof of (4.8.1).

4.9 Some properties of the pentad group

Since the centre of N [2] is trivial, N [2] maps isomorphically onto its inner auto-morphism group. Let G[2] be the image of G[2] in the automorphism group of

Some properties of the pentad group 75

N [2]. Then

G[2] ∼= G[2]/CG[2](N [2])

by the definition and

G[2] ∼= 23+12 · (L3(2) × Sym5)

by (4.8.5 (iii)). In what follows we call G[2] the pentad group and identify thesubgroups of N [2] (including N [2] itself) with their images in the pentad group.

Lemma 4.9.1 Let S7 and S3 be subgroups of order 7 and 3 from L[2], respect-ively such that S3 normalizes S7 and let S5 be a Sylow 5-subgroup in G[2]. Forp = 3, 5, 7 let Np be the normalizer of Sp in the pentad group G[2]. Then

(i) Q[2]/Z [2] ∼= 212, as a module for G[2]/Q[2] ∼= L3(2) × Sym5 is isomorphicto the tensor product of the natural module of L3(2) and the naturalmodule of PΣL2(4) ∼= Sym5;

(ii) N7 ∼= F 37 × Sym5;

(iii) N3/S3 ∼= 26 : Sym5, O2(N3/S3) is a uniserial module forN3/(S3O2(N3)) ∼= Sym5 with orbit lengths on the non-zero vectors 1,12, 15, 15, and 20;

(iv) N5 ∼= (23 · L3(2)) : F 45 is the semidirect product of a subgroup X iso-

morphic to the non-split extension of 23 by L3(2) and the Frobenius groupY of order 20, CY (X) ∼= D10 and N5 maps onto Aut X.

Proof (i) is immediate from (4.7.4). Since Z [2] is the dual natural module forL[2], we have CN [2](S7) = 1 and (ii) follows from (i) and a Frattini argument.By (4.8.1) N3 is an extension of 26 = N3 ∩ N [2] by Sym5 and by (i) the exten-sion splits. The module structure follows from (4.8.5 (ii)). By (i) and (4.8.5(ii)) CN [2](S5) is a non-split quasi-complement on which an F 4

5 -subgroup of N5induces an outer automorphism, hence (iv) follows (notice that by (4.4.5) Out Xis of order 2).

Exercises

1. Let L ∼= L3(2) and U be the natural module for L. Show that up toisomorphism there exists a unique non-split extension 23 · L3(2) of U by L.

2. Prove the isomorphism Sym5∼= PΣL2(4).

3. Performing direct calculations in the amalgam G show that N [4] = 1.

5

TOWARDS 21+12+ · 3 · AUT (M22)

In the chapter we show that N [3] is the extraspecial group Q[3] ∼= 21+12+ extended

by a group R of order 3 acting fixed-point freely on Q[3]/Z [3] (where Z [3] isthe centre of Q[3]) and that G[3] is the automorphism group Aut (M22) of thesporadic Mathieu group M22. We use Shpectorov’s geometric characterizationof M22 in terms of a Petersen type geometry. Notice that the sporadic Mathieugroup M22 appears here before we have even started considering any specificcompletions of G.

5.1 Automorphism group of N [3]

The ultimate goal of this chapter is to calculate the image of G[3] in the (outer)automorphism group of N [3]. It is natural first to ask what the automorphismgroup of N [3] is.

Lemma 5.1.1 The group N [3] is the extraspecial group Q[3] ∼= 21+12+ with centre

Z [3] extended by the subgroup R of order 3 which acts fixed-point freely onQ[3]/Z [3].

Proof Everything is immediate from (2.1.2) except possibly the type of Q[3]

as an extraspecial group. By the proof (4.2.5) Q[0] ∩ Q[3] is of order 27 andthe intersection is elementary abelian since Q[0] is already elementary abelian.By (1.6.7) Q[3] is of plus type.

Lemma 5.1.2 The automorphism group of N [3] is the semidirect product ofQ[3] = Q[3]/Z [3] ∼= 212 and ΓU6(2), where the latter group acts on Q[3] as on thenatural unitary module.

Proof Since the orders of R and Q[3] are coprime, Aut (N [3]) acts faithfully onQ[3] and by a Frattini argument Aut (N [3]) is a split extension of Q[3] (which isthe image of Q[3] in Inn (N [3])) and M := NAut (N [3])(R). Furthermore, since Q[3]

is the Frattini factor of Q[3], the action of M on Q[3] is also faithful. Since Q[3] ∼=21+12+ , by (1.6.1) we have O := Out Q[3] ∼= O+

12(2) and hence M = NO(R), whereR is identified with its image in O. Let f and q be the symplectic and associatedquadratic forms on Q[3] defined as in (1.6.1). Then O is the automorphism groupof the orthogonal space (Q[3], f, q) ∼= V +

12 . The action of R on Q[3] defines on thelatter two GF (4)-space structures preserved by CO(R) as follows. Let α be one

76

A Petersen-type amalgam in G[3] 77

of the two isomorphisms

α : GF (4)∗ → R,

where GF (4)∗ is the multiplicative group of the field of four elements. Then forv ∈ Q[3] and ω ∈ GF (4)∗ we define the scalar multiplication by ωv = vα(ω),where on the right-hand side we have the image of v under the element α(ω) ∈ Racting by conjugation. Then q becomes a non-singular Hermitian form on Q[3]

where the latter is now treated as a 6-dimensional GF (4)-space. The stabilizerof this form in GL6(4) is the general unitary group GU6(2). By the definition theunitary group preserves q and the latter in turn determines the homology classof 2-cocycles responsible for the extension of Z [3] by Q[3]. Therefore GU6(2) actson Q[3] as an automorphism group. The centre of GU6(2) is of order 3 actingfixed-point freely on Q[3] and hence this centre can be identified with R. ThusCO(R) ∼= GU6(2). Finally the non-identity automorphism of GF (4) inverts Rand also acts on Q[3] permuting the two GF (4)-space structures preserved by R.This automorphism extends CO(R) ∼= GU6(2) to ΓU6(2).

5.2 A Petersen-type amalgam in G[3]

We would like to start the section with the following remark. We know exactlythe structure of the amalgam G = G[0], G[1] and the location of N [3] inside thisamalgam. In view of (5.1.2) this enables us to write down a collection of generat-ors for G[3] as semilinear transformations of the 6-dimensional GF (4)-space Q[3].Then one way or another we should be able to identify the subgroup in ΓU6(2)generated by these transformations. In the last resort we could apply computercalculations. But we prefer to operate less explicitly and to apply Shpectorov’scharacterization (11.4.2) of the Mathieu group M22 and its automorphism groupin order to identify G[3] as a completion of the amalgam formed by the imagesin G[3] of the subgroups G[i3] taken for i = 0, 1, and 2.

Up to some extend the above remark applies to G[2] as well in the sensethat the calculations could be performed more explicitly in the automorphismgroup of N [2], but this way we would probably learn less about the structureof G[2].

As above let

ϕ : G → G

be a faithful generating completion of the amalgam G = G[0], G[1]. The struc-ture of G[3] depends only on the amalgam, so there is no harm of taking G tobe the universal completion. For i = 0, 1, and 2 let G[i3] be the image of therestriction to G[i3] of the natural homomorphism of G[3] onto G[3]. Let

A[3] = G[i3] | 0 ≤ i ≤ 2

78 Towards 21+12+ · 3 · Aut (M22)

be the subamalgam in G[3] formed by these subgroups. Since N [3] is containedin each of the G[i3]’s, it is by the definition that

G[i3] ∼= G[i3]/(N [3]CG[i3](N [3])).

Proposition 5.2.1 Let A[3] be the above defined subamalgam in G[3]. Then

(i) G[03] ∼= G[03]/N [3] ∼= 23 : L3(2) × 2;

(ii) G[13] ∼= G[13]/N [3] ∼= (21+4+ : Sym3 × 2) : 2;

(iii) G[23] ∼= 25 : Sym5;(iv) A[3] corresponds to the diagram

P3 :1 P

2

2.

(v) G[3] is isomorphic to either Aut (M22) or 3 · Aut (M22).

Proof Since G[0] is identified with H [0], while

H [03] = Q[3]L[3](R[3] ∩ H [0]),

where R[3] ∩ H [0] ∼= 23 : L3(2) and since N [3] = O2(K [3]) = Q[3](O2(L[3])),(i) follows. From (i) we immediately obtain the structure of G[013] having index2 in G[13], so (ii) follows. The subgroup N [3] is contained in N [2] (this inclusion iseasily seen in terms of the locally projective graph Λ(G, ϕ, G) and the geometricsubgraphs in that graph). Furthermore we can assume without loss of general-ity that the subgroup R of order 3 from N [3] is contained in the complementL[2] ∼= L3(2) to Q[2] in N [2], so that R is a Sylow 3-subgroup S3 in L[2] as inSection 4.8. Now (iii) follows from (4.9.1 (iii)), since CQ[3](R) = Z [3]. Recall thatG[3] is the stabilizer in G of a geometric subgraph Γ[3] in Λ(G, ϕ, G) of valency 7.Furthermore, G[3] induces on Γ[3] a locally projective action of type (3, 2) withkernel N [3]. Finally G[03], G[13], and G[23] are stabilizers in G[3] of incident ver-tex, edge, and cubic geometric subgraph in Γ[3], respectively. By (4.8.1) and the

above proof of (iii) we observe that O2(G[23]) is the image of G[23] ∩G[01] in G[3],(iv) follows from (iii). Finally (v) is what (11.4.2) is for.

Applying (4.9.1 (ii)) or otherwise, it is easy to see that the following holds.

Lemma 5.2.2 A Sylow 7-subgroup S7 of G[03] acts fixed-point freely on Q[3] =Q[3]/Z [3].

Lemma 5.2.3 G[3] ∼= Aut (M22).

Proof By (5.2.1 (v)) either the assertion holds or the image X of G[3] inAut (N [3]) is of the form 212 : 32 · Aut (M22). Suppose the latter and let

A Petersen-type amalgam in G[3] 79

Q = O2(X) and Y be a Sylow 3-subgroup of O2,3(X). Since Q is the imageof Q[3] in the automorphism group of N [3], it is elementary abelian of order 212.We claim that Y is elementary abelian of order 9. In fact, otherwise it would bethe cyclic group of order 9. Since Y contains the image of a conjugate of R, it actsfixed-point freely on Q and therefore endows it with a GF (64)-vector space struc-ture of dimension 2. Since clearly M22 does not possess 2-dimensional projectiverepresentations, this is impossible. Thus Y is elementary abelian and hence itcontains a subgroup P of order 3 whose action on Q is not fixed-point free.In other terms both CQ(P ) and [Q, P ] are non-trivial. By a Frattini argumentthe commutator subgroup C of CX(P ) is either M22 or a non-split central exten-sion of M22. By (5.2.2) C acts non-trivially both on CQ(P ) and on [Q, P ]. On theother hand 11 divides the order of M22 and the minimal faithful representationover GF (2) of a cyclic group of order 11 has dimension 10. Since

dim CQ(P ) + dim [Q, P ] = dim Q = 12,

this leads to a contradiction.

Let G[3] denote the image of G[3] in the automorphism group of N [3]. Then(unlike in the G[2]-case) the natural homomorphism of N [3] into G[3] has non-trivial kernel, which is Z [3] = Z(N [3]).

Lemma 5.2.4 The group G[3] is a semidirect product of Q[3] ∼= 212 which is theimage of Q[3] and a subgroup

K = N˜G[3](R) ∼= 3 · Aut (M22),

where R is the image of R. The action of K on Q[3] is irreducible.

Proof We follow notation of the proof of (5.2.3) only that here Y is of order3 identified with the image of R. All we have to show is that NX(Y ) doesnot split over Y . This is the case, since otherwise the commutator subgroupof NY (X) would be isomorphic to M22. By (5.2.2) the irreducible constituentsof the commutator subgroup in X are all non-trivial. This contradicts the listof 2-modular representations of M22 calculated in James (1973): the minimaldimension of a faithful representation is 10 and the second minimal is 34. Alsoby James (1973) 12 is the minimal dimension of faithful GF(2)-module for 3·M22.Hence the irreducibility of K on Q[3] follows.

Lemma 5.2.5 For i = 0, 1, and 2 G[i3] is the image in G[3] of NG[i](R).

Proof For i = 0 and 1 CG[i](N [3]) = Z [3] and Q[3] = [Q[3], R] so the assertion

holds for i = 0 and 1. Since G[23] is generated by the images of G[023] and G[123],for i = 2 the assertion also holds.

80 Towards 21+12+ · 3 · Aut (M22)

Let N [i] be the image of NG[i](R) in K for i = 0, 1, and 2. By (5.2.5) theamalgam

N = N [i]/R | 0 ≤ i ≤ 2

(where the intersections are chosen in the obvious way) is isomorphic to theamalgam A[3] and hence also to the amalgam Z from (11.4.2). We would liketo find a ‘complement’ K to R in the amalgam N [i] | 0 ≤ i ≤ 2. Since eachof the three N [i]’s split over R in N [i], the only problem is that there are threecomplements K [i] for each i. The complements have to be chosen consistently sothat a Sylow 2-subgroup of K [0] is contained in both K [1] and K [2]. This can beachieved in the following geometric way.

Assume for a moment that G is the universal completion of G, so thatΛ(G, ϕ, G) is a tree containing the complete family of geometric subgraphs. Thenthe geometric subgraph Γ[3] of valency 7 stabilized by G[3] is contained in threegeometric subgraphs of valency 15 and the subgraph Γ[4] stabilized by G[4] is oneof them. These three subgraphs are transitively permuted by R. For every i = 0,1, and 2 the subgroup NG[i](R) induces the symmetric group Sym3 on the set ofgeometric subgraphs properly containing Γ[3]. This suggests a way to maintainthe consistency.

Lemma 5.2.6 Let K [i] be the image in K of NG[i](R)∩G[4]. Then the amalgamK = K [i] | 0 ≤ i ≤ 2 is isomorphic to the amalgam Z in (11.4.2) and K is theuniversal completion of K.

5.3 The 12-dimensional module

In this section we discuss some properties of the GF (2)-module Q[3] of the group

K ∼= 3 · Aut (M22).

According to James (1973) this is the unique 12-dimensional irreducible GF (2)-module of K. In James (1973) the existence of this module is justified by theembedding of M22 into U6(2).

A possible way to see this embedding is in terms of the Leech lattice (cf.Conway (1969), Wilson (1989), and chapter 4 in Ivanov (1999) for details). LetΛ be the Leech lattice, Λ = Λ/2Λ and C ∼= Co1 be the first Conway sporadicsimple group, acting naturally on Λ ∼= 224. We use the standard notation forvectors in the Leech lattice, assuming their coordinates to be in a basis of pairwiseorthogonal vectors of squared length 1/8 (cf. section 4.4 in Ivanov (1999)). Let

µ0 = ( 4, 4, 0, 021),µ1 = ( 0, 4, 4, 021),ν = (−3, −3, 5, 121).

The 12-dimensional module 81

Let X ∼= 22 be the subspace in Λ spanned by the images of µ0 and µ1, andlet Y ∼= 23 be the subspace spanned by the images of µ0, µ1, and ν. Then bylemma 4.10.6 in Ivanov (1999)

CC(X) ∼= U6(2).

According to the paragraph after lemma 4.10.10 in Ivanov (1999)

CC(Y ) ∼= M22

which gives the required embedding.The 12-dimensional module of K was studied intensively mostly in the

context of J4.

Lemma 5.3.1 Let G[3] ∼= 212 : 3 · Aut (M22), let Q[3] ∼= Q[3]/Z [3] ∼= 212 be theimage of Q[3] ∼= 21+12

+ and let

K = N˜G[3](R) ∼= 3 · Aut (M22).

Let f and q be the symplectic and associated quadratic forms on Q[3] defined as in(1.6.1). Then K acting on the non-zero vectors of Q[3] has exactly three orbits Ψ1,Ψ2, and Ψ3, such that the following assertions hold (where the singularity/non-singularity of vectors in Q[3] is defined with respect to q):

(i) Ψ1 is of length 693, it consists of singular vectors and K [2] ∼= 25 : Sym5is the stabilizer of a vector from Ψ1;

(ii) Ψ2 is of length 1386, it consists of singular vectors with stabilizers of theform 24 : PGL2(5);

(iii) Ψ3 is of length 2016, it consists of the non-singular vectors with stabilizersisomorphic to PGL2(11).

Proof The orbits on the non-zero vectors and the stabilizers were first determ-ined in Janko (1976). The image W of Z [2] in Q[3] ∼= Q[3]/Z [3] is a 2-subspacenormalized by the image of NG[2](R) in K. Therefore K [2] stabilizes a vectorfrom W and this vector is in Ψ1.

By (5.2.6) K [2] is the stabilizer of a point in the rank 3 Petersen type geo-metry G(3 · M22) of which K is the automorphism group. By (5.3.1 (i)) thereis a bijection β of the point-set of G(3 · M22) onto Ψ1 which commutes withthe action of K. This bijection induces what is called a representation of thegeometry G(3 · M22). In Proposition 4.4.6 in Ivanov and Shpectorov (2002)the following intersection diagram of the collinearity graph of G(3 · M22) wasgiven.

82 Towards 21+12+ · 3 · Aut (M22)

1 2 · 15

23 · 15 22 · 15

24 · 10 25 · 10

2

30 19

4

16

1

3

2 4

91 30

3

9

12 6

15

x Γ11(x)

Γ12(x) Γ1

3(x) Γ14(x)

Γ22(x) Γ2

3(x)

8

6

16

6

16

3

Lemma 5.3.2 In the above terms if x and y are points of G(3 · M22), then

f(β(x), β(y)) = 1

if and only if y ∈ Γ23(x). In particular for every v ∈ Ψ1 there are exactly 320

vectors u ∈ Ψ1 such that v ∈ u⊥ (where perp is with respect to f).

Proof This is Corollary 4.4.7 in Ivanov and Shpectorov (2002).

5.4 The triviality of N [4]

Recall that N [4] is the largest subgroup in G[014] which is normal in both G[04]

and in G[14]. By (4.2.1 (v)) N [4] ≤ N [3] ∼= 21+12+ : 3. Since H [4] ∼= 28 : O+

8 (2), itis easy to see that

G[04] = H [04] ∼= 24+4 : 26 : L4(2)

so that the chief factors of G[04] inside O2(G[04]) are 4- and 6-dimensional. There-fore N [4] is a 2-group of order 2m, where m ∈ 0, 4, 6, 8 and N [4] is containedin Q[3].

On the other hand, clearly

NG[0](R) ∩ G[4] ≤ G[04]

and

NG[1](R) ∩ G[4] ≤ G[14]

The triviality of N [4] 83

and since G(3 · M22) is connected, K is generated by K [0] and K [1] (compare

(5.2.6)) while K [i] = ˜K [i] ∩ G[4] for i = 0 and 1. By (5.2.4) K acts on Q[3]/Z [3]

irreducibly. So from this point of view N [4], if non-trivial, must be Q[3] or Z [3].This, compared with the above spectrum of possible orders of N [4] gives therequired result.

Lemma 5.4.1 The subgroup N [4] is trivial.

Since N [4] is trivial one can not guarantee the existence of the geometricsubgraph Γ[4] for an arbitrary completion of G (compare (4.2.1 (iv))). In factthis subgraph is missing in the graph Λ(G, ϕ, J4). Notice also that because of thetriviality of N [4] the action of G[4] on the graph Λ[4] = Λ(G[4], ϕ[4], G[4]) is faithful.This action is locally projective of type (4, 2) corresponding to the amalgamG[4] = G[04], G[14]. This is the amalgam A(4)

4 in Table 4 in Section 9.11.

Exercises

1. Use the embedding of M22 into U6(2) in terms of the Leech lattice to showthat the orbits of 3 · Aut (M22) on the non-zero vectors of its irreducible12-dimensional module are as in (5.3.1).

2. Establish the isomorphism G3 ∼= Aut (M22) performing direct calculations inthe amalgam G.

6

THE 1333-DIMENSIONAL REPRESENTATION

In this chapter we prove the existence of a completion of the amalgam G which isconstrained at level 2. We achieve this by studying the non-trivial representationsof G (the representations of the universal completion group of G) of minimaldegree. This degree turns out to be 1333 which is the famous minimal characterdegree of J4.

6.1 Representations of rank 2 amalgams

Let ϕ : G → A be a faithful completion of the amalgam G = G[0], G[1] whereA = GLm(C). Such a completion will be called a faithful representation of Gin dimension m over the complex numbers or simply a representation of G. Thecomplex-valued function on the element set of G defined by

χ(g) = tr(ϕ(g))

will be called the character of ϕ.Let ϕ[0], ϕ[1], and ϕ[01] be the restrictions of ϕ to G[0], G[1], and G[01],

respectively. Then ϕ[x] is a group representation for x = 0, 1, and 01 and bothϕ[0] and ϕ[1], restricted to G[01] equal to ϕ[01]. On the other hand, having a pairof group representations

ϕ[0] : G[0] → GLm(C) and ϕ[1] : G[1] → GLm(C)

whose restrictions to G[01] coincide, we can reconstruct the whole completionmap ϕ.

Lemma 6.1.1 Let χ[0] and χ[1] be characters of G[0] and G[1], respectively. Thenfor the existence of a representation (GLm(C), ϕ) such that the restrictions ofϕ to G[0] and G[1] afford χ[0] and χ[1], respectively, it is necessary and sufficientfor the restrictions to G[01] of χ[0] and χ[1] to coincide.

Proof The necessity is clear by considering the characters of the representationsϕ[0] and ϕ[1] in the paragraph before the lemma. To establish the sufficiencysuppose that χ[0] and χ[1] satisfy the hypothesis. Let ψ[0] and ψ[1] be repres-entations affording these characters. Since the restrictions of the characters toG[01] coincide, there is l ∈ GLm(C), such that ψ[0](a) = l−1ψ[1](a)l for everya ∈ G[01]. Define ϕ : G → GLm(C) by putting ϕ(a) = ψ[0](a) for a ∈ G[0]

and ϕ(a) = l−1ψ[1](a)l for a ∈ G[1]. Then it only remains to check that ϕ isinjective. Since both χ[0] and χ[1] are faithful, the only obstacle is that the whole

84

Representations of rank 2 amalgams 85

of l−1ψ[1](G[1])l might be in ψ[0](G[0]). But this is impossible since G[01] is normalin G[1] and self-normalized in G[0].

The next lemma, which is the main result of (Thompson 1981) supplies thenumber k(χ[0], χ[1]) of equivalence classes of representations corresponding toa pair of characters satisfying the hypothesis of (6.1.1).

Lemma 6.1.2 Let (GLm(C), ϕ) be a representation of G = G[0], G[1] suchthat the restrictions of ϕ to G[0] and G[1] afford χ[0] and χ[1], respectively. Thenthe equivalence classes of representations corresponding to the pair (χ[0], χ[1]) arein one-to-one correspondence with the (C [0], C [1])-double cosets in C [01], where

C [x] = CGLm(C)(ϕ(G[x])) for x = 0, 1, 01.

In particular there is precisely one equivalence class if and only if C [01] =C [0]C [1].

By Schur’s lemma the isomorphism type of C [x] is determined by the multipli-cities of the irreducible constituents of χ[x]. Suppose that χ[01] is multiplicity-free.Then C [01] is the set of ordered sequences of elements from C∗, whose compon-ents are indexed by the irreducible constituents of χ[01] and the multiplicationis componentwise. Then for i= 0 and 1 the subgroup C [i] consists of thesequences from C [01] subject to the condition that components, correspondingto χ[01]-constituents fused in χ[i], are equal. In this case the number k(χ[0], χ[1])is uniquely determined by the characters and easy to calculate.

Next we are aiming to deduce an existence condition for representations interms of the characters of G[0] only. If I is a conjugacy class of G[0] and χis a character of G[0] then χ(I) is the value χ(i) for some (and hence for all)i ∈ I. Since G[1] contains G[01] as a normal subgroup (of index 2) the formeracts on the conjugacy classes and on the irreducible characters of the latter. Lett ∈ G[1] \ G[01], let I be a conjugacy class of G[01] and let λ be a character ofG[01]. Put

It = t−1it | i ∈ I, λt : g → λ(t−1gt) for g ∈ G[01].

Then I, It and λ, λt are G[1]-orbits on classes and characters of G[01] oflengths 1 or 2. Clearly λt is irreducible if and only if λ is such. The followingresult is rather standard (cf. section 7.15 in Conway et al. (1985)).

Lemma 6.1.3 Let µ be a character of G[01].

(i) If µ is the sum of two irreducibles transposed by G[1] then there existsa unique character ν of G[1] whose restriction to G[01] equals to µ,this character is defined by ν(a) = µ(a) if a ∈ G[01] and ν(a) = 0 ifa ∈ G[01] \ G[1].

(ii) If µ is irreducible stable under G[1], then there are exactly two characters νof G[1] such that ν restricted to G[01] equals to µ; for these two charactersthe values on the elements from G[1] \ G[01] are negatives of each other.

86 The 1333-dimensional representation

Proof Let λ be an irreducible constituent of µ and θ be the character of G[1]

induced from λ. By the Frobenius reciprocity ν is a constituent of θ. Evaluatingthe inner product of θ with itself, we deduce that in (i) θ is irreducible andhence equal to ν. In (ii) θ is the sum of two irreducibles, say θ1 and θ2 andthe restriction of each to G[01] is µ. Since θ(a) = 0 for every a ∈ G[1] \ G[01],θ1(a) = −θ2(a).

Lemma 6.1.4 Let χ[0] be a character of G[0]. Then for the existence of a rep-resentation (GLm(C), ϕ), such that the restriction of ϕ to G[0] affords χ[0] itis necessary and sufficient that for every conjugacy class I of G[01] the equal-ity χ[0](K) =χ[0](L) holds, where K and L are the conjugacy classes of G[0]

containing I and It, respectively (where t ∈ G[1] \ G[01]).

Proof Since the classes I and It are fused in G[1], the necessity is obvious. If χ[0]

satisfies the hypothesis, then the restriction of χ[0] to G[01] is stable under theaction of G[1]. Therefore we can write this restriction as a sum of characters µ asin (6.1.3). By that lemma the restriction of χ[0] to G[01] extends to a characterof G[1] and the result follows.

In practice the action of t on the classes of G[01] can be determined by calcu-lating the automorphism group of the character table of G[01] and by identifyingthe image of t in this automorphism group.

6.2 Bounding the dimension

In this section we prove the following.

Proposition 6.2.1 Let m =m(G) be the smallest positive integer such that thereexists a faithful completion ϕ : G → A where A = GLm(C). Then

m(G) ≥ 1333.

Further down in this chapter we will construct a completion of G inGL1333(C). This shows that in (6.2.1) the equality is attained. Originally thisequality was established in Ivanov and Pasechnik (2004) by checking the neces-sary and sufficient conditions in (6.1.4) using explicit computer calculations withcharacter tables.

Before proceeding to the proof of (6.2.1) we recall a few basic facts aboutrepresentations and group characters. For a group F by m(F ) we denote theminimal degree of a faithful complex character of F (equivalently the smallestdimension of a faithful representation over the complex numbers).

Lemma 6.2.2 Let Q be an elementary abelian 2-group and P be a subgroup ofindex 1 or 2 in Q. Let χ[P ] be the function on Q such that χ[P ](q) = 1 if q ∈ Pand χ[P ](q) = −1 otherwise. Then χ[P ] is an irreducible character (of degreeone) of Q and all the irreducible characters of Q can be obtained in this way.

The following result is a direct consequence of (6.2.2).

Bounding the dimension 87

Lemma 6.2.3 Let F be a group which contains an elementary abelian 2-groupQ as a normal subgroup and l be the length of the shortest orbit on the set ofhyperplanes (that are subgroups of index 2) in Q of F acting by conjugation.Then l ≤ m(F ).

Lemma 6.2.4 Let E ∼= 21+2m+ be an extraspecial group of order 22m+1 and of

plus type. Let Z = 〈z〉 be the centre of E and x ∈ E \ Z be an involution. Letχ be a faithful irreducible character of E. Then

(i) the degree of χ is 2m;(ii) χ(z) = −2m;(iii) χ(x) = 0;(iv) χ is unique.

Proof Let U be a maximal elementary abelian subgroup in E. By (1.6.7) |U | =2m+1 and U contains Z. Let ψ be an irreducible character of U of degree 1 whosekernel is disjoint from Z. Let χ be the character of E induced from ψ. Then itis well known and easy to check that χ is as stated.

Proof of Proposition 6.2.1 Let ϕ : G → A be a faithful completion of G,where A = GLm(C) and let χ be the group character afforded by the restrictionof ϕ to Q[3] ∼= 21+12

+ . Then

χ = χ0 + χ1 + χ2,

where χ0 is the sum of the faithful constituents, χ1 is the sum of the constituentswith kernel Z [3] and χ2 is the sum of the trivial constituents. Let z be thegenerator of Z [3] and x ∈ Z [2] \ Z [3] (in particular x is an involution). Thenx ∈ Q[3], z and x are conjugate in G[2] and hence χ(x) =χ(z). By (6.2.4) theirreducible constituents in χ0 are all equal to the unique irreducible faithfulcharacter of Q[3] of degree 64; if n is the multiplicity of this character, then

χ0(1) = 64n, χ0(z) = −64n, χ0(x) = 0.

Since χ2 is a trivial character, χ2(x) = χ2(z) = χ(1). Thus for the equalityχ(x) = χ(z) to hold the constituent χ1 must be non-zero. By (6.2.2) χ1 is thesum of linear characters of Q[3] = Q[3]/Z [3] (considered as characters of Q[3]).By (5.2.4) the group G[3] induces on Q[3] the group 3 · Aut (M22), whose orbitson the non-zero vectors are described in (5.3.1). Notice that Q[3] is self-dualand the orbit lengths on the hyperplanes are the same as on the set of non-zerovectors. Combining (6.2.3) and (the dual version of) (5.3.1) we observe thateither χ1(1) ≥ 1386 or χ1(1) = 693 and χ1 is the sum of 693 linear characterscorresponding to the orbit Ψ1 in (5.3.1). Since x ∈ Z [2], we conclude fromstatement (5.3.1) and its proof that the image of x in Q[3] belongs to Ψ1. By(5.3.2) this gives

χ1(x) = 693 − 2 · 320 = 53.


In this case, since χ0(x) = 0,

53 = χ0(z) + χ1(z) = −64n + 693,

which gives n = 10. Hence

χ(1) ≥ χ0(1) + χ1(1) ≥ 64 · 10 + 693 = 1333.

As an immediate consequence of the above proof we obtain the following.

Corollary 6.2.5 Let ϕ : G → A be a faithful completion, where A = GL1333(C)and χ is the corresponding character. Let z be a generator of Z [3]. Then thefollowing assertions hold:

(i) the restrictions of χ to Q[3] does not contain trivial constituents;(ii) χ(z) = 53.

6.3 On irreducibility of induced modules

In this section we prove a standard result from the representation theory of finitegroups.

Let F be a finite group, let Σ be an F -module over the complex numbers.Let Q be an elementary abelian normal 2-subgroup in F which acts fixed-pointfreely on Σ, that is, CΣ(Q) = 0. Let R be the set of hyperplanes in Q havingnon-zero centralizers in Σ:

R = R | R ≤ Q, [Q : R] = 2, CΣ(R) = 0.

For R ∈ R put ΣR = CΣ(R).

Proposition 6.3.1 In the situation described in the paragraph before theproposition suppose that F (acting by conjugation) transitively permutes thehyperplanes in R. Then

(i) the module Σ is isomorphic to the one induced from the NF (R)-module ΣR;(ii) Σ is irreducible provided that the action of NF (R) on ΣR is irreducible;(iii) suppose that ΣR is the direct sum of k pairwise non-isomorphic con-

stituents of dimensions d1, . . . , dk, then Σ is a direct sum of k pairwisenon-isomorphic irreducible constituents of dimensions d1 · m, . . . , dk · m,respectively, where m = |R| = [F : NF (R)].

Proof By the definition ΣR is a constituent of Σ restricted to NF (R). By thehypothesis the space Σ is spanned by the images of ΣR under F . By the Frobeniusreciprocity Σ is a quotient of the module I induced from the NF (R)-module ΣR.Since both I and Σ have dimension

dim ΣR · [F : NF (R)] = dim ΣR · |R|,(i) follows.

Representing G[0] 89

Suppose that the action of NF (R) on ΣR is irreducible. Let Θ be a non-zero F -submodule in Σ. By Maschke theorem Θ is a direct sum of irreducibleQ-submodules. Since CΣ(Q) = 0, by (6.2.2) each such submodule is non-trivialand its kernel is a hyperplane of Q. Therefore ΘP : =CΘ(P ) is non-trivial forsome hyperplane P in Q. Since CΘ(P ) ≤ CΣ(P ), we have P ∈ R and withoutloss of generality we assume that P =R. Clearly ΘR ≤ΣR. Since Θ is an F -submodule in Σ, NF (R) stabilizes ΘR. Since the action of NF (R) on ΣR isirreducible, we conclude that ΘR = ΣR and (ii) follows.

To establish (iii) we consider an irreducible constituent of NF (R) in ΣR, takeits images under F and consider the subspace in Σ spanned by these images. By(i) and (ii) such a subspace is an irreducible F -submodule in Σ. Furthermore,arguing as in the above paragraph, it is easy to see that every constituent appearsin this way. Finally the irreducible constituents of F in Σ are pairwise non-isomorphic since those of NF (R) in ΣR are such.

6.4 Representing G[0]

In the next five sections (including this one) we construct a faithful completionϕ : G → GL1333(C). Actually we will construct eight such completions and theresult will be summarized in Section 6.9. Suppose first that such a completionexists. Let Π[0] be a 1333-dimensional vector space over the field C of complexnumbers which supports the restriction of ϕ to G[0]. By (6.2.5) the character χof ϕ satisfies χ(z) = 53, where z is the generator of Z [3]. In terms of (3.6.2) theelement z is in the orbit Q1 of length 155 of the action of L[0] on Q[0].

Consider the eigenspace decomposition of Π[0] with respect to Q[0]. Let Pbe the set of hyperplanes (subgroups of index 2) in Q[0]. Then by (3.6.1) underthe natural action by conjugation P splits into two G[0]-orbits P1 and P2 withlengths 155 and 868, respectively. For P ∈ P put

Π[0]P = CΠ[0](P ).

Notice that q ∈ Q[0] acts on Π[0]P as the identity operator if q ∈ P and as

the (−1)-scalar operator otherwise. Combining (3.6.3), (6.2.2), and (6.2.5) andapplying direct calculations we obtain the following.

Lemma 6.4.1 Let Π[0] be the G[0]-module which supports the restriction to G[0]

of a 1333-dimensional representation of G. Then

(i) CΠ[0](Q[0]) = 0;(ii) if P ∈ P1 then dim Π[0]

P = 3;(iii) if P ∈ P2 then dim Π[0]

P = 1.

By (6.4.1) Π[0] is the direct sum of the Π[0]P ’s taken for all P ∈ P. Furthermore,

if Π[0]1 is the sum of the subspaces Π[0]

P taken for all P ∈ P1 and Π[0]2 is the sum of

such subspaces taken for all P ∈ P2 then both Π[0]1 and Π[0]

2 are G[0]-submodules


in Π[0] and

Π[0] = Π[0]1 ⊕ Π[0]

2 .

Finally, dim Π[0]1 = |P1|·3 = 465 and dim Π[0]

2 = |P2|= 868 (notice that 465+868 =1333).

Let P1 ∈ P1, P2 ∈ P2. Let F1 and F2 be the stabilizers in G[0] of P1 and P2,respectively. Put Π1 = Π[0]

P1and Π2 = Π[0]

P2. Then Π1 is a 3-dimensional module for

F1, while Π2 is a 1-dimensional module for F2. By the (6.3.1 (i)) Π[0]1 is induced

from Π1 and Π[0]2 is induced from Π2.

The above discussion suggests the following strategy to produce Π[0]:

(a) calculate the stabilizers F1 and F2;(b) describe a 3-dimensional C-module Π1 of F1 and a 1-dimensional module

Π2 of F2;(c) define Π[0]

1 and Π[0]2 as the G[0]-modules induced from Π1 and Π2,

respectively;(d) put Π[0] = Π[0]

1 ⊕ Π[0]2 .

First we define Π1 and Π2 up to two possibilities each. The adjustment willbe made when we restrict Π[0] to G[01] and require the restricted action to beextendable to an action of G[1].

In due course we will use the following direct consequence of (6.4.1)and (3.6.3).

Corollary 6.4.2 Let y be an element from the orbit of length 868 of the actionof L[0] on Q[0]. Then

χ(y) = −11.

In terms introduced in the paragraph after (3.6.1), if P1 =P (U3) thenF1 =G[02] and if P2 =P (U1, f) then F2 =G[0](f) is the stabilizer in G[0] of asymplectic form f of rank 4 on U5, whose radical is U1. Equivalently, F2 is thestabilizer in G[04] of the non-singular symplectic form f on U5/U1 induced by f .Therefore

[G[04] : G[04](f)] = 28, O2(G[04]) ≤ G[04](f) and

G[04](f)/O2(G[04]) ∼= S4(2) ∼= Sym6.


(i) G[02]/O2(G[02]) ∼= (L[0] ∩ G[2])Q[2]/Q[2] ∼= Sym3 × L3(2);(ii) if S3 is a Sylow 3-subgroup of O2,3(G[02]), then [O2(G[02]), S3] = Q[2] =

O2(N [2]);(iii) Q[0] ∩ Q[2] = P1 = P (U3).

Representing G[0] 91

Proof Statement (i) is immediate, since G[02] is the stabilizer of U3 in G[0]. Inorder to establish (ii), notice that

G[02]/Q[2] ∼= (Sym3 × 2) × L3(2),

and that Z [2] is the commutator subgroup of Q[2]. Furthermore, Q[2]/Z [2], asa module for

H [2]/Q[2] ∼= O+4 (2) × L3(2)

is isomorphic to the tensor product W4 ⊗ U3, which shows that S3 acts onQ[2]/Z [2] is fixed-point freely. Without loss we assume that S3 ≤ L[0]. Thenelementary calculations in Q[0] (which is the exterior square of U5) show thatCQ[0](S3) is 1-dimensional, which gives (iii).

The following lemma was known since the time of A. Hurwitz and F. Klein.


(i) the smallest faithful complex representation of L3(2) is 3-dimensional;(ii) L3(2) possesses exactly two complex irreducible characters of degree 3,

say λ1 and λ2, which are algebraically conjugate and transposed by theouter automorphism group of L3(2);

(iii) if S ∼= Sym4 is a maximal parabolic subgroup in L3(2) then the restrictionto S of a representation affording the character λ1 or λ2 is irreducible.

For the remainder of the chapter we fix a complex character λ of L[2] ∼= L3(2)of degree 3. In terms of (6.4.4) λ is either λ1 or λ2.

Let us turn to the description of the modules Π1 and Π2 of G[02] and G[04](f),respectively. First we consider the possibilities for the kernels K1 and K2 of thesemodules.

Lemma 6.4.5 Let K1 be a normal subgroup in G[02] such that

G[02]/K1 = Q[0]K1/K1 × L[2]K1/K1 ∼= 2 × L3(2).

Then K1 is one of two particular subgroups K(1)1 and K

(2)1 distinguished by

the condition that the equality K(i)1 = P1(K

(i)1 ∩ L[0]) holds for i= 1 but not

for i = 2.

Proof Since K1 is normal in G[02] and by considering the order of K1 we con-clude that K1 contains a Sylow 3-subgroup S3 of O2,3(G[02]). By (6.4.3 (iii))Q[2] = [O2(G[02]), S3]. Therefore K1 contains Q[2]. We know that G[02]/Q[2] ∼=(Sym3 × 2) × L3(2) and that Q[0]Q[2]/Q[2] is the centre of order 2 in G[02]/Q[2].Therefore K1/Q[2] ∼= Sym3, while by the above G[02]/Q[2] contains exactly twonormal subgroups isomorphic to Sym3. On other hand, (L[0] ∩ G[2])Q[2]/Q[2] ∼=Sym3 × L3(2) contains just one such normal subgroup. Hence the result.


If K(i)1 is as in (6.4.5), then by (6.4.3 (iii))

K(i)1 ∩ Q[0] = P1 = P (U3)

both for i = 1 and 2.Let Π1 = Π(i)

1 , where i = 1 or 2, be the 3-dimensional module for G[02] overthe field of complex numbers whose kernel is K

(i)1 and the action of L[2] ∼= L3(2)

affords the character λ as defined after (6.4.4). Since Q[0]K(i)1 /K

(i)1 is the centre

of order 2 of G[02]/K(i)1

∼= 2 × L3(2), every element from Q[0] \ K(i)1 = Q[0] \ P1

acts on Π(i)1 as the (−1)-scalar operator. Because of the factorization

G[02] = K(i)1 Q[0]L[2],

the above conditions define the module Π(i)1 up to isomorphism for i= 1

and 2.

Lemma 6.4.6 The group G[04](f) contains exactly two normal subgroups K(1)2

and K(2)2 of index 2 intersecting Q[0] in P2. Furthermore the notation can chosen

so that L[0](f) = Sf∼= 24 : S4(2) is contained in K

(1)2 but not in K

(2)2 .

Proof Since G[04](f)/P2 ∼= 2 × 24 : S4(2), the quotient X of G[04](f) over thecommutator subgroup of G[04](f) is elementary abelian of order 4. One of theorder 2 subgroups in X is of course the image of Q[0], hence the result.

Let Π2 = Π(i)2 be a 1-dimensional module for G[04](f) over the field of complex

numbers, whose kernel is K(i)2 for i = 1 and 2.

There are just eight possibilities for the modules Π1 and Π2 (taking intoaccount the two possibilities for the action of L[2] on Π1). To specify thesemodules we have to choose K1 between K

(1)1 and K

(2)1 and also to choose K2

between K(1)2 and K

(2)2 . Inducing from G[02] and G[04](f) to G[0] and taking the

direct sum, we obtain a G[0]-module Π[0], whose dimension is

1333 = 155 × 3 + 868 = [G[0] : G[02]] × dim Π1 + [G[0] : G[04](f)] × dim Π2.

6.5 Restricting to G[01]

The G[0]-module Π[0] from the previous section is 1333-dimensional over thefield of complex numbers and it is the direct sum of two submodules Π[0]

1 andΠ[0]

2 of dimension 465 and 868, respectively. The module Π[0]1 is induced from a

3-dimensional module Π1 = Π(i)1 for G[02] and the module Π[0]

2 is induced froma 1-dimensional module Π2 = Π(j)

2 of G[04](f), where i, j ∈ 1, 2. In this sec-tion we discuss the restriction of Π[0] from G[0] to G[01]. Here the choice betweenpossibilities for Π[0] is irrelevant.

If P ∈ P1 then P = P (V3) for a 3-dimensional GF (2)-subspace V3 in U5,and the module Π[0]

P is 3-dimensional (over the complex numbers) by (6.4.1 (ii)).

Restricting to G[01] 93

The stabilizer of V3 in G[0] coincides with NG[0](Π[0]P ) and this stabilizer acts

on Π[0]P as G[02] does on Π1. Namely, these two actions are conjugate by an

element of G[0] which maps U3 onto V3. If P ∈ P2, then P = P (V1, f), Π[0]P is

1-dimensional and the stabilizer of P in G[0] acts on Π[0]P in the way G[04](f)

does on Π2.

Lemma 6.5.1 The G[0]-modules Π[0]1 and Π[0]

2 are irreducible.

Proof The result is immediate by (6.3.1).

Next we restrict Π[0] from G[0] to G[01]. Recall that the latter is the stabilizerin G[0] of the hyperplane U4 in U5.

Lemma 6.5.2 Put

P(1)11 = P (V3) | V3 < U4, P(1)

12 = P (V3) | V3 < U4,

P(1)21 = P (V1, f) | V1 < U4, P(1)

22 = P (V1, f) | V1 < U4.

Then

(i) P(1)11 and P(1)

12 are the G[01]-orbits on P1;(ii) P(1)

21 and P(1)22 are the G[01]-orbits on P2;

(iii) |P(1)11 | = 15, |P(1)

12 | = 140, |P(1)21 | = 420, |P(1)

22 | = 448.

Proof Since L5(2) has two orbits on the set of pairs (V3, V4) where V3 and V4are 3- and 4-dimensional subspaces in the natural module U5, (i) follows. LetSf be the stabilizer of f in L[0]. Then by (1.5.1) Sf

∼= 24 : S4(2). In order toestablish (ii) we have to determine the orbits of Sf on the set of hyperplanes inU5. By (1.5.3) the orbit of such a hyperplane V4 is determined by whether ornot V4 contains V1 and by the isomorphism type of (V4, f |V4). Since V1 is theradical of f , it is easy to see that the symplectic form (V4, f |V4) has rank 2 if V4contains V1 and it is non-singular otherwise. The assertion (iii) now follows fromelementary geometric calculations.

Lemma 6.5.3 For a hyperplane P ∈ P of Q[0] the stabilizer G[01](P ) of P inG[01] acts non-trivially and irreducibly on Π[0]

P .

Proof An equivalent reformulation of the assertion is that for any pair (V4, P )where V4 is a hyperplane in U5 and P is a hyperplane in Q[0], the stabilizerG[0](V4, P ) of this pair in G[0] acts non-trivially and irreducibly on Π[0]

P . Bythe proof of (6.5.2), G[0] has four orbits on the set of pairs (V4, P ) as above.We take P to be P1 = P (U3) or P2 = P (U1, f) and vary V4. Thus we haveto check that G[0](V4, Pi) acts irreducibly on Πi. Since Π2 is 1-dimensional andQ[0] acts on Π2 non-trivially, and G[0](V4, Pi) contains Q[0], the result is obviousin the case i = 2. Therefore we consider the case i = 1. First suppose that


V4 contains U3. There are three hyperplanes in U5 containing U3. These hyper-planes are transitively permuted by a Sylow 3-subgroup S3 of O2,3(G[02]). HenceG[0](V4, P1)K1 =G[02], which implies that the actions induced on Π1 by G[02]

and G[0](V4, P1) coincide, giving the assertion. If V4 does not contain U3, thenas always we have Q[0] ≤ G[0](V4, P1) and also without loss we may assume thatL[2] ∩ G[0](V4, P1) is the stabilizer of the 2-dimensional subspace V4 ∩ U3 in thenatural module U3 of L[2]. Therefore G[0](V4, P1) induces 2×Sym4 on Π1 andthe action is irreducible by (6.4.4 (iii)).

Lemma 6.5.4 For a, b ∈ 1, 2 put

Π[01]ab =

⊕P∈P(1)

ab

Π[0]P .

Then the Π[01]ab ’s are the irreducible G[01]-submodules in Π[0] of dimensions 45,

420, 420, and 448 for ab = 11, 12, 21, and 22, respectively.

Proof By (6.5.2) the Π[01]ab ’s are G[01]-submodules. Arguing as in the proof

of (6.5.1) and making use of (6.5.3) it is easy to check that each of the foursubmodules is irreducible.

6.6 Lifting Π[01]11

In terms of Section 3.7 we have G[01] = Q[1]L[1], where Q[1] = O2(G[01]), L[1] ∼=L4(2), and U4 is the natural module of L[1]. Furthermore, Q[1] = Z [1]A[1]B[1]

and there are L[1]-module isomorphisms α : U4 → A[1], β : U4 → B[1] andζ :∧ 2U4 → Z [1], such that the key commutator relation reads as

[α(w1), β(w2)] = ζ(w1 ∧ w2),

where w1, w2 ∈ U4. As above put D[1] = 〈α(w)β(w) | w ∈ U4〉 (recall that D[1]

is elementary abelian of order 210 and that D[1] contains Z [1]).The family P(1)

11 consists of 15 hyperplanes of Q[0] which are naturally indexedby the 3-dimensional subspaces in U4. The group G[01] induces on this fam-ily the natural permutation action of L4(2) ∼= L[1] of degree 15 with kernel Q[1].The hyperplane P1 corresponds to the subspace U3. Let L[1](U3) ∼= 23 : L3(2)be the stabilizer of U3 in L[1]. Then

G[012] = G[01] ∩ G[02] = Q[1]L[1](U3)

is the stabilizer of U3 in G[01].Consider the action of G[012] on ΠP1 (the latter being canonically identified

with Π1). By (6.3.1 (i)) the action of G[01] on Π[01]11 is induced from that action.

As above let K1 denote the kernel of G[02] on Π1 (so that K1 is K(1)1 or K

(2)1

in terms of (6.4.5)) and put R1 = K1 ∩ G[012]. By the proof of (6.5.3) we have[K1 : R1] = 3 and K1 = R1S3 for a Sylow 3-subgroup S3 of O2,3(G[02]). On the

Lifting Π[01]11 95

other hand, R1 contains Q[2] and by the structure of G[01], we have

Q[2] = Z [1]α(U3)β(U3)O2(L[1](U3)).

Since G[012]/R1 ∼= G[02]/K1 ∼= 2 × L3(2), while G[012]/Q[2] ∼= 22 × L3(2), weconclude that R1/Q[2] is a normal subgroup of order 2 in G[012]/Q[2]. One theother hand, the normal subgroups of order 2 in G[012]/Q[2] are A[1]Q[2]/Q[2],B[1]Q[2]/Q[2], and D[1]Q[2]/Q[2].

By the above K1 = R1S3 where S3 is a Sylow 3-subgroup of O2,3(G[02]) andwe will assume without loss of generality that S3 ≤ L[0]. If R1 = A[1]Q[2] thenR1 (and hence K1 as well) contains Q[0] = Z [1]A[1] which is not allowed. Onthe other hand B[1]S3 ≤ L[0] and B[1]S3Q

[2]/Q[2] ∼= Sym3. Therefore in terms of(6.4.5) we obtain:

R(1)1 := K

(1)1 ∩ G[012] = Q[2]B[1],

R(2)1 := K

(2)1 ∩ G[012] = Q[2]D[1].

Suppose first that R1 = R(1)1 = Q[2]B[1]. Then Z [1]B[1] is in K1. Since Z [1]B[1]

is normal in G[01] and G[01] permutes transitively the subspaces Π[0]P for P ∈ P(1)

11

the action of Z [1]B[1] on the whole of Π[01]11 is trivial. Since the action of Z [1]A[1]

is non-trivial and G[1] contains an element t1 which conjugates Z [1]A[1] ontoZ [1]B[1], the action of G[01] on Π[01]

11 cannot be lifted to that of G[1] in this case,and we have established the following result.

Lemma 6.6.1 If the kernel of G[02] on Π1 is K(1)1 , then the action of G[01] on

Π[01]11 cannot be lifted to an action of G[1].

Now suppose that R1 = R(2)1 = Q[2]D[1]. Then D[1] ≤ K1. Since D[1] is nor-

mal in G[01], we conclude that D[1] acts trivially on the whole of Π[01]11 . Therefore

the action of G[01] on Π[01]11 is isomorphic to

G[01]/D[1] ∼= 24 : L4(2).

On the other hand, D[1] stays normal in G[1] and the element t1 (defined inSection 3.7 which extends G[01] to G[1]) centralizes G[01] modulo D[1]. Therefore

G[1]/D[1] ∼= 24 : L4(2) × 2

and the action of G[01] on Π[01]11 can be lifted to that of G[1] simply by declaring

that t1 acts as a (±1)-scalar operator.

Proposition 6.6.2 If the kernel of G[02] on Π1 is K(2)2 then there are exactly two

ways to lift the action of G[01] on Π[01]11 to an action of G[1]. There is an element

t1 ∈ G[1] \ G[01] commuting with D[1], which acts as the identity operator in oneof the liftings and as the (−1)-scalar operator in the other one.


6.7 Lifting Π[01]22

Recall that for P ∈ P(1)22 the stabilizer of P in G[01] coincides with the stabilizer

of a pair (V1, f), where V1 is a 1-dimensional subspace in U5, disjoint from U4 andf is a non-singular symplectic form on U4 (notice that in the considered case U4is canonically isomorphic to U5/V1). In terms of the development of Sections 3.4and 3.7 we assume that V1 = U5 ∩ W2 and V1 = 〈u〉, so that the stabilizer of V1in G[01] is Z [1]A[1]L[1] and

G[01](P ) = G[01](V1, f) = Z [1]A[1]S

where S ∼= S4(2) ∼= Sym6 is the stabilizer of f in L[1]. As an L[1]-module thesubgroup Z [1] is isomorphic to the exterior square of U4 and also to the exteriorsquare of U∗

4 . By (1.4.3) the elements of Z [1] are the symplectic forms on U4.Therefore f is an element of Z [1] and its perp (with respect to the non-singularsymplectic form on Z [1] preserved by L[1] ∼= Ω+

6 (2)) is an S-invariant hyperplanein Z [1]. We denote this hyperplane by Z

[1]f . Then P = Z

[1]f A[1], and if C is the

commutator subgroup of G[01](V1, f), then

C = Z[1]f A[1]T,

where T ∼= Alt6 is the commutator subgroup of S ∼= Sym6. Let R2 be the kernelof the action of G[01](V1, f) on the subspace Π[0]

P . Then R2 contains C withindex 2 and R2 does not contain Z [1]. This leaves us with two possibilities forR2 (notice that the choice between these two possibilities specifies the choicebetween K

(1)2 and K

(2)2 in (6.4.6)).

In any event we have R2 ∩ Z [1] = Z[1]f . Let

Π(f) = CΠ[01]22

(Z [1]f ).

By the above, Π(f) is the sum of the subspaces Π[0]Q taken for all hyperplanes

Q = P (V1, f), where V1 is a 1-dimensional subspace in U5 disjoint from U4 andf is the above form on U5/V1 ∼= U4.

Lemma 6.7.1 dim Π(f) = 16 and the action of G[01] on Π[01]22 is induced from

the action of

CG[01](f) = Z [1]A[1]B[1]S

on Π(f).

Proof Since there are exactly 16 1-dimensional subspaces in U5 disjoint fromU4, we get the dimension of Π(f). Since

448 = dim Π[01]22 = dim Π(f) × [G[01] : CG[01](f)] = 16 × 28,

the second assertion is by (6.3.1 (i)).

Lifting Π[01]22 97

Put F [01] = CG[01](f)/Z [1]f and adopt the bar notation for the images in F [01]

of elements and subgroups of CG[01](f), and also for the homomorphisms inducedby α, β and ζ.


(i) Z [1]A[1]B[1] = O2(F [01]) is extraspecial of type 21+8+ with centre Z [1];

(ii) S ∼= S4(2) ∼= Sym6 is a complement to O2(F [01]) in F [01];(iii) Z [1]A[1], Z [1]B[1] and D[1] are the elementary abelian normal subgroups

of order 25 in F [01]; the former two are semisimple while the latter isindecomposable as S-modules.

Proof The key commutation of G[01] in F [01] becomes

[α(w1), β(w2)] = zf(w1,w2),

where z is the generator of Z [1], which gives (i). The remaining assertions areobvious and/or follow from the above commutator relation.

Consider the action of O2(F [01]) on Π(f). This action is induced from thelinear representation of the subgroup Z [1]A[1] (of index 16 in O2(F [01])) on ΠP

with kernel A[1]. Therefore the action is faithful and since 16 = dim Π(f) is thedimension of the unique faithful irreducible representation of O2(F [01]) ∼= 21+8

+ ,we have the following lemma.

Lemma 6.7.3 As a module for O2(F [01]) ∼= 21+8+ , the space Π(f) is isomorphic

to the unique irreducible faithful module of that group.

Let F [1] =CG[1](f)/Z [1]f . Then F [1] is a semidirect product of F [01] and a

group of order 2 generated by the image t1 of t1 in F [1], where t1 is the involutionwhich extends G[01] to G[1] as in Section 3.7. We require a detailed informationabout the action of t1 on F [01].

The action of t1 on O2(F [01]) is clear: it transposes α(w) onto β(w) forevery w ∈ U#

4 .

Lemma 6.7.4 The element t1 can be chosen so that t1 centralizes S.

Proof Consider L[1] ∼= Alt8 as the alternating group of an 8-element set P8.Then Z [1] is the heart Pe

8/Pc8 of the GF (2)-permutation module of L[1] on P8

(compare Section 3.3). Since

〈Z [1]L[1], t1〉 ∼= (P8/Pc8) : L[1],

there are 64 candidates for t1 and these candidates are indexed by the partitionsof P8 into two odd subsets. In these terms the subgroup S ∼= Sym6 is the stabil-izer in L[1] of a 2-element subset a, b ⊂ P8. The partition ρ = (a, b, P8\a, b)(considered as an element of Z [1]) is the only non-identity element in Z [1] stabil-ized by S and therefore ρ corresponds to the symplectic form f on U4, stabilized


by S; in particular ρ ∈ Z[1]f . Suppose that t1 corresponds to the partition

(a, P8 \ a). Then by (3.3.3)

CL[1](t1) = L[1](a) ∼= Alt7

and [t1, T ] = 1, where T ∼= Alt6 is the commutator subgroup of S. Let π be thetransposition (a, b). Then S = 〈T, π〉 and [t1, π] = ρ. Since ρ ∈ Z

[1]f ; the result

follows.

At this point we feel it is appropriate to explain our further strategy.By (6.7.3) the action of O2(F [01]) on Π(f) is determined uniquely (up toisomorphism). Our goal is to lift this action to that of the whole of F [1], subjectto the condition that the restriction to F [01] is known (up to the choices). SinceD[1] is normal in F [1] it is natural to study the eigenspace decomposition of Π(f)

with respect to D[1]. Since the action of O2(F [01]) on Π(f) is faithful irreducibleand z is an involution in the centre of O2(F [01]), we observe (by Schur’s lemmaor otherwise) that z acts on Π(f) as the (−1)-scalar operator. Since D[1] containsz, this implies that D[1] acts fixed-point freely on Π(f). Hence the kernel of everyeigenspace is a hyperplane C in D[1] which does not contain z. There are 16 suchhyperplanes and it is easy to deduce from (1.6.7 (iii)) that O2(F [01]) acting onD[1] by conjugation permutes these hyperplanes transitively. Since 16 is also thedimension of Π(f), every hyperplane not containing z is the kernel of an actionon an eigenspace and every eigenspace is 1-dimensional.

Let C be a hyperplane in D[1] which does not contain z. Then by the abovethe action of O2(F [01]) on Π(f) is induced from the linear action of D[1] withkernel C. Put

I = NF [1](C).

Then by (6.3.1 (i)) the action of F [1] on Π(f) is induced from a linear rep-resentation of I. Therefore our immediate goal is to calculate I. By (1.6.7(ii)), I ∩O2(F [01]) = D[1] and since O2(F [01]) acts transitively on the set ofhyperplanes in D[1] not containing z, IO2(F [01]) = F [1] and so by (6.7.4)

I/D[1] ∼= S4(2) × 2.

In order to accomplish our goal we study the classes of Alt6-subgroups in F [01]

and the action of t1 on these classes.For X =A, B, or D, the group Y = Z [1]X [1]T is the semidirect product

of T ∼= Alt6 and an extension of the trivial 1-dimensional module Z [1] by thenatural symplectic module Z [1]X [1]/Z [1] of S restricted to T (compare (6.7.2)).By (3.2.6 (iv)) Y contains two classes of Alt6-subgroups. Let TX denote sucha subgroup not conjugate to T .


(i) F [01] contains exactly four classes of Alt6-subgroups with representativesT , TA, TB, and TD;

Lifting Π[01]12 ⊕ Π[01]

21 99

(ii) D[1] is semisimple as a module for TA or TB and indecomposable asa module for T or TD;

(iii) the representatives TA and TB can be chosen in such a way that

D[1]TA = D[1]TB ;

(iv) the element t1 stabilizes the classes containing T and TD and transposesthe classes of TA and TB.

Proof By (6.7.2) O2(F [01]/Z [1]) is the direct sum of two copies of the naturalsymplectic module of S and by (3.2.6 (iv)) H1(T , O2(F [01]/Z [1])) ∼= 22 whichgives (i). Now (ii) is by (3.2.7 (iii)) while (iii) follows from the definition. Finally(iv) is by (6.7.4) and the paragraph before that lemma.

By (6.7.5 (ii)), TA stabilizes a direct sum decomposition

D[1] = Z [1] ⊕ C,

where C is a hyperplane in D[1] disjoint from Z [1]. Clearly this decomposition isalso stabilized by NF [01](TA). By (3.2.6) we have H1(S, A[1]) ∼= H1(T , A[1]) ∼= 2,and therefore

NF [01](TA) ∼= Sym6 × 2.

And now by the order reason the following hold.

Lemma 6.7.6 I = D[1]NF [01](TA)〈t1〉.

Thus the quotient of I over the commutator subgroup of I is elementaryabelian of order 23 and hence there are exactly four linear representations ofI whose kernels do not contain Z [1]. Therefore we can construct four requiredrepresentations of F [1] on Π(f). For each such representation we consider theeigenspace decomposition with respect to Z [1]A[1] and observe that the restrictionto F [01] is induced from a linear representation of

Z [1]A[1]NF [01](TA) = Z [1]A[1]NF [01](T ) ∼= (2 × 24) : Sym6

as required. Thus we obtain the final result of the section.

Proposition 6.7.7 For each of the two choices of the kernel K2 of G[04](f) onΠ2, there are two ways to lift the action of G[01] on Π[01]

22 to an action of G[1].

6.8 Lifting Π[01]12 ⊕ Π[01]

21

At this stage it is appropriate to calculate the eigenspace decomposition of Πwith respect to Z [1].

Lemma 6.8.1 Let f and h be elements in Z [1] contained, respectively in theorbits of length 28 and 35 of the action of L[1] on Z [1]. Let Z

[1]f and Z

[1]h be the

hyperplanes in Z [1] formed by the vectors perpendicular to f and h, respectively,


with respect to the unique non-zero symplectic form on Z [1] preserved by L[1].Then the following assertions hold:

(i) CΠ(Z [1]) = Π[01]11 and [Π, Z [1]] = Π[01]

12 ⊕ Π[01]21 ⊕ Π[01]

22 ;(ii) C[Π,Z[1]](Z

[1]f ) ≤ Π[01]

22 and dim C[Π,Z[1]](Z[1]f ) = 16;

(iii) C[Π,Z[1]](Z[1]h ) ≤ Π[01]

12 ⊕ Π[01]21 ;

(iv) dim CΠ[01]12

(Z [1]h ) = dim CΠ[01]

21(Z [1]

h ) = 12.

Proof The orbit of length 35 of L[1] on Z [1] is the intersection Q1 ∩Z [1] and theelements in this orbit are indexed by the 2-dimensional subspaces containedin U4, while the orbit of length 28 is the intersection Q2 ∩ Z [1]. Keeping inmind this observation it is easy to deduce the assertions from (3.6.3) in view of(6.5.2) and (6.5.4). Alternatively one can make use of the equalities χ(h) = 53and χ(f) = −11 implied by (6.2.5) and (6.4.2).

Let h be as in (6.8.1) so that h is a symplectic form of rank 2 on U4, whoseradical will be denoted by W . Notice that h is uniquely determined by W andthat

CG[01](h) = Z [1]A[1]B[1]M,

where M := L[1](W ) ∼= 24 : (Sym3 × Sym3) is the stabilizer of W in L[1]. WithZ

[1]h as in (6.8.1) put

Π(h)12 = CΠ[01]

12(Z [1]

h ), Π(h)21 = CΠ[01]

21(Z [1]

h ),

and

Π(h) = Π(h)12 ⊕ Π(h)

21 .

Since

420 = dim Π[01]α = dim Π(h)

α × [G[01] : CG[01](h)] = 12 × 35,

for α = 12 and 21, we have the following result analogous to (6.7.1).

Lemma 6.8.2 For α = 12 and 21 the action of G[01] on Π[01]α is induced from

the action of CG[01](h) on Π(h)α .

6.8.1 24-dim representations of CG[1](h)In this subsection we put

E[1] = CG[1](h)/Z [1]h ,

and describe a class of 24-dimensional representation of E[1] over the field of com-plex numbers. These representations can be considered as that of CG[1](h) withkernel Z

[1]h . After that we restrict a representation from this class to CG[01](h)

and compare the restriction with the action of CG[01](h) on Π(h)12 ⊕ Π(h)

21 .

Lifting Π[01]12 ⊕ Π[01]

21 101

First of all we investigate the structure of E[1]. Similarly to the previoussection we adopt the bar convention for the images in E[1] of elements andsubgroups of CG[01](h), and also for the homomorphisms induced by α, β, and ζ.Then in E[1] the key commutator relation of G[01] becomes

[α(w1), β(w2)] = zh(w1,w2),

where w1, w2 ∈ U4, z is the generator of Z [1] = Z [1]/Z[1]h and h is treated as

a symplectic form on U4.We have seen in Section 3.7 that G[1] is a semidirect product of Q[m] ∼= 211

and A[1]L[1] ∼= 24 : L4(2). The relevant action is described in (3.8.1) in terms ofthe Mathieu group M24 and its irreducible Todd module C11. Since Z

[1]h ≤ Q[m],

E[1] is a semidirect product of Q[m] =Q[m]/Z[1]h

∼= 26 and A[1]M ∼= A[1]M , where

M = L[1](W ) = NL[1](Z [1]h ) ∼= 24 : (Sym3 × Sym3).

Thus the structure of E[1] is completely determined by the action of A[1]M onQ[m]. One could determine the action implementing (3.8.1) and conducting relev-ant calculations in the Todd module, although here we follow a more elementaryapproach. First we describe the action of A[1] on Q[m].

Lemma 6.8.3 The following assertions hold where z denotes the generatorof Z [1]:

(i) Q[m] = 〈z, α(w)β(w), t1 | w ∈U4〉 ∼= 26 and [Z [1], A[1]] = 1;(ii) if v, w ∈U4 then [α(v), α(w)β(w)] is trivial unless 〈v, w〉 is a

2-dimensional subspace in U4 disjoint from W , in which case thecommutator is z;

(iii) [α(v), t1] = α(v)β(v) for every v ∈ U#4 .

Proof (i) is immediate from the definition of Q[m] before (3.7.1), while (ii)follows from the basic commutator relation. Since t1 (acting by conjugation)transposes α(v) and β(v) for every v ∈ U#

4 , (iii) follows.

We follow the notation for subgroups and elements of M =L[1](W ) introducedbetween the proofs of (3.3.5) and (3.3.6) (we still follow the bar convention).In particular T ∼= T (1) ×T (2) is a Sylow 3-subgroup of M , which stabilizes adirect sum decomposition U4 = W (1) ⊕ W (2), such that W (1) = W and W (i) =CU4(T

(3−i)); R is a Sylow 2-subgroup of NM (T ), R# = r(1), r(2), r(12) and〈r(i), T (3−i)〉 is abelian of order 6 for i= 1 and 2. The following lemma describesthe action of M on Q[m].


(i) [Z [1], M ] = 1;(ii) if v ∈ U4, m ∈M , and m(v) is the image of v under m, then m conjugates

α(v)β(v) onto α(m(v))β(m(v));


(iii) CM (t1) = O2(M)T 〈r(12)〉 ∼= 24 : 32 : 2, in particular [M : CM (t1)] = 2;(iv) [r(1), t1] = [r(2), t1] = z.

Proof Statement (i) is quite clear since Z [1] is normal in G[1], while (ii) is bythe definition of Q[m]. Finally (iii) and (iv) follow from (3.3.6). Notice that t1commutes with O2(M) modulo Z

[1]h .

The following corollary is a direct consequence of (6.8.3) and (6.8.4).

Corollary 6.8.5 The group A[1]M acts on the non-identity elements of Q[m]

with five orbits of lengths 1, 3, 3, 24, and 32. The orbits are the following:

Υ1 = z, Υ2 = α(w)β(w) | w ∈ W#,

Υ3 = α(w)β(w)z | w ∈ W#,

Υ4 = α(v)β(v), α(v)β(v)z | v ∈ U4 \ W and

Υ5 = α(v)β(v)t1, α(v)β(v)t1z | v ∈ U4.

If V =CQ[m](O2(A[1]M)), then V # = Υ1 ∪ Υ2 ∪ Υ3; D[1] is the only index 2subgroup in Q[m] normalized by A[1]M and D[1] = Q[m] \ Υ5.

Next we would like to enumerate the hyperplanes in Q[m] (treating Q[m]

as a 6-dimensional GF (2)-vector space) disjoint from Z [1] and calculate theirstabilizers.

Lemma 6.8.6 Let q be an orthogonal form on U4 associated with h in the sensethat

h(w, v) = q(w) + q(v) + q(w + v)

for all w, v ∈ U4 and let p ∈ 0, 1. Then every hyperplane in Q[m] disjoint fromZ [1] is of the form

0 ∪ α(v)β(v)zq(v), α(v)β(v)t1zq(v)+p | v ∈ U4.

Proof Let P be a hyperplane in Q[m] disjoint from Z [1]. Since Q[m] = P Z [1],for every v ∈ U4 exactly one of α(v)β(v) and α(v)β(v)z is in P . Since

α(v)β(v)α(w)β(w) = α(v + w)β(v + w)zh(v,w)

the mapping q : U4 → GF (2) defined by

q(v) =

0, if α(v)β(v) ∈ P ;1, if α(v)β(v)z ∈ P .

is an orthogonal form on U4 associated with h. This orthogonal form uniquelydetermines the intersection P ∩ D[1]. Since z, t1, t1z is the set of non-zerovectors of a 2-dimensional subspace in Q[m], P contains either t1 or t1z butnot both. Since there are 16 = 2dim U4 choices for q (compare (1.1.1)) and two

Lifting Π[01]12 ⊕ Π[01]

21 103

choices for p, the above construction gives the totality of 32 hyperplanes in Q[m]

disjoint from Z [1].

Lemma 6.8.7 The group A[1]M has exactly two orbits P(0) and P(1) on the setof hyperplanes in Q[m] disjoint from Z [1], described as follows:

(i) |P(0)| = 8, if q(0) is the form on U4, such that q(0)(v) = 0 if and only ifv ∈ W , then the two hyperplanes corresponding to q(0) are in P(0) andthe stabilizer of every one of them is

S(0) = α(W )O2(M)T 〈r(12)〉;

(ii) |P(1)| = 24, if q(1) is the form on U4, such that there is exactly onew

(1)1 ∈ W such that q(1)(w(1)

1 ) = 0 and q(1)(v) = 1 for every v ∈ W (2),then the two hyperplanes corresponding to q(1) are in P(1) and (assumingthat r(12) stabilizes w

(1)1 ) the stabilizer of each of the two hyperplanes is

S(1) = α(W )O2(M)T2〈r(12)〉;

(iii) the stabilizer of the pair of hyperplanes in (ii) as a whole is

S(2) = α(W )O2(M)T2R.

Proof It is straightforward to check that both q(0) and q(1) are orthogonalforms on U4 associated with h. Therefore by (6.8.6) in order to specify the cor-responding hyperplane we only have to decide whether t1 or zt1 is in thehyperplane. The stabilizers are easy to calculate using (6.8.3) and (6.8.4). Sincethen |P(i)|= [A[1]M : S(i)] for i = 0 and 1 and |P(0)| + |P(1)| = 8 + 24 = 32 isthe total number of hyperplanes in question, (i) and (ii) follow. The statement(iii) is also quite clear.

Let Σ be a 24-dimensional C-module for E[1] = CG[1](h)/Z [1]h such that

CΣ(Q[m]) = 0 and CΣ(P ) is 1-dimensional for every P ∈ P(1). Since

24 = dim Σ = |P(1)| × dim CΣ(P ),

by (6.3.1 (i)) Σ is induced from a 1-dimensional linear representation of thestabilizer of P in E[1]. By (6.8.7) the stabilizer is the semidirect product of Q[m]

and S(1). Therefore the structure of Σ is uniquely determined by the kernel KP

of the action of Q[m]S(1) on CΣ(P ). A subgroup KP in Q[m]S(1) is suitable forbeing the kernel if and only if

[Q[m]S(1) : KP ] = 2,

KP ∩ Q[m] = P .

By the former of the two conditions KP contains the commutator subgroup ofQ[m]S(1). The next lemma, which is a direct consequence of (6.8.7), describesthis commutator subgroup.


Lemma 6.8.8 Suppose that P ∈ P(1) is determined by the orthogonal form q(1)

as in (6.8.7 (ii)). Then the elements z, t1, α(w(1)1 ), r(12) generate in Q[m]S(1)

an elementary abelian subgroup of order 16 which complements the commutatorsubgroup of Q[m]S(1).

By the following lemma there are four possibilities for the kernel KP .

Lemma 6.8.9 Suppose that P is as in (6.8.7 (ii)) and that P contains t1. Thenin order to generate KP one should take

(a) the commutator subgroup of Q[m]S(1);(b) the element t1; and(c) one of the four subgroups of order 22 in 〈z, α(w(1)

1 ), r(12)〉 which aredisjoint from z.

6.8.2 Structure of Π(h)12

In this section we study the action of CG[01](h) on Π(h)12 and relate it to the action

of CG[1](h) on Σ described in Section 6.8.1.We follow notation introduced above in this section. To refresh the memory

we suggest the reader to take a note of the paragraph before (6.8.4). In addi-tion recall that d denotes the L[0]-invariant injection of the set of 2-dimensionalsubspaces of U5 into Q[0] ∼=

∧2U5 and that u is the unique vector from U5 \ U4

which is stabilized L[1] ∼= L4(2).Let V3 = 〈W, u〉. Then V3 is 3-dimensional and V3 ∩ U4 = W . Let P (V3)

be the hyperplane in Q[0] which corresponds to V3 and let z = d(W (2)). SinceV3 ∩ W (2) = 0, (3.6.3 (iii)) implies that z ∈ P (V3). According to the barconvention adopted since the beginning of Section 6.8.1 z is the generator of Z [1].

Notice that 〈T (2), r(2)〉 ∼= L2(2) ∼= Sym3 is the largest subgroup in L[0] whichstabilizes the subspace V3 vectorwise and the subspace W (2) as a whole. LetM (3) ∼= L3(2) be the subgroup of L[0](V3) which stabilizes W (2) vectorwise.

Let K1(V3) be the kernel of the action of G[0](V3) on Π[0]P (V3)

. Then by (6.4.5)and (6.6.2) we have the following:

K1(V3) = P (V3)O2(L[0](V3))T (2)〈zr(2)〉.

Furthermore both z and r(2) act on Π[0]P (V3)

as the (−1)-scalar operator, whileM (3) acts irreducibly. This implies the following.

Lemma 6.8.10 Let W be a 2-dimensional subspace in U4 and let h be the sym-plectic form on U4 whose radical is W . Let X3 be a 3-dimensional subspace inU5 such that X3 ∩ U4 = W . Then

K1(X3) ∩ Z [1] = P (X3) ∩ Z [1] = Z[1]h ,

in particular Π[0]P (X3)

< Π(h)12 .

Lifting Π[01]12 ⊕ Π[01]

21 105

Elementary calculations in the projective geometry of U5 show that for a fixed2-dimensional subspace W in U4 there are exactly four 3-dimensional subspacesX3 which satisfy the hypothesis of (6.8.10). Since Π[0]

P (X3)is 3-dimensional over

C and since by (6.8.1 (iv)) Π(h)12 is 12-dimensional, we obtain the following.

Lemma 6.8.11 The space Π(h)12 is the direct sum of four 3-dimensional

C-subspaces Π[0]P (X3)

taken for all 3-dimensional GF (2)-subspaces X3 in U5 suchthat X3 ∩ U4 = W .

So far we were able to get away without giving explicit names to all theelements of W = W (1) and W (2). It is getting to be more and more difficult, sowe are giving up. For i = 1 and 2 let

w(i)j | 1 ≤ j ≤ 3

be the set of non-zero vectors of the 2-dimensional subspace W (i).Let V

(0)3 = V3 and V

(j)3 = 〈W, u + w

(2)j 〉 for j = 1, 2, 3. Then it is easy to see

that

X = V(j)3 | 0 ≤ j ≤ 3

is the set of 3-dimensional subspaces in U5 intersecting U4 in W . Notice thatevery vector from U5 \U4 is contained in exactly one of the V

(j)3 ’s. For 0 ≤ k ≤ 3

put Φ(k) = Π[0]

P (V (k)3 )

. Then by (6.8.11) we have

Π(h)12 = Φ(0) ⊕ Φ(1) ⊕ Φ(2) ⊕ Φ(3).

Our next objective is to describe the action of D[1] on Π(h)12 and to calculate the

eigenspaces of D[1] in Π(h)12 . We know that z acts as the (−1)-scalar operator.

Next lemma describes the action of A[1] on Π(h)12 .


(i) for 1 ≤ j ≤ 3 the element α(w(1)j ) acts trivially on Π(h)

12 ;

(ii) for 1 ≤ j ≤ 3 the element α(w(2)j ) acts trivially on Φ(0) ⊕ Φ(j) and as the

(−1)-scalar operator on Φ(k) ⊕ Φ(l), where j, k, l = 1, 2, 3.

Proof By the definition, for w ∈ U#4 we have

α(w) = d(〈u, w〉).On the other hand, if X3 ∈ X and 〈u, w〉∩X3 is non-zero then α(w) ∈ P (X3) andα(w) acts trivially on Π[0]

P (X3); if 〈u, w〉∩X3 = 0 then α(w) acts on Π[0]

P (X3)as the

(−1)-scalar operator. Now the result follows from the definition of the w(l)j ’s.

Next we turn to the action of B[1] on Π(h)12 .


Lemma 6.8.13 Let 1 ≤ j ≤ 3. Then

(i) β(w(1)j ) stabilizes Φ(k) as a whole and acts on it with one eigenvalue 1

and two eigenvalues −1 for every 0 ≤ k ≤ 3;(ii) β(w(2)

j ) transposes Φ(0) and Φ(j), and also Φ(k) and Φ(l), where j, k, l =1, 2, 3.

Proof If w ∈ U#4 then β(w) is the transvection from L[0] whose centre is w and

whose axis U4. Put

Y = 〈T (1), β(w(1)j ) | 1 ≤ j ≤ 3, r(1)〉.

Then Y ∼= Sym4 and for every X3 ∈ X the subgroup Y stabilizes X3 as a wholeand acts faithfully on the vector-set of X3. This means that Y acts faithfully onΦ(k) for every 0 ≤ k ≤ 3. In addition, Y is a maximal parabolic subgroup inM (3). Now (i) follows from elementary properties of the 3-dimensional C-moduleΦ(k) of M (3) restricted to Y . The element β(w(2)

j ) is the transvection of U5 with

centre w(2)j and axis U4. Therefore it transposes u and u+w

(2)j , and also u+w

(2)k

and u + w(2)l , which gives (ii).

Since r(1) and β(w(1)1 ) are conjugate in M (3) we observe that r(1) acts on

Φ(k) with one eigenvalue 1 and two eigenvalues −1 for every 0 ≤ k ≤ 3.For 1 ≤ j ≤ 3 and 0 ≤ k ≤ 3 put Φ(k)

j = CΦ(k)(β(w(1)j )). Then by the proof

of (6.8.13 (i)) for 0 ≤ k ≤ 3

Φ(k) = Φ(k)1 ⊕ Φ(k)

2 ⊕ Φ(k)3

is the eigenspace decomposition of Φ(k) with respect to 〈β(w(1)j ) | 1 ≤ j ≤ 3〉.

For 1 ≤ j ≤ 3 put

Φj = Φ(0)j ⊕ Φ(1)

j ⊕ Φ(2)j ⊕ Φ(3)

j .

Since B[1] is abelian, Φj is a B[1]-submodule in Π(h)12 , and the dimension of Φj

is 4 (because of (6.8.11 (ii), (iii))). This gives the following.

Lemma 6.8.14 As a module for B[1], the subspace Π(h)12 possesses the direct

sum decomposition

Π(h)12 = Φ1 ⊕ Φ2 ⊕ Φ3,

whose summands are 4-dimensional.

We choose a basis in Π(h)12 with respect to which the elements of D[1] are

presented by monomial matrices. We achieve this by choosing consistently anon-zero vector e

(k)j from each of the Φ(k)

j ’s. We choose the e(0)j arbitrary and

take e(k)j to be the image of e

(0)j under β(w(2)

k ).The next lemma describes the action of D[1] on Φ1.

Lifting Π[01]12 ⊕ Π[01]

21 107


(i) α(w(1)1 )β(w(1)

1 ) acts on Φ1 trivially;(ii) α(w(1)

j )β(w(1)j ) acts on Φ1 as the (−1)-scalar operator for j = 2, 3;

(iii) the actions of the α(w(2)j )β(w(2)

j )’s for j = 1, 2, and 3 in the basis e(k)1 |

0 ≤ k ≤ 3 are described, respectively, by the following matrices:

0 1 0 01 0 0 00 0 0 −10 0 −1 0

,

0 0 1 00 0 0 −11 0 0 00 −1 0 0

and

0 0 0 10 0 −1 00 −1 0 01 0 0 0

.

Proof Statements (i) and (ii) follow from (6.8.12 (i)) and (6.8.13 (i)), while (iii)follows from (6.8.12 (ii)), (6.8.13 (ii)) and the definition of the e

(k)j ’s.

Calculating with the matrices in (6.8.15 (iii)) one checks that

f(0)1 = −e

(0)1 + e

(1)1 + e

(2)1 + e

(3)1

is an eigenvector of the action of D[1] on Π(h)12 . Generalising this observation we

obtain the following.

Lemma 6.8.16 For 1 ≤ j ≤ 3 and 0 ≤ k ≤ 3 put f(k)j =

∑3l=0(−1)δ(l,k)e

(k)j ,

where δ is the Kronecker delta function. Then Π(h)12 =

⊕3j=1⊕3

k=0 〈f (k)j 〉 is the

eigenspace decomposition of Π(h)12 with respect to the action of D[1].

The next and final lemma in this subsection relates the action of CG[01](h)on Π(h)

12 with that of CG[1](h) on Σ. Notice that A[1]M is both a complement toD[1] in CG[01](h) and a complement to Q[m] in CG[1](h) (compare (6.8.7), (6.8.8),and (6.8.9)). In the next lemma we follow notations used in (6.8.7)

Lemma 6.8.17 Let P (1) and P (2) be the hyperplanes of Q[m] as in (6.8.7 (ii))and let C be the commutator subgroup of D[1]S(1). Then

(i) CD[1](f (0)1 ) = P (1) ∩ D[1] = P (2) ∩ D[1];

(ii) the normalizer of CD[1](f (0)1 ) in A[1]M is S(2);

(iii) CA[1]M (f (0)1 ) = 〈C, α(w(1)

1 ), r(12), zr(1)〉.

Proof By (6.8.15) if 1 ≤ l ≤ 2 and 1 ≤ j ≤ 3 then α(w(l)j )β(w(l)

j ) negates f(0)1

unless l = j = 1 which gives (i). Clearly C centralizes f(0)1 . In addition, α(w(1)

1 )centralizes the whole of Π(h)

12 , in particular it centralizes f(0)1 . It was mentioned at

the beginning of this subsection that r(2) acts on Π(h)12 as the (−1)-scalar operator,

in particular r(2) negates f(0)1 . By the remark after the proof of (6.8.13), it follows

that for 0 ≤ k ≤ 3 the element r(1) acts on Φ(k) with one eigenvalue 1 and twoeigenvalues −1. Since r(1) transposes 〈f (k)

2 〉 and 〈f (k)3 〉, r(1) must negate f

(1)1 .


Hence r(12) = r(1)r(2) centralizes f(0)1 . Keeping in mind that z negates every

vector in Π(h)12 we observe that the proof is complete.

6.8.3 Structure of Π(h)21

In this subsection we turn our attention to the action of CG[01](h) on Π(h)21 . This

action will also be related to that of CG[1](h) on Σ as in Section 6.8.1.In the above terms let V1 be a 1-dimensional subspace in U4 and let g be a

non-singular symplectic form on U5/V1, which we consider as a symplectic formon U5 with radical V1 (compare Section 1.4). Then the restriction g|U4 of g toU4 is a non-zero symplectic from, whose radical is also non-zero (clearly thisradical contains V1). By (1.1.5 (i)) the radical of g|U4 is 2-dimensional. In thissubsection we consider the situation when this 2-dimensional subspace is W or,equivalently, that g|U4 = h.

Lemma 6.8.18 Let V1 be a 1-dimensional subspace in W and g be a symplecticform on U5 such that V1 is the radical of g and W is the radical of the restrictionof g to U4. Then

P (V1, g) ∩ Z [1] = Z[1]h ,

in particular Π[0]P (V1,g) < Π(h)

21 .

Proof The result follows from (3.6.3 (iv)) along with elementary geometricalconsiderations.

Let us describe the symplectic forms g on U5 such that the restriction of gto U4 coincides with h. As above let u denote the vector in U5 \ U4 stabilized byL[1]. Then g is uniquely determined by the values

g(u, w(l)j ) for 1 ≤ l ≤ 2, 1 ≤ j ≤ 3.

Lemma 6.8.19 Let g be a symplectic form on U5 whose radical is a1-dimensional subspace contained in W and whose restriction to U4 coincideswith h. Then

(i) g(u, w(1)j ) = 0 for exactly one j ∈ 1, 2, 3;

(ii) g(u, w(2)j ) = 0 either for all or for exactly one j ∈ 1, 2, 3.

Proof Since w(l)j | 1 ≤ j ≤ 3 is the set of non-zero vectors in a 2-dimensional

subspace, either all or exactly one of the three values g(u, w(l)j ) for 1 ≤ j ≤ 3 is

zero (here l ∈ 1, 2). Only if g(u, w(1)j ) would be zero for all j ∈ 1, 2, 3, the

whole of W = W (1) would be in the radical of g, which is not allowed.

Thus there are 12 forms g satisfying the hypothesis of (6.8.19) (and 12 is alsothe dimension of Π(h)

21 ). For 0 ≤ k ≤ 3 and 1 ≤ j ≤ 3 let g(k)j denote a form

Lifting Π[01]12 ⊕ Π[01]

21 109

satisfying the hypothesis of (6.8.19) and subject to the following:

g(k)j (u, w

(1)i ) = 0 if and only if i = j;

g(0)j (u, w

(2)i ) = 0 for all i ∈ 1, 2, 3;

if k = 0 then g(k)j (u, w

(2)i ) = 0 if and only if i = k.

For k and j from the above ranges put

Ψ(k)j = Π[0]

P (V1,g(k)j )

,

where V1 = 〈w(1)j 〉 is the radical of g

(k)j . Then by (6.8.18) Π(h)

21 is the direct sum of

the twelve Ψ(k)j ’s. We are going to use this decomposition to describe the action

of D[1] on Π(h)21 . We proceed in the usual manner starting with the remark that

z negates every vector in Π(h)21 .

Lemma 6.8.20 If v ∈ U#4 then the element α(v) centralizes Ψ(k)

j if g(k)j (u, v) =

0 and negates the vectors in Ψ(k)j otherwise.

Proof Recall that α(v) is the image in Q[0] of the 2-dimensional subspace 〈u, v〉under the L[0]-invariant injection d. By (3.6.3 (iv)) α(v) ∈ P (V1, g) if and onlyif 〈u, v〉 is totally singular with respect to g. Hence the result.


(i) if 1 ≤ j ≤ 3 then β(w(1)j ) stabilizes Ψ(k)

i as a whole for every 0 ≤ k ≤ 3and 1 ≤ i ≤ 3;

(ii) if j ∈ 1, 2, 3 and i ∈ 1, 2, 3 then β(w(2)j ) transposes Ψ(0)

i and Ψ(j)i ,

and also Ψ(l)i and Ψ(k)

i , where j, l, k = 1, 2, 3.

Proof Recall that for v ∈ U#4 the element β(v) is the transvection from L[0]

whose centre is v and whose axis is U4. Then the action of β(v) on the set ofsymplectic forms (satisfying the hypothesis of (6.8.19)) is quite clear. In fact, ifg is such a form and g(v) is the image of g under β(v), then

g(v)(u, w) = g(u + v, w) = g(u, w) + g(v, w) = g(u, w) + h(v, w)

for every w ∈ U4. Since W is the radical of h we immediately get (i). Now (ii)follows easily from the definition of the g

(k)j ’s.

To spare a letter let g(k)j denote also a non-zero vector in Ψ(k)

j . We assume

that for every 1 ≤ j ≤ 3 the vector g(k)j is the image under β(w(2)

k ) of the

vector g(0)j (compare the paragraph after (6.8.14)).


Recall that there are actually two possibilities for the module Π[0]2

corresponding to the two possible kernels K(1)2 and K

(2)2 of F1 = G[04](f) on Π2.

Furthermore in (6.4.6) we have chosen our notation so that

K(1)2 = P (V1, f)L[0](V1, f).

For 1 ≤ i ≤ 3 let Ψi = Ψ(0)i ⊕Ψ(1)

i ⊕Ψ(2)i ⊕Ψ(3)

i . Then the Ψi’s are D[1]-modulesand the next lemma (which is analogous to (6.8.15)) describes the action ofD[1] on Ψ1.

Lemma 6.8.22 Suppose that the kernel of G[04](f) on Π2 is K(ε+1)2 where ε = 0

or 1. Then the following assertions hold:

(i) α(w(1)1 )β(w(1)

1 ) acts on Ψ1 as (−1)ε-scalar operator;(ii) α(w(1)

j )β(w(1)j ) acts on Ψ1 as the (−1)ε+1-scalar operator for j = 2, 3;

(iii) the actions of the α(w(2)j )β(w(2)

j )’s for j = 1, 2, and 3 in the basis g(k)1 |

0 ≤ k ≤ 3 are described by the matrices as in (6.8.15 (iii)).

Proof The assertions follow from (6.8.20) and (6.8.21) in view of the observationthat the elements β(v) for v ∈ U#

4 are transvections from L[0] and therefore theyare contained in a G[0]-conjugate of K

(1)2 , but not of K

(2)2 .

Comparing (6.8.15) and (6.8.22) we notice that if ε = 1, then the elementα(w(1)

1 )β(w(1)1 ) has different characters on Π(h)

12 and Π(h)21 . Since this elements is

centralized by t1 and t1 must transpose Π(h)12 and Π(h)

21 , we arrive to the followingconclusion.

Proposition 6.8.23 In the considered situation, unless the kernel of F2 on Π2

is P (V1, f)L[0](V1, f), the action of G[01] on Π[01]12 ⊕ Π[01]

21 cannot be extended toan action of G[1].

From now on we assume that

K2 = P (V1, g)L[0](V1, f).

Then in (6.8.22) we have ε = 0, which makes the analogy between (6.8.15) and(6.8.22) is even more striking. The vector

h(0)1 = −g

(0)1 + g

(1)1 + g

(2)1 + g

(3)1

is an eigenvector of the action of D[1] on Π(h)21 and we are ready to prove the

following result analogous to (6.8.17).

Lemma 6.8.24 Let P (1) and P (2) be the hyperplanes of Q[m] as in (6.8.7 (ii))and let C be the commutator subgroup of D[1]S(1). Then

(i) CD[1](h(0)1 ) = P (1) ∩ D[1] = P (2) ∩ D[1];

(ii) the normalizer of CD[1](h(0)1 ) in A[1]M is S(2);

Lifting Π[01]12 ⊕ Π[01]

21 111

(iii) CA[1]M (h(0)1 ) = 〈C, α(w(1)

1 ), r(12), r(1)〉.

Proof Statements (i) and (ii) are immediate from (6.8.22). The element r(1)

is a transvections from L[0](g(k)j ) for 0 ≤ k ≤ 3, therefore it centralizes h

(0)1 ,

because of (6.8.23). We assume that r(2) stabilizes w(2)1 and transposes w

(2)2 and

w(2)3 . Then r(2) is a transvection from L[0] which stabilises g

(0)1 and g

(1)1 . Since

r(2), acting by conjugation, transposes β(w(2)2 ) and β(w(2)

3 ), we conclude the r(2)

transposes g(2)1 and g

(3)1 . Overall this shows that r(2) centralizes h

(0)1 and the

result follows.

6.8.4 Gluing

In this section we show that the action of CG[01](h) on Π(h)12 ⊕ Π(h)

21 extendsuniquely to an action of CG[1](h). With (6.8.17) and (6.8.24) in hands it is rathereasy to achieve.

Lemma 6.8.25 As above let t1 ∈ Q[m] \ D[1]. Then the following assertionshold:

(i) the subgroup Q[m]S(2) is normalized by t1;(ii) acting by conjugation, t1 transposes the centralizers in CG[01](h) of f

(0)1

and h(0)1 ;

(iii) t1 transposes (the characters of) the actions of CG[01](h) on Π(h)12 and

Π(h)21 .

Proof The action of t1 on CG[01](h), described in (6.8.3) and (6.8.4), gives (i).Compare (6.8.17 (iii)) and (6.8.24 (iii)). By (6.8.4 (iv)) t1 conjugates r(1) ontor(1)z; by (6.8.4 (iii)) t1 centralizes r(12) and finally, by (6.8.3 (iii)) t1 conjug-ates α(w(1)

1 ) onto β(w(1)1 ). Since α(w(1)

1 )β(w(1)1 ) centralizes both f

(0)1 and h

(0)1

(compare (6.8.7), (6.8.17 (i)), and (6.8.24 (i))), we obtain (ii). Now the actionsof CG[01](h) on Π(h)

12 and Π(h)21 are induced from 1-dimensional representations of

D[1]S(2) and by (ii) the element t1 acting by conjugation transposes the kernelsof these 1-dimensional representations.

Alternatively we can arrive to the same conclusion from the other side.

Proposition 6.8.26 Let Σ be a 24-dimensional module for CG[1](h) inducedfrom a 1-dimensional module of Q[m]S(1), whose kernel contains t1, α(w(1)

1 ) andr(12). Then the restriction of this module to CG[01](h) is isomorphic to the directsum Π(h)

12 ⊕ Π(h)21 .

Proof Just compare (6.8.9), (6.8.17 (iii)), and (6.8.24 (iii)).

Corollary 6.8.27 There is a unique G[1]-module Π[1]2 , whose restriction to G[01]

is isomorphic to Π[01]12 ⊕ Π[01]

21 . Furthermore the module Π[1]2 is induced from the

24-dimensional module Σ of CG[1](h) as in (6.8.26).


Proof The result is immediate from (6.8.2), (6.8.25), and (6.8.26).

6.9 The minimal representations of GAbove in this section we have classified the representation of the amalgamG = G[0], G[1] of minimal dimension 1333. Before stating the result of theclassification we recall a few definitions.

Let U3 be the 3-dimensional subspace of U5 stabilized by G[02]; let f be asymplectic form on U5 whose radical is the 1-dimensional subspace U1 stabilizedby G[04]. Let P1 = P (U3) and P2 = P (U1, f) be the corresponding hyperplanesin Q[0]. Let L[2] ∼= L3(2) be a complement to Q[2] in N [2], let z ∈ Q[0] \ P (U3)and let r be an involution from L[0](U3) ∼= 26 : (L3(2) × Sym3) which commuteswith L[2].

Let Π1 = Π1(λ) be a 3-dimensional module for G[02] whose kernel is

K1 = P (U3)O2,3(L[0](U3))〈zr〉,

and in which the action of L[2] affords the character λ. Let Π[0]1 = Π[0]

1 (λ) bethe G[0]-module induced from the G[02]-module Π1. Let Π2 be the 1-dimensionalmodule for G[0](f) = G[04](f) whose kernel is

K2 = P (U1, f)L[0](f).

Let Π[0]2 be the G[0]-module induced from the G[0](f)-module Π2.

Next we are going to describe a few G[1]-modules. We refer to Sections 3.7and 4.1 for definition and basic properties of some important subgroups in G[1].Notice that there is an action of G[1] on U4, where

O2(G[1]) = Q[1]〈t1〉 = Q[m]A[1] = Q[m]B[1]

is the kernel and L[1] ∼= L4(2) acts naturally.Recall that U3 is the 3-dimensional subspace in U4, stabilized by G[12] and

v ∈ U4 \ U3. The commutator subgroup of G[12] is N [2] = Q[2]L[2], where

Q[2] = Z [1]α(U3)β(U3)O2(L[1]),

and the elements α(v), β(v), and t1 generate a D8-subgroup which complementsN [2] in G[12].

For ε1 ∈ 0, 1 let Σ1 = Σ1(µ, ε1) be a 3-dimensional module for G[02] whosekernel is

〈Q[2], α(v)β(v), t1−ε11 , (β(v)t1)ε1〉

and the action of L[2] affords the character µ (the latter must be an irreduciblecharacter of degree 3). Let Π[1]

1 = Π[1]1 (µ, ε1) be the G[1]-module induced from

the G[12]-module Σ1.It is worth mentioning that t1 acts on Π[1]

1 (µ, ε1) as the (−1)ε1-scalar operator.Let h be a symplectic form on U4 whose radical is 2-dimensional. We con-

sider h as an element of Z [1] ∼=∧2

U4. Let Σ be a 24-dimensional module

The minimal representations of G 113

for G[1](h) = CG[1](h) described in Section 6.8.1, such that in terms of (6.8.9)the kernel KP contains α(w(1)

1 ) and r(12). Notice that the latter condition meansthat KP is the semidirect product of P and the stabilizer of P in A[1]L[1]. LetΠ[1]

2 be the G[1]-module induced from the G[1](h)-module Σ. Since Σ is inducedfrom a 1-dimensional module for the stabilizer in Q[m] of a hyperplane, the wholemodule Π[1]

2 is also induced from this representation.Let f be a non-singular symplectic form on U4. Then f also can be considered

as an element of Z [1] ∼=∧2

U4. The subgroup G[1](f) stabilizes in Q[m] exactlytwo hyperplanes Q

(0)f and Q

(1)f disjoint from f . For ε2 ∈ 0, 1 let Π[1]

3 = Π[1]3 (ε2)

be the G[1]-module induced from the 1-dimensional G[1](f)-module, whosekernel is

Q(ε2)f A[1]L[1](f).

Proposition 6.9.1 If λ = µ then the G[0]-module

Π[0]1 (λ) ⊕ Π[0]

2

and the G[1]-module

Π[1]1 (µ, ε1) ⊕ Π[1]

2 ⊕ Π[1]3 (ε2)

have isomorphic restrictions to G[01].

Proof The result is by (6.6.2), (6.7.7), and (6.8.26).

Proposition 6.9.2 Let m = m(G) be the smallest positive integer such thatthere exists a faithful completion ϕ : G → A, where A = GLm(C). Then

(i) m = 1333;(ii) for every completion ϕ as above the restrictions of ϕ to G[0] and G[1]

turn the underlying vector space into the modules

Π[0]1 (λ) ⊕ Π[0]

2 and Π[1]1 (λ, ε1) ⊕ Π[1]

2 ⊕ Π[1]3 (ε2),

respectively, where λ is one of the two 3-dimensional irreducible charac-ters of L[2] ∼= L3(2) and ε1, ε2 ∈ 0, 1;

(iii) up to conjugacy there are exactly eight representations of G of the minimaldimension 1333.

Proof Combining (6.2.1) and (6.9.1) we obtain (i). The statement (ii) followfrom (6.4.1), (6.6.1), and (6.8.23). Finally (iii) follows from the above in view of(6.4.2). In fact, in the considered situation

(a) χ[0] is a sum of two irreducible characters with degrees 465 and 868;(b) χ[1] is a sum of three irreducible characters with degrees 45, 840, and 448;(c) χ[01] is a sum of four distinct irreducible characters with degrees 45, 420,

420, and 448.


The obvious fusion pattern immediately shows that C [01] is factorized by C [0]

and C [1].

6.10 The action of G[2]

Let ϕ : G → GL1333(C) be a representation of G of the minimal degree 1333.By (6.9.1) and (6.9.2) we assume that ϕ = ϕ(λ, ε1, ε2) in the sense that therestrictions of ϕ to G[0] and G[1] turn the underlying 1333-dimensional vectorspace Π (over the complex numbers) into the modules Π[0] = Π[0]

1 (λ) ⊕ Π[0]2 and

Π[1] = Π[1]1 (λ, ε1) ⊕ Π[1]

2 ⊕ Π[1]3 (ε2), respectively. As usual put

G[2] = 〈ϕ(G[02]), ϕ(G[12])〉.

In the next section we show that under a suitable choice of ε1 and ε2 the com-pletion ϕ = ϕ(λ, ε1, ε2) is constrained at level 2. Recall that by the definitionthe latter means that

CG[2](N [2]) ≤ N [2].

Since the centre of N [2] is trivial the completion is constrained at level 2 if andonly if

C [2] := CG[2](N [2])

is the identity subgroup.As in (4.2.2) put

G[2] = G[2]/(N [2]CG[2](N [2])).

Since ϕ is a completion of G, by (4.8.5 (iii)), independently of the particularchoice of λ, ε1 and ε2, we have G[2] ∼= Sym5. Let us adopt the hat conventionfor the images in G[2] of elements and subgroups of G[2].


(i) G[02] ∼= G[02]/N [2] ∼= Sym3 × 2 ∼= D12;

(ii) G[12] ∼= G[12]/N [2] ∼= D8;

(iii) G[012] ∼= G[012]/N [2] ∼= 22;(iv) if t1 ∈ Q[m] \ D[1] then t1 belongs to the commutator subgroup of G[2]

isomorphic to Alt5.

Proof Statements (i) to (iii) are implicit in the first paragraph of Section 4.7.Statement (iv) is an immediate consequence of the following elementary observa-tion: every involution which stabilizes as a whole but not vertexwisely an edge ofthe Petersen graph is an even permutation (of the underlying 5-element set).

The action of G[2] 115

We are going to study the restriction to G[02] of the module Π[0] = Π[0](λ, ε1).First we consider the action of G[02] on the set of hyperplanes in Q[0] and provethe following result, which is analogous to (6.5.2).

Lemma 6.10.2 Put

P(2)11 = P (U3), P(2)

12 = P (V3) | dim (U3 ∩ V3) = 2,

P(2)13 = P (V3) | dim (U3 ∩ V3) = 1;

P(2)21 = P (V1, f) | f |U3 = 0, P(2)

22 = P (V1, f) | V1 < U3 and f |U3 = 0,

P(2)23 = P (V1, f) | V1 < U3.

Then

(i) P(2)11 , P(2)

12 and P(2)13 are the G[02]-orbits on P1;

(ii) |P(2)11 | = 1, |P(2)

12 | = 42, |P(1)13 | = 112;

(iii) the subgroup N [2]

(a) acts transitively on P(2)11 ;

(b) has three orbits on P(2)12 of length 14 each;

(c) acts transitively on P(2)13 ;

(iv) P(2)21 , P(2)

22 , and P(2)23 are the G[02]-orbits on P2;

(v) |P(2)21 | = 84, |P(2)

22 | = 112, |P(2)23 | = 672.

Proof It is a direct consequence of the main theorem of projective geometrythat the orbit under a general linear group L of a pair of subspaces from thenatural module of L is uniquely determined by the dimensions of the subspacestogether with the dimension of their intersection. This gives (i) and reduces (ii)to direct calculations.

Let us turn to (iii). Statement (a) is obvious. Let M be the kernel of theaction of L[0](U3) on the quotient U5/U3. Then N [2]Q[0] = MQ[0] and hencein order to establish (b) and (c) it is sufficient to calculate the orbits of Mon the set of 3-dimensional subspaces V3 in U5. First suppose that U3 ∩ V3 is2-dimensional. Then 〈U3, V3〉 is a hyperplane in U5 containing U3. On the otherhand, it is easy to see that M stabilizes each of the three hyperplanes in U5

containing U3. Therefore N [2] has at least three orbits on P(2)12 . Suppose that

〈U3, V3〉 = U4. The subgroup M induces on U4 the maximal parabolic subgroupisomorphic to 23 : L3(2). Clearly this subgroup acts transitively on U4\U3. SinceV3 is generated by U3 ∩ V3 and a vector from V3 \ (V3 ∩ U3), we conclude thatthere are exactly three orbits of N [2] on P(2)

12 and (b) follows. Finally, supposethat U3 ∩ V3 is 1-dimensional. Then V3 = 〈v, V2〉, where v is the non-zero vectorin U3 ∩ V3 and V2 is a complement to 〈v〉 in V3 which is also a complement toU3 in U5. It is easy to check that O2(M) ∼= 26 fixes all the vectors in U3 andacts regularly on the set of 2-dimensional subspaces in U5 disjoint from U3. Thisgives (c).


By the version (1.5.3) of Witt’s theorem the orbit of a subspace W underthe stabilizer of a symplectic form f is determined by the intersection of W withthe radical of f together with the restriction f |W of f to W . This gives (iv) andreduces (v) to some straightforward calculations (one might find it even easier tocompute the orbits of G[0](f) on the set of 3-dimensional subspaces in U5).

For a ∈ 1, 2 and b ∈ 1, 2, 3 put

Π[02]ab =

⊕P∈P(2)

ab

Π[0]P .

It is clear that Π[02]ab is a G[02]-module whose dimension is 3δ(1,a) · |P(2)

ab |.

Lemma 6.10.3 The following assertions holds:

(i) Π[02]11 is stable under G[2];

(ii) both G[02] and G[2] induce on Π[02]11 the group L3(2) × 2;

(iii) if ε1 = 1 then the completion ϕ is not constrained at level 2.

Proof By the paragraph before the lemma the subspace Π[02]11 is stable under

G[02]. On the other hand, if t1 is as in (6.10.1 (iv)) then G[2] = 〈G[02], t1〉.By (6.6.2) t1 acts on Π[01]

11 as a (±1)-scalar operator. Since Π[02]11 ≤ Π[01]

11 , (i)follows. Since Π[02]

11 is canonically isomorphic to the G[02]-module Π1 (from whichthe module Π[0]

1 was induced), we have (ii). Suppose that the completion isconstrained at level 2. Then

G[2] ∼= 23+12 · (L3(2) × Sym5)

(the pentad group). Hence in this case G[2] possesses a unique homomorphismonto a group of order 2 and by (6.10.1 (iv)) the element t1 is in the kernel ofthis homomorphism. On the other hand, if ε1 = 1 then t1 acts on Π[02]

11 as the(−1)-scalar operator. Hence (iii) follows.

Lemma 6.10.4 Let b = 2 or 3. Let V(b)3 ∈ P(2)

1b , so that

dim (U3 ∩ V(b)3 ) = 4 − b,

and let P (b) = P (V (b)3 ). Then

(i) both G[02](V (2)3 ) and N [2](V (2)

3 ) induce on ΠP (2) an irreducible action ofSym4 × 2;

(ii) G[02](V (3)3 ) induces on ΠP (3) an irreducible action of Sym4 × 2, while

N [2](V (3)3 ) induces on ΠP (3) an elementary abelian group of order 23;

(iii) under the action of G[02] the module Π[02]1b is irreducible of dimension

3 · |P(2)1b |;

The action of G[2] 117

(iv) under the action of N [2] the module Π[02]1b is the direct sum of three irre-

ducible pairwise non-isomorphic irreducible modules of dimension |P(2)1b |

each.

Proof Let G(b) = G[0](V (b)), let N (b) be the largest normal subgroup in G(b)

such that G(b)/N (b) is solvable, and let Q(b) = O2(N (b)). Then an element ofG[0] which maps U3 onto V

(b)3 conjugates N [2] onto N (b) and Q[2] onto Q(b).

Let K(b)d , K

(b)u , and K

(b)l be the kernels of the actions of G(b) on V

(b)3 , U5/V

(b)3 ,

and ΠP (b) , respectively. By the structure of the parabolic subgroups in L[0] ∼=L5(2) we have

G(b)/K(b)d

∼= L3(2), G(b)/K(b)u

∼= L2(2) ∼= Sym3,

while

G(b)/K(b)l

∼= L3(2) × 2, G(b)/N (b) ∼= Sym3 × 2

(compare (6.4.5) and the first paragraph of Section 4.7). Since

G(b)/Q(b) ∼= Sym3 × 2 × L3(2)

and Q(b) is contained in each of K(b)d , K

(b)u , K

(b)l , and N (b), we conclude that K

(b)d

contains K(b)l with index 2, and K

(b)u contains N (b) with index 2. In addition,

since

N (b) ∩ Q[0] = K(b)l ∩ Q[0] = P (b),

we have

K(b)d = Q[0]K

(b)l and K(b)

u = Q[0]N (b).

In the above terms in order to establish (i) and (ii) we have to calculate

(G[02] ∩ G(b))/(G[02] ∩ K(b)l ) and (N [2] ∩ G(b))/(N [2] ∩ K

(b)l )

for b = 2 and 3. Since P (U3) and P (b) are different hyperplanes in Q[0] andP (U3) < N [2] < G[2], by the above for X = G[02] or N [2]

(K(b)d ∩ X)/(K(b)

l ∩ X) ∼= (K(b)u ∩ X)/(N (b) ∩ X) ∼= 2.

In view of the obvious symmetry between V(b)3 and U3 we observe that N [2] is

a subgroup of index 2 in the kernel of the action of G[02] on U5/U3. Now, easygeometrical considerations show that for b = 2 and 3 the subgroup G[02] inducesSym4 on the vector-set of V

(b)3 , while N [2] induces on this set Sym4 if b = 2 and

the elementary abelian group of order 22 if b = 3. This gives the assertions (i)and (ii).

Now (iii) and (iv) follow from (i), (ii), (6.10.2 (i), (ii), (iii)), and (6.3.1 (ii),(iii)). Notice that the modules in (iv) have pairwise different kernels, thereforethey are clearly pairwise non-isomorphic.


Lemma 6.10.5 For every b ∈ 1, 2, 3 the action of G[02] on Π[02]2b is irreducible.

Proof The result immediately follows from (6.10.2 (iv), (v)) and (6.3.1 (ii)).

Let us take a closer look at Π, considering it as a 1333-dimensional module forN [2]. Let us discuss the irreducible constituents of N [2] involved in Π and theircharacters. By (6.10.3) Π[02]

11 is such a constituent of dimension 3. By (6.10.4 (iv))each of Π[02]

12 and Π[02]13 is the sum of three pairwise non-isomorphic irreducible

constituents of dimension 42 and 112, respectively. The group G[2] acts on N [2]

by automorphisms. This action induces an action on the set of conjugacy classesand hence also on the set of irreducible characters of N [2] (clearly N [2] is thekernel of the latter action).

For b = 2 and 3 let χ(b) be the character of an irreducible constituent of N [2]

in Π[02]1b . Then by (6.10.4 (iv))

χ(2)(1) = 42 and χ(3)(1) = 112.

Let X(b) be the set of images of χ(b) under the action of G[2]. Since the action ofN [2] on Π is the restriction of an action of G[2] we conclude that every characterfrom X(2) ∪ X(3) is afforded by an irreducible constituent of N [2] in Π. Further-more the irreducible constituents in Π[02]

1b correspond to three different charactersin X(b). Comparing (6.10.4 (iii)) and (6.10.4 (iv)), we conclude that these threecharacters in X(b) are transitively permuted by G[02]. Thus G[2] ∼= Sym5 actsfaithfully on X(b) for b = 2 and 3. In particular each of X(2) and X(3) containsat least 5 elements. By (6.10.5) for c ∈ 1, 2, 3 and b ∈ 2, 3 either all ornone of the characters of irreducible constituent of N [2] in Π[02]

2c are contained inX(b). Since |X(3)| ≥ 5 and the characters in X(3) have degree 112, we concludethat the restriction of Π[02]

23 to N [2] is the sum of six irreducible constituents ofdegree 112, whose characters are contained in X(3). By a divisibility conditionand since |X(2)| ≥ 5 we conclude that Π[02]

22 is the sum of two irreducible con-stituents, whose characters are contained in X(2). Finally it is easy to concludethat the character of Π[02]

22 must also be in X(3). In fact, otherwise it wouldbe impossible to define an action of G[2] ∼= Sym5 on X(3), whose restriction toG[02] ∼= Sym3 ×2 has an orbit of length 3 and an orbit of length 6. Thus we haveestablished the following main result of the section.

Proposition 6.10.6 Define the following G[02]-submodules in Π:

Π[2]1 = Π[02]

11 , Π[2]2 = Π[02]

12 ⊕ Π[02]21 , and Π[2]

3 = Π[02]13 ⊕ Π[02]

22 ⊕ Π[02]23 .

Let Ω be a 5-element set on which G[2] ∼= Sym5 acts naturally as the symmetricgroup. Then

Π = Π[2]1 ⊕ Π[2]

2 ⊕ Π[2]3

is a decomposition of Π into irreducible G[2]-submodules. Furthermore,

The centralizer of N [2] 119

(i) Π[2]1 restricted to N [2] stays irreducible;

(ii) Π[2]2 restricted to N [2] is the direct sum of five pairwise non-isomorphicirreducible constituents of dimension 42 each;

(iii) Π[2]3 restricted to N [2] is the direct sum of ten pairwise non-isomorphicirreducible constituents of dimension 112 each;

(iv) G[2] permutes the irreducible constituents in Π[2]2 as it permutes the

elements of Ω;(v) G[2] permutes the irreducible constituents in Π[3]

3 as it permutes the2-element subsets of Ω.

One can conclude from the above proposition that N [2]

(a) has two irreducible constituents in Π[02]21 of dimension 42 each;

(b) acts irreducibly on Π[02]22 ;

(c) has six irreducible constituents in Π[02]23 of dimension 112 each.

We could have deduced this conclusion earlier by computing the orbits ofN [2] on P2 (compare Exercise 3 at the end of the chapter).

6.11 The centralizer of N [2]

The completion ϕ(λ, ε1, ε2) is constrained at level 2 if and only if the centralizerC [2] of N [2] in G[2] is trivial. In order to get hold of C [2] it is useful to calculatethe centralizer of N [2] in the general linear group of Π or even in the ring ofall (possibly non-invertible) linear transformations of Π. The structure of thesecentralizers is immediate from (6.10.6) in view of Schur’s lemma. Before statingthe result we slightly improve our notation.

As in (6.10.6) let Ω be a set of 5-elements on which G[2] ∼= Sym5 acts as thesymmetric group. Let

Π[2]2 =

⊕ω∈Ω

Υ(2)ω and Π[2]

3 =⊕

π∈(Ω2)

Υ(3)π

be the direct sum decompositions into (pairwise non-isomorphic) irreducibleconstituents of N [2] as in (6.10.6 (iv)) and (6.10.6 (v)), respectively. Also putΠ[2]

1 = Υ(1)0 . Then

I = 0 ∪ Ω ∪(Ω

2)

is the index-set of the irreducible constituents of N [2] in Π and every such com-ponent is Υ(a)

ι for some a ∈ 1, 2, 3 and some ι ∈ I (notice that the possiblevalues of ι depend on a). There is a natural action of G[2] on I, where N [2]C [2]

is the kernel and G[2] = G[2]/N [2]C [2] ∼= Sym5 acts naturally.

Lemma 6.11.1 Let CL be the centralizer of N [2] in the general linear group ofΠ and CM be the centralizer of N [2] in the ring of all (possibly non-invertible)linear transformations of Π. Then


(i) as a module for G[2] the centralizer CM is isomorphic to the space ofcomplex-valued functions on I;

(ii) CL consists of the invertible elements in CM ;(iii) C [2] is a subgroup of CL.

Proof As an N [2]-module, Π is the sum of 16 = 1 + 5 + 10 pairwise non-isomorphic irreducible modules indexed by the set I. Hence the result isimmediate from Schur’s lemma.

In what follows an element f of CM which acts on Υ(a)ι as the f(ι)-scalar

operator will be identified with C-valued function on I which sends ι onto f(ι).Below we indicate the explicit fusion of the irreducible constituents of N [2]

into those of G[01]. For this purpose put Ω = 1, 2, 3, 4, 5 and suppose that G[02]

is the stabilizer in G[2] of 1, 2 while G[12] is the stabilizer of 1, 2, 3, 4.Then by (6.10.4), (6.10.5), (6.10.6), and the remark after (6.10.6) we have thefollowing:

Π[01]11 = Υ(1)

0 ⊕ Υ(2)5 ;

Π[01]12 = Υ(2)

3 ⊕ Υ(2)4 ⊕ Υ(3)

34 ⊕ Υ(3)35 ⊕ Υ(3)

45;

Π[01]21 = Υ(2)

1 ⊕ Υ(2)2 ⊕ Υ(3)

12 ⊕ Υ(3)15 ⊕ Υ(3)

25;

Π[01]22 = Υ(3)

13 ⊕ Υ(3)14 ⊕ Υ(3)

23 ⊕ Υ(3)24.

The elements t1 as in (6.10.1 (iv)) can be taken to act on Ω as the permutation(1, 3)(2, 4).

6.12 The fundamental group of the Petersen graph

As above let ϕ = ϕ(λ, ε1, ε2) be the completion map of G into A = GL1333(C).Let G be the subgroup of A generated by the image of ϕ, and let

Γ = Λ(G, ϕ, G)

the corresponding coset graph. Then Γ is connected of valency 31 and the naturalaction of G on Γ is locally projective of type (5, 2). Let Γ[2] be the geometricsubgraph in Γ of valency 3, stabilized by G[2] (this subgraph is connected by thedefinition).

The subgroup N [2] is the kernel of the action of G[2] on Γ[2]. Put G[2] =G[2]/N [2] and adopt the tilde convention for the images in G[2] of elements andsubgroups of G[2]. Let Γ[2] be the graph on the set of orbits of C [2] on Γ[2] inwhich two orbits are adjacent if there is an edge of Γ[2] which joins them. Let

ψ : Γ[2] → Γ[2]

be the mapping which assigns to a vertex of Γ[2] its orbit under C [2].

The fundamental group of the Petersen graph 121

Lemma 6.12.1 Let E = G[02], G[12] and E = G[02], G[12] be subamalgamsin G[2] and G[2], respectively. Then

(i) E ∼= E;(ii) Γ[2] ∼= Λ(E , id, G[2]) and Γ[2] ∼= Λ(E , id, G[2]), where id denotes the

identity map;(iii) Γ[2] is isomorphic to the Petersen graph;(iv) ψ is a covering of graphs which commutes with the action of G[2].

Proof Statement (i) follows from (6.10.1), while (ii) follows directly from thedefinition. Since G[2] ∼= Sym5, (iii) follows from (ii) and the definition of thePetersen graph. Finally, (iv) is by (i) and (ii).

The subgroup C [2] ∼= C [2] is the kernel of the natural homomorphism of G[2]

onto G[2] associated with the covering ψ. By (6.11.1 (iii)) C [2] is abelian. On theother hand by (6.12.1) C [2] is a factor group of the fundamental group of thePetersen graph. Thus we have the following.

Lemma 6.12.2 The subgroup C [2] is an abelian factor group of the fundamentalgroup of the Petersen graph.

Let us recall some basic facts about fundamental groups of graphs and theirquotients over commutator subgroups. These standard results can be found inany textbook on algebraic topology, for instance in (Vick 1994), although I preferthe brief summary in (Venkatesh 1998).

Let Ξ be an undirected connected graph which has n = |V (Ξ)| vertices andm = |E(Ξ)| edges. Consider every edge x, y as a pair of directed arcs (x, y)and (y, x). Let Θ be a spanning tree of Ξ. By the definition Θ is a connectedsubgraph in Ξ with no cycles which contains all the vertices. Then Θ containsexactly n − 1 edges. Let x be a vertex of Ξ called a base vertex. With every edgey, z of Ξ which is not in Θ we associate a cycle T by the following rule:

(a) choose an arc corresponding to y, z, say (y, z);(b) take the only path (x = x0, x1, . . . , xk = y) which joins x with y in Θ;(c) take the only path (xk+1 = z, xk+2, . . . , xl = x) which joins z with x;(d) the resulting cycle is T = (x0, x1, . . . , xk, xk+1, . . . , xl = x0) (T contains

the arcs (xi, xi+1) for 0 ≤ i ≤ l − 1).

The cycles obtained by the above procedure are called fundamental cycles.Thus there are

|E(Ξ)| − |V (Ξ)| + 1 = m − n + 1

fundamental cycles.The so-called Hurewicz homomorphism (cf. proposition 4.21 inVick (1994))

establishes an isomorphism between the quotient of the fundamental group of Ξ


over its commutator subgroup and the first homology group H1(Ξ) of Ξ. Thisisomorphism commutes with the natural action of the automorphism group ofΞ. In turn the homology group H1(Ξ) can be defined in the following way. LetEZ be the free abelian group on the set of arcs of Ξ modulo the identification(x, y) = −(y, x). Let VZ be the free abelian group on the set of vertices of Ξ. Let

∂ : EZ → VZ

be the homomorphism defined by ∂ : (x, y) → y − x. The following result isstandard.


(i) the kernel of ∂ is (isomorphic to) the homology group H1(Ξ) of Ξ;(ii) the sum hT of arcs over a cycle T in Ξ is an element of ker ∂;(iii) the elements hT taken for all fundamental cycles T (with respect to a

spanning tree and a base vertex) freely generate H1(Ξ) ∼= ker ∂.

From now on (till the end of the section) we assume that Ξ is the Petersengraph and that X = Aut (Ξ) ∼= Sym5 is the automorphism group of Ξ (thus Ξand X are aliases of Γ[2] and G[2], respectively).

If x is a (base) vertex of Ξ then a spanning tree Θ in Ξ can be obtained byremoving the six edges contained in Ξ2(x). This way we obtain six fundamentalcycles of length 5. By (6.12.3) this means that H1(Ξ) is a free abelian groupof rank 6 with free generators indexed by the cycles of length 5 in Ξ passingthrough x. In order to recover the X-module structure of H1(Ξ) we take all thetwelve cycles of length 5 in Ξ as generators (these generators are no longer free,of course). For each such 5-cycle T we fix one of the two possible orientationsof T and obtain an element hT of H1(Ξ) ∼= ker ∂ as in (6.12.3 (ii)). The sub-group HT in H1(Ξ) generated by hT is the infinite cyclic group (isomorphic tothe additive group Z of integers). The group X acts on H1(Ξ) permuting thesubgroups HT in the way it permutes the corresponding cycles of length 5 in Ξ.The stabilizer of HT with respect to this action coincides with X(T ) ∼= D10.The subgroup O2(X(T )) has order 5 and it centralizes HT ; an element fromX(T ) \ O2(X(T )) negates HT

∼= Z. This gives the following result.

Lemma 6.12.4 Let T be a 5-cycle in the Petersen graph and X(T ) ∼= D10 bethe stabilizer of T in

X = Aut (Ξ) ∼= Sym5.

Let HT be the 1-dimensional Z-module for X(T ) such that O2(X(T )) acts trivi-ally, while every element from X(T ) \ O2(X(T )) negates HT . Let T be theX-module induced from the X(T )-module HT .Then the following assertions hold


(where HT is canonically identified with a X(T )-submodule in T ):

(i) there is an X-homomorphism

λ : T → H1(Ξ);

(ii) the restriction of λ to HT is an isomorphism.

By (6.12.2) there is an X-homomorphism

µ : H1(Ξ) → CL,

whose image is C [2] ∼= C [2]. Let ν be the composition of λ and µ.

Lemma 6.12.5 If T is a 5-cycle in Ξ and f = ν(hT ) then f(ι) = ±1 for everyι ∈ I, in particular the order of ν(HT ) is at most 2.

Proof By (6.11.1) CM is the C-permutation module of X = G[2] ∼= Sym5 onI. It is elementary to check that both X(T ) and O2(X(T )) act on I with threeorbits of length 5 and one fixed element (the latter element is of course 0).Therefore we have the equality

CCM (X(T )) = CCM (O2(X(T ))).

In plain words the equality means that whenever an element from CM is cent-ralized by O2(X(T )), it is centralized by the whole of X(T ). On the other hand,every element g ∈ X \O2(X) negates hT and therefore it sends f onto its inversein CL. By the above g also centralizes f . Hence

f(ι) = f(ι)−1

for every ι ∈ I and the result follows.

Since T is generated by the images of HT under X, as an immediateconsequence of (6.12.5) we obtain the following crucial result.

Corollary 6.12.6 Let P be the subgroup in CL generated by the (±1)-valuedfunctions. Then

(i) P is an elementary abelian 2-group;(ii) as a GF (2)-module for G[2], P is isomorphic to

P1 ⊕ P5 ⊕ P10,

where P1, P5 and P10 are the GF (2)-permutation modules of G[2] on0, Ω and

(Ω2), respectively;

(iii) C [2] is isomorphic to a G[2]-submodule in P.


In order to restrict C [2] even further we once again make use of the fact thatit is the image of H1(Ξ) under an X-homomorphism µ. By (6.12.5) and (6.12.6)µ(h2) is the identity element for every h ∈ H1(Ξ). Put

H[2]1 (Ξ) = H1(Ξ)/〈h2 | h ∈ H1(Ξ)〉.

Then H[2]1 (Ξ) is the largest elementary abelian factor-group of H1(Ξ) and in view

of (6.12.6) µ induces an X-homomorphism

µ[2] : H[2]1 (Ξ) → P,

where P ∼= P1 ⊕ P5 ⊕ P10.The above definition of H1(Ξ) in terms of the map ∂ can be easily transfered

into the following definition of H[2]1 (Ξ). Let 2E(Ξ) be the power-space of E(Ξ).

This means that the elements of 2E(Ξ) are the subsets of E(Ξ) and the additionis performed by the symmetric difference operator. Let 2V (Ξ) be the power-spaceof V (Ξ) and ∂[2] be the map defined by

∂[2] : x, y → x ∪ y.

Then H[2]1 (Ξ) = ker ∂[2].


(i) H[2]1 (Ξ) is elementary abelian of order 26;

(ii) considering H[2]1 (Ξ) as an X-module,

(a) there are five orbits on the non-zero elements with lengths 6, 10, 12,15 and 20;

(b) there is a unique composition series

0 < M (1) < M (2) < H21 (Ξ),

where M (1) is the natural module of X ∼= PΣL2(4) and the remainingtwo composition factors are trivial 1-dimensional modules;

(c) the orbit of length 15 is in M (1) while the orbits of length 6 and 10constitute M (2) \ M (1).

Proof From the definition of H[2]1 (Ξ) it is easy to deduce that a collection F

of edges of Ξ is contained in ker ∂[2] = H[2]1 (Ξ) if and only if every vertex

x ∈ V (Ξ) is incident to an even number nF (x) of edges from F . Since Ξ is cubic,nF (x) ∈ 0, 2 and therefore F is either empty or the edge-set of a union ofdisjoint cycles. Elementary calculations in the Petersen graph Ξ show that F (ifnon-empty) is one of the following:

(1) a 5-cycle (there are 12 of them);(2) a union of two disjoint 5-cycles (there are 6 of them);(3) a 6-cycle (there are 10 of them);


(4) an 8-cycle (there are 15 of them);(5) a 9-cycle (there are 20 of them).

Furthermore, each of (1) to (5) is an X-orbit, which gives (ii) (a). With theexplicit form of H

[2]1 (Ξ) in hand the remaining statements are easy to check.

It is worth mentioning that H[2]1 (Ξ) is dual to the universal representation

module of the geometry of edges and vertices of the Petersen graph (comparesection 3.9 in Ivanov and Shpectorov (2002)).

Lemma 6.12.8 Let µ[2] : H[2]1 (Ξ) → P be an X-homomorphism. Then Im µ[2]

is either zero or a 1-dimensional submodule in

R := P(c)1 ⊕ P(c)

5 ⊕ P(c)10

(where P(c)m is the 1-dimensional submodule in Pm consisting of the constant-

valued functions).

Proof Let T1 and T2 be disjoint 5-cycles in Ξ, T0 = T1 ∪ T2. Let Fi be theedge-set of Ti for i = 0, 1, 2. Then the Fi’s are elements of H

[2]1 (Ξ). Let Xi be

the setwise stabilizer of Fi in X ∼= Sym5 for i = 0, 1, 2. Then

X1 = X2 ∼= D10, X0 ∼= F 45 and X1, X2 ≤ X0.

Therefore F1 and F2 are in the orbit of length 12 of X on H[2]1 (Ξ), while F0 is in

the 6-orbit. Since X0 acts transitively both on Ω and on(Ω2), we conclude that

µ[2](F0) ≤ R.

By (6.12.7 (ii)) this implies that µ[2](M (2)) ≤ R and either Im µ[2] ≤ R or(Im µ[2])R/R is a 1-dimensional X-submodule in P/R. It is easy to check thatthere are no 1-dimensional submodules in P/R. Hence µ[2](F1) ≤ R. SinceH

[2]1 (Ξ) is generated by the orbit of F1 under X and X centralizes R we conclude

that Im µ[2] is generated by µ[2](F1).

The final result of the section is the following proposition.

Proposition 6.12.9 Suppose that G[2] is defined with respect to ϕ = ϕ(λ, ε1, ε2).Then (depending on the particular choice of ϕ) one of the following holds:

(i) G[2] ∼= Sym5 and ϕ is constrained at level 2;(ii) G[2] ∼= Sym5 × 2 and C [2] is a 1-dimensional submodule in R.

Proof By the definition G[2] is an extension of C [2] by G[2] ∼= Sym5. By (6.12.8)

C [2] is of order 1 or 2. Since G[12] ∼= G[2] ∼= D8 is a Sylow 2-subgroup of G[2] theextension splits by Gaschutz’s theorem.


6.13 Completion constrained at level 2

In this section we complete the analysis of the subgroup G[2] corresponding toa completion ϕ = ϕ(λ, ε1, ε2) of G. We are interested in completions which areconstrained at level 2. Therefore in view of (6.10.3 (iii)) we assume that ε1 = 0.Let us make explicit a remark made in Section 6.9 after defining Π[1]

1 .

Lemma 6.13.1 Let t1 ∈ Q[m]\D[1] and suppose that ε1 = 0. Then t1 centralizesΠ[1]

1 .

Since N [2] ≤ G[1], Π[1]1 is an N [2]-module. Since dim Π[1]

1 = 45, we deducefrom (6.10.6) that

Π[1]1 = Υ(1)

0 ⊕ Υ(2)ω

for some ω ∈ Ω.

Lemma 6.13.2 There is a unique element ω ∈ Ω such that Υ(2)ω is normalized

by t1. Furthermore, Υ(2)ω is centralized by t1 if ε1 = 0 and it is negated by t1

otherwise.

Proof By (6.10.1 (iv)) t1 acts on Ω as an even involution, therefore it hasa unique fixed element. The second assertion is by (6.13.1).

At this stage it is convenient to make use of the fact that G[2] splits overN [2]. Let

N7 ∼= F 37

be the normalizer of a Sylow 7-subgroups in N [2]. For a subgroup Y of G[2]

containing N [2] let Y7 denote a complement to N7 in NY (N7)Y7.By the proof of (4.9.1) N7 is self-normalized in N [2]. By a Frattini argument

this implies that Y ∼= Y7. By (6.12.9) this gives the following result.

Lemma 6.13.3 If ϕ is constrained at level 2 then G[2]7

∼= Sym5, otherwise G[2]7

∼=Sym5 × 2.

Thus C[2]7 is of order 1 or 2. Our next goal is to show that provided ε1 = 0

the action of C[2]7 on

Υ(1)0

⊕ω∈Ω

Υ(2)ω

is trivial. Without loss of generality we assume that t1 commutes with N7. Weneed the following preliminary result.

Lemma 6.13.4 Suppose that G[2]7

∼= Sym5 × 2. Then

Completion constrained at level 2 127

(i) G[02]7

∼= Sym3 × 2 is contained in a Sym5-complement to C[2]7 in G

[2]7 ;

(ii) t1 is not contained in a Sym5-complement to C[2]7 in G

[2]7 .

Proof First notice that Sym5 ×2 contains exactly two subgroups isomorphic toSym5. Let E7 = G

[02]7 , G

[12]7 be the subamalgam in G

[2]7 . Then Γ[2] is canonically

isomorphic to Λ(E7, id, G[2]7 ). Since G

[2]7

∼= Sym5×2, Γ[2] is the standard bipartitedoubling of Ξ. This means that V (Γ[2]) = V (Ξ) × 0, 1 and (x, α), (y, β) ∈E(Γ[2]) if and only if x, y ∈ E(Ξ) and α = β. This graph is bipartite and thestabilizer of a part is a Sym5-complement, hence (i) follows. By (6.10.1 (iv)) t1is contained in O2(G[2]

7 ) ∼= Alt5, therefore it is contained in either both or noneof the Sym5-complements. Since G

[2]7 = 〈G[02]

7 , t1〉, (ii) follows.

Lemma 6.13.5 If ε1 = 0 then the subgroup C[2]7 acts trivially on Υ(1)

0 .

Proof By (6.10.3 (ii)) we know that G[2] induces on Υ(1)0 the group L3(2) × 2.

Clearly N [2] induces L3(2), therefore G[2]7 induces a group of order 2. Suppose

that G[2]7

∼= Sym5 × 2 (otherwise the claim is obvious). Then the kernel of theaction of G

[2]7 on Υ(1)

0 is isomorphic either to Sym5 or to Alt5 × 2. Since t1 iscontained in the kernel by (6.13.1) and by (6.13.4 (ii)) it is not contained in aSym5-subgroup, the kernel is Alt5 × 2 and in particular it contains C

[2]7 .

Lemma 6.13.6 If ε1 = 0 then the subgroup C[2]7 acts trivially on

Υ(2) :=⊕ω∈Ω

Υ(2)ω .

Proof Suppose the contrary. Then in view of (6.13.3) G[2]7

∼= Sym5 × 2 and thegenerator σ of C

[2]7 acts on Υ(2) as the (−1)-scalar operator.

We follow the explicit form of Ω and of the actions on Ω of G[02], G[12] andt1 as at the end of Section 6.12. In particular

t1 = (1, 3)(2, 4)

(we follow the hat convention for the images in G[2]). Let s1 be the uniqueelement from G

[02]7 such that

s1 = (1, 2)(3, 5).

Let us identify t1 and s1 with their images under ϕ and put r = s1t1. Thenr ∈ G[2],

r = (1, 5, 3, 2, 4)

and exactly one of the following holds:

(a) r5 centralizes Υ(1), r3 conjugates the action of t1 on Υ(2) onto that of s1,C [2] acts trivially on Υ(2);


(b) r5 = σ acts on Υ(2) as the (−1)-scalar operator, r3 conjugates the actionof t1 on Υ(2) onto that of s1σ, C [2] ∼= 2 and C [2] acts faithfully on Υ(2).

By (6.13.2) Υ(2)5 is the only irreducible constituent of N [2] in Υ(2) which is

stabilized by t1 and it is centralized by t1 (since ε1 is assumed to be 0). Thereforea conjugate of t1 in G[2] centralizes the only irreducible constituent of N [2] whichit stabilizes, while a conjugate of t1σ negates every vector in the constituent itstabilizes.

On the other hand, Υ(2)4 is the only irreducible constituent in Υ(2) stabilized

by s1. Comparing (a) and (b) above, we conclude that s1 either centralizes ornegates every vector in Υ(1)

4 in the respective cases (a) and (b). Thus in order toestablish the claim it is sufficient to show that s1 centralizes at least one vectorfrom Υ(1)

4 .Since t1 and s1 are conjugate in G[2], comparing the paragraph before (6.6.2)

we conclude that s1 is in the kernel K1 of G[02] on Π1. Let V2 be a 2-dimensionalsubspace which complements U3 in U5 and let v be a non-zero vector from V2.Suppose that N7 stabilizes the direct sum decomposition

U5 = U3 ⊕ V2.

Then V2 is the only 2-dimensional subspace in U5 which is stabilized by N7. Theelement s1 is the product τδ, where τ is the transvection from L[0] with centrev and axis 〈v, U3〉 while δ is the image of V2 under the L[0]-invariant mapping dof the set of 2-dimensional subspaces of U5 into Q[0] (compare the beginning ofSection 6.9).

Now we need to find a 3-dimensional subspace V3 in U5 such that

(1) Π[0]P (V3)

is contained in Υ(2)4 ;

(2) s1 centralizes Π[0]P (V3)

.

By (6.10.2) in order to satisfy (1) we must have dim (U3 ∩ V3) = 2 andV4 := 〈U3, V3〉 must be an s1-invariant hyperplane in U5. Put V4 = 〈U3, v〉 andtake V3 to be a 3-dimensional subspace in V4 which does not contain v and whichintersects U3 in a 2-dimensional subspace. An easy counting shows that such a3-dimensional subspace V3 exists. Under our choice V2 is disjoint from U3. By(3.6.3 (iii)) this means that δ = d(V2) is in Q[0] \ P (V3). Since V3 ≤ V4 and V4is the centre of the transvection τ , we have s1 = τδ ∈ G[0](V3). Furthermore,s1 is in the kernel of the action of G[0](V3) on Π[0]

P (V3)(compare the beginning of

Section 6.9). Hence the condition (2) is also satisfied and the result follows.

Now it only remains to show that for at least one of the completions ϕ(λ, 0, ε2)the centre C [2] of G[2] does not project onto

Υ(3) :=⊕

π∈(Ω2)

Υ(3)π .

Completion constrained at level 2 129

For accomplishing this we still have one life line left in the sense that we canchoose ε2 between 0 and 1.

As a direct consequence of (6.4.3 (ii)) and (6.7.7) we have the following usefulresult.

Lemma 6.13.7 Let ρ denote the element which centralizes

Π[01]11 ⊕ Π[01]

12 ⊕ Π[01]21

and negates every vector in Π[01]22 . Let t

(ε2)1 denote the image of t1 under

ϕ(λ, 0, ε2). Then

t(1−ε2)1 = t

(ε2)1 ρ.

Lemma 6.13.8 There is exactly one ε(0)2 ∈ 0, 1 such that the completion

ϕ(λ, 0, ε(0)2 )

is constrained at level 2.

Proof Let s1 ∈ G[0] be as in the proof of (6.13.6). As above we identify s1 withits image under the completion ϕ(λ, 0, ε2). Let γ be the element from P whichnegates every vector from Υ(3) and centralizes Υ(1)

0 ⊕ Υ(2). Then, arguing as inthe proof of (6.13.6), we observe that

(s1t(ε2)1 )5 = 1

if the completion ϕ(λ, 0, ε2) is constrained at level 2 and

(s1t(ε2)1 )5 = γ

otherwise. Applying (6.13.7) we check that

(s1t(1−ε2)1 )5 = (s1t

ε21 )5 ·

∏p∈〈 r 〉

ρp = (s1t(ε2)1 )5γ.

This immediately gives the result.

Thus we have a pair of completions ϕ(λ, 0, ε(0)) which are constrained atlevel 2. These completion differ by the choice of the irreducible character λ ofL[2] ∼= L3(2) of degree 3 among two algebraically conjugate such characters.Thus we have arrived to the following main result of the chapter.

Proposition 6.13.9 The amalgam G = G[0], G[1] possesses a faithful gener-ating completion which is constrained at level 2, that is a completion

ϕ : G → G,

such that

G[2] := 〈ϕ(G[02]), ϕ(G[12])〉 ∼= 23+12 · (L3(2) × Sym5).


The completion in (6.13.9) is ϕ(λ, 0, ε(0)) for λ being either λ1 or λ2. By theMain Theorem the completion group of a generating completion of G whichis constrained at level 2 is isomorphic to the fourth Janko group J4. Thusϕ(λ1, 0, ϕ(0)) and ϕ(λ2, 0, ϕ(0)) induce the algebraically conjugate pair of faithfulcomplex representations of J4 of the minimal degree 1333. This implies that theimage of ϕ(λ, 1, 1 − ε

(0)2 ) is J4 × 2 (compare Exercise 4 below).

Exercises

1. Prove (6.1.2), (6.4.1) and (6.4.2).2. Complete the proof of (6.2.4).3. Extend argument in the proof of (6.10.2) to calculate the N [2]-orbits

on P2.4. With ε

(0)2 as in (6.13.8) decide what is the group generated by the image of

ϕ(λ, 0, 1 − ε(0)2 ).

5. Determine the dimension of the minimal representation of the amalgam H =H [0], H [1] from O+

10(2). What are the groups generated by the correspondingimages?

7

GETTING THE PARABOLICS TOGETHER

From now on we assume that G is the completion group of G which is constrainedat level 2. We identify the third geometric subgroup with the famous involutioncentralizer

21+12+ · 3 · Aut (M22)

in J4. We also recover another important subgroup in G which is

211 : M24.

This enables us to associate with G a coset geometry D(G) which eventually willbe identified with the Ronan–Smith geometry for J4.

7.1 Encircling 21+12+ · 3 · Aut (M22)

Let ϕ : G → G be a faithful generating completion of the amalgam G whichis constrained at level 2. The existence of such a completion is guaranteed by(6.13.9). First we assume that ϕ : G → G is universal among the completionswhich are constrained at level 2. Since the centre of N [2] = K [2] is trivial we candefine such a completion in the following way (compare Section 2.5).

Let ϕ : G → G be the universal completion of G, ϕ : G → G be an arbitrarycompletion which is contrained at level 2, ψ : G → G be the correspondinghomomorphism of completions and Y be the kernel of ψ. Then the restrictionof ψ to

G[2] = 〈ϕ(G[02]), ϕ(G[12])〉

is a homomorphism onto

G[2] = 〈ϕ(G[02]), ϕ(G[12])〉

with kernel Y [2] = Y ∩ G[2]. Since ϕ : G → G is constrained at level 2, we have

CG[2](ϕ(N [2])) ≤ Z(ϕ(N [2])) = 1.

On the other hand, the restriction of ψ to ϕ(N [2]) is an isomorphism onto ϕ(N [2])and therefore,

Y [2] = CG[2](ϕ(N [2])).

If we want G to be the ‘largest’ completion group subject to the property thatit is constrained at level 2 we must take Y to be the smallest normal subgroup

131

132 Getting the parabolics together

in G which intersects G[2] in Y [2]. This means that Y should be taken to be thenormal closure in G of CG[2](ϕ(N [2])).

Alternatively we can define G to be the universal completion of the rank 3amalgam

J = G[0], G[1], G[2].

It is worth mentioning that the existence of the amalgam J is independent ofthe existence of completions of G which are constrained at level 2. In fact, J isthe amalgam

ϕ(G[0]), ϕ(G[1]), G[2]

factorised over CG[2](ϕ(N [2])). On the hand, J possesses a faithful completion ifand only if G possesses a completion constrained at level 2.

From now on (unless explicitly stated otherwise)

ϕ : G → G

is assumed to be an arbitrary faithful completion of G which is constrained atlevel 2. The amalgam G will be identified with its image in G under ϕ, so thatwe can plainly write

G[i] = 〈G[0i], G[1i]〉.

By (4.2.6) and (4.2.8) N [2] and N [3] are non-trivial, so by (4.2.1 (iv)) G[2] andG[3] are proper subgroups in G. On the other hand by (5.4.1) N [4] = 1 and inSection 7.4 we will show that G[4] is in fact the whole of G.

Let Γ = Λ(G, ϕ, G) be the coset graph corresponding to the completion ϕ :G → G. Let x and x, y be defined as in the paragraph before (4.1.1) so that

G = G(x), Gx, y.

Let Γ[2] and Γ[3] be the geometric subgraphs in Γ induced by the images of xunder G[2] and G[3], respectively (compare (4.2.1 (iii))). Since Γ[2] is of valency 3and G[2] induces on the vertex set of Γ[2] an action of G[2]/N [2] ∼= Sym5 on thecosets of G[02]/N [2] ∼= Sym3 × Sym2 the following statement is an immediateconsequence of the definition of the Petersen graph.

Lemma 7.1.1 Γ[2] is isomorphic to the Petersen graph.

Since the action of G[3] on Γ[3] is locally projective of type (3, 2) and Γ[2]

is a geometric cubic subgraph in Γ[3], (7.1.1), (5.2.3), and (11.4.3) imply thefollowing.

Lemma 7.1.2 One of the following two possibilities takes place:

(i) Γ[3] is the octet graph Γ(M22), CG[3](N [3]) = Z(N [3]) = Z [3] ∼= 2,G[3]/N [3] ∼= Aut (M22) and

G[3] ∼= 21+12+ · 3 · Aut (M22);

Tracking 211 : M24 133

(ii) Γ[3] is the Ivanov–Ivanov–Faradjev graph Γ(3 · M22), CG[3](N [3]) ∼= 2 × 3,G[3]/N [3] ∼= 3 · Aut (M22) and

G[3] ∼= (21+12+ × 3) · 3 · Aut (M22).

It will be proved in Section 7.4 that the possibility (7.1.2 (i)) takes place.Clearly G[3] is a completion of the amalgam

G[3] = G[03], G[13], G[23].

Since the completion ϕ : G → G is constrained at level 2, it is rather straight-forward to check that

CG[2](N [3]) ≤ Z(N [3]) = Z [3]

and therefore the amalgam G[3] = G[03]/N [3], G[13]/N [3], G[23]/N [3] is iso-morphic to the amalgam A[3] defined before (5.2.1). Hence by (5.2.1) G[3] isisomorphic to the amalgam Z as in (11.4.1).

Lemma 7.1.3 Let C [3] be the universal completion of G[3]. Then C [3]/N [3] isthe universal completion of G[3] ∼= Z, therefore C [3]/N [3] ∼= 3 · Aut (M22).

Proof Let K ∼= 3 · Aut (M22) be the universal completion of Z and let usidentify Z with its image in K. Let α be a homomorphism of G[3] onto Z whichis the composition of the canonical homomorphism g → gN [3] of G[3] onto G[3]

and an isomorphism of G[3] onto Z. Let

C [3] × K = (c, k) | c ∈ C [3], k ∈ K

be the direct product of C [3] and K and let X be the subset of C [3]×K consistingof the pairs (c, k), such that α(c) = k. Then X is isomorphic to G[3]. Furthermoreif X is the subgroup in C [3] × K generated by X then the restriction to X ofthe canonical homomorphism of C [3] × K onto K is surjective and the claimfollows.

By (7.1.3) if G[3] is the universal completion of G[3] then the possibility (ii) in(7.1.2) takes place. Therefore there is no way we can get down to the possibility(i) looking at the amalgam G only and some further subgroups of G should bebrought into play.

7.2 Tracking 211 : M24

In Sections 3.7 and 3.8 we have seen that G[1] is a semidirect product ofQ[m] ∼= 211 and A[1]L[1] ∼= 24 : L4(2). The relevant action is isomorphic to theaction of the octad stabilizer in M24 on the irreducible Todd module C11. InSection 7.4 we will prove the following.

Proposition 7.2.1 Let G[m] be the subgroup in G generated by the normalisersof Q[m] in G[1], G[2], and G[3]. Then

G[m] ∼= 211 : M24,


more specifically G[m] is the semidirect product of Q[m] ∼= C11 and M24 withrespect to the natural action.

For i = 1, 2 and 3 put G[mi] = NG[i](Q[m]), G[m] = G[mi] | 1 ≤ i ≤ 3 andG[m] = G[mi]/Q[m] | 1 ≤ i ≤ 3, so that G[m] is the quotient of G[m] over Q[m].Notice that G[m1] = G[m].


(i) G[m1]/Q[m] ∼= 24 : L4(2)(ii) G[m2]/Q[m] ∼= 26 : (L3(2) × Sym3);(iii) either

(a) (7.1.2 (i)) takes place and G[m3]/Q[m] ∼= 26 : 3 · Sym6, or(b) (7.1.2 (ii)) takes place and G[m3]/Q[m] ∼= (26 × 3) : 3 · Sym6.

Proof Statement (i) follows is directly from (3.8.1). In order to establish (ii) welocate Q[m] inside G[2]. It is clear that Q[m] is contained in G[2] (for instancebecause [G[1] : G[12]] = 15 is odd and Q[m] is a normal 2-subgroup in G[1]).By (4.2.7) |N [2] ∩ Q[m]| = 29 and the image of Q[m] in G[2]/N [2] ∼= Sym5 is anelementary abelian subgroup of order 4, which stabilizes an edge of Γ[2] as awhole but not vertexwisely. This means that Q[m]N [2]/N [2] is contained in thecommutator subgroup of G[2]/N [2], isomorphic to Alt5.

Let S7 be a Sylow 7-subgroup in G[2], C ∼= Sym5 be the complement to S7 inCG[2](S7) (compare (4.9.1)) and R be the elementary abelian subgroup of order4 in C, such that RN [2]/N [2] = Q[m]N [2]/N [2]. We claim that R is containedin Q[m]. In fact, by (3.7.1) G[1]/Q[m] ∼= 24 : L4(2) and since [G[1] : G[12]] = 15is not divisible by 7, Q[m] is normalized by a Sylow 7-subgroup in G[2]. BySylow’s theorem without loss we assume that this subgroup is S7. By (4.9.1) (ii)CQ[2](S7) = 1. Therefore

CQ[m](S7)N [2]/N [2] = Q[m]N [2]/N [2]

and the claim follows. Next we claim that Q[m] = RCQ[2](R). Since R ≤ Q[m]

and Q[m] is abelian, Q[m] is obviously in the centralizer of R in Q[2] and hencewe only have to show that |CQ[2](R)| is at most 29. The subgroup CG[2](R)contains S7, therefore CQ[2](R) is normalized by S7. Clearly CQ[2](R) containsZ [2] and therefore every dent of Q[2] is either completely contained in CQ[2](R)or intersects CQ[2](R) in Z [2]. In addition, since R commutes with S7 and everydent is the direct sum of two non-isomorphic S7-modules, whenever R normalizesa dent, it necessarily centralizes it. Now it only remains to recall that by (4.7.4)C acts on the set of dents as it acts on the edge-set of the Petersen graph Γ[2].Finally, R stabilizes exactly three edges of Γ[2] (these edges form the antipodaltriple containing x, y).

By the above paragraph the number of conjugates of Q[m] in G[2] is equal tothe number of conjugates of R in C (which is five). Since

NC(R) ∼= Sym4∼= 22 : Sym3,

(ii) follows.

Tracking 211 : M24 135

The stabilizer in G of an edge e = u, v of Γ is a conjugate of G[1] andby (3.7.1) this stabilizer contains a unique normal elementary abelian subgroupQe of order 211, which is of course a conjugate of Q[m]. By the above paragraphwhenever two edges e and f are contained in a common geometric cubic subgraphΣ ∼= Γ[2] and are antipodal in the line graph of Σ, the equality

Qe = Qf

holds (notice that there are 15 edges in Σ and only 5 different conjugates ofQ[m] in G[2]). Let Φ = ΦΓ be the local antipodality graph of Γ, so that Φ is agraph on the edge-set of Γ in which two edges are adjacent if they are containedin a common geometric cubic Petersen subgraph Σ and are antipodal in theline graph of Σ. Then Qe = Qf whenever e and f are in the same connectedcomponent of Φ.

Let us turn to (iii). It is clear that Q[m] ≤ G[3]. On the other hand, sinceQ[3] = O2(G[3]) ∼= 21+12

+ is extraspecial, while Q[m] is elementary abelian,|Q[m] ∩ Q[3]| ≤ 27 by (1.6.7). Let Ψ = ΦΓ[3] be local antipodality graph ofΓ[3] and Ψc be the connected component of Ψ containing x, y. Since Γ[3] iseither the octet graph or the Ivanov–Ivanov–Faradjev graph, by (11.4.4) andthe paragraph after that lemma Ψc contains 15 or 45 edges of Γ[3] dependingon whether we are in case (a) or (b). By the above paragraph the stabilizer Sof Ψc in G[3] is contained in G[m3]. Furthermore, S contains N [3] and S/N [3]

is 24 : Sym6 and (24 × 3) · Sym6 in the respective cases (a) and (b). SinceQ[m]N [3]/N [3] = O2(S/N [3]), using the well-known fact that Kh = NK(O2(Kh))for the stabilizer Kh

∼= 24 : Sym6 of a hexad in K ∼= Aut (M22), we concludethat S is the whole of G[m3], which completes the proof of (iii).

Lemma 7.2.3 Suppose that (7.1.2 (i)) takes place. Then

(i) the coset geometry corresponding to the embedding into G[m]/Q[m] of theamalgam

G[mi]/Q[m] | 1 ≤ i ≤ 3

is described by the locally truncated diagram

Ct4 :

2

2

2 ;

(ii) G[m]/Q[m] ∼= M24;(iii) G[m] splits over Q[m].

Proof First notice that the assertion (7.2.2 (iii) (a)) holds. Calculating theintersections of the G[mi]’s we obtain (i). Now (ii) is by (i) and (11.2.1), while(iii) is by (ii), (3.7.1) and Gaschutz’s theorem.


7.3 P -geometry of G[4]

In this section for a subsequence α of 0123 we denote the subgroup G[α4]

by F [α]. This convention also applies when α is empty, so that F = G[4].Let F = F [0], F [1] be the corresponding subamalgam in F , let Ξ = Λ[4] =Λ(G[4], ϕ[4], F ) be the coset graph associated with the completion

ϕ[4] : G[4] → F

(which is the restriction of ϕ to G[4]). At this stage we do not know yet that Fis the whole of G, but at any event the action of F on Ξ is faithful (since N [4] istrivial by (5.4.1)) and locally projective of type (4, 2). Let u, v be the edge ofΞ such that

F = F (u), Fu, v,

where F is identified with its image in F under ϕ[4]. For i = 2 and 3 let Ξ[i] bethe geometric subgraph in Ξ induced by the images of u under F [i] and let I [i]

be the vertexwise stabilizer of Ξ[i] in F .


(i) F [0] = G[04] ∼= 24+4 : 26 : L4(2);(ii) F [1] = 〈G[014], t1〉 ∼= 26+5+6 · (L3(2) × 2) ∼= 211 : 21+6

+ : L3(2);(iii) F [2] ∼= 23+12 · (Sym4 × Sym5), Ξ[2] is the Petersen graph and I [2] ∼=

23+12 × Sym4;(iv) F [3] ∼= 21+12

+ · 3 · Aut (M22), Ξ[3] is the Ivanov–Ivanov–Faradjev graphand I [3] ∼= 21+12

+ .

Proof Since F [0] and F [1] are the stabilizers of U1 in G[0] and G[1], respectively(i) and (ii) are quite clear. In terms of Section 3.8 F [1] is a semidirect product ofQ[m] ∼= 211 and the stabilizer of U1 in A[1]L[1] ∼= 24 : L4(2). The latter stabilizercoincides with the centralizer of a central involution in L[0] ∼= L5(2), isomorphicto 21+6

+ : L3(2). We know that F [2] is the subgroup in G[2] generated by G[024] andG[124]. The set P of geometric subgraphs of valency 7 in Γ containing Γ[2] is ofsize 7 (of course Γ[3] ∈ P). The action (isomorphic to L3(2)) of N [2] on P inducesa structure of the projective plane of order 2. Then F [2] is the stabilizer in G[2]

of a line in that projective plane structure which gives (iii).The subgroup F [3] is generated by G[034] and G[134]. Since Q[3] = O2(N [3]) =

N [3] ∩ G[014] we immediately conclude that

I [3] ≥ Q[3] ∼= 21+12+ .

On the other hand, the whole of N [3] could not be in I [3] since it is not evenin G[014]. The action of F [3] on Ξ[3] is locally projective of type (3, 2) andby (ii) Ξ[2] is a geometric cubic subgraph in Ξ[3] isomorphic to the Petersengraph. By (11.4.3) this implies that Ξ[3] is either the octet graph Γ(M22) of the

P -geometry of G[4] 137

Ivanov–Ivanov–Faradjev graph Γ(3 · M22). Let

χ : G[3] → Out Q[3]

be the natural homomorphism. By (5.2.4) the image of χ is isomorphic to3 · Aut (M22). For α = 0 and 1 the subgroups G[α34] and N [3] ∼= 21+12

+ : 3factorize G[α3] and hence by (5.2.1 (i), (ii)) we have

χ(F [03]) ∼= 23 : L3(2) × 2, χ(F [13]) ∼= (21+4+ : Sym3 × 2) · 2.

Since χ(G[3]) does not split over O3(χ(G[3])),

3 · Aut (M22) ∼= χ(G[3]) = χ(F [3]) = 〈χ(F [03]), χ(F [13])〉

and therefore F [3]/I [3] possesses a homomorphism onto 3 · Aut (M22). Thus (iv)follows.

Let G(G[4]) be the geometry, whose elements of type 1 are the vertices of Ξ,the elements of type 2 are the edges of Ξ, the elements of type 3 are the geometriccubic subgraphs in Ξ and the elements of type 4 are the geometric subgraphs ofvalency 7 in Ξ; the incidence relation is via inclusion. As a direct consequenceof (7.3.1) we obtain the following

Proposition 7.3.2 The geometry G(G[4]) is a P -geometry of rank 4 with thediagram

P4 :1 P

2

2

2.

The group G[4] acts on G(G[4]) faithfully and flag-transitively. The residue inG(G[4]) of an element of type 4 is isomorphic to the geometry G(3 · M22).

In Section 7.4 we will show that G[4] is the whole of G and by the MainTheorem the latter is J4. Therefore the geometry in (7.3.2) is the P -geometryG(J4) of J4 first constructed in (Ivanov 1987).

For 1 ≤ i ≤ 3 put F [mi] = F [i] ∩ Q[m] and F [m] = 〈F [mi] | 1 ≤ i ≤ 3〉.


(i) F [m1]/Q[m] ∼= 21+6+ : L3(2) and F [m1] splits over Q[m];

(ii) F [m2]/Q[m] ∼= 26 : (Sym4 × Sym3);(iii) F [m3]/Q[m] ∼= 26 : 3 · Sym6;(iv) the coset geometry M corresponding to the embedding of the amalgam

F [mi]/Q[m] | 1 ≤ i ≤ 3 into F [m]/Q[m] is described by the tilde diagram

T3 :2 ∼

2

2;

(v) F [m]/Q[m] ∼= M24 and M ∼= G(M24);(vi) Q[m] is the irreducible Todd module C11;(vii) F [m] splits over Q[m].


Proof A mere comparison of (7.2.2) and (7.3.1) gives (i) to (iii). The diagram ofM can be recovered by direct calculating the intersection of the F [mi]’s. Altern-atively one can employ the following combinatorial realization of M. Let Υ = ΦΞbe the local antipodality graph of Ξ. Then, arguing as in the proof of (7.2.2), onecan see that F [m] coincides with the stabilizer in F of the connected componentΥc of Υ containing u, v. The elements of M are the vertices of Υc (which areedges of Ξ), the intersections of the vertex set of Υc with edge-sets of geometricsubgraphs of valency 3 and 7. Then by (7.3.2) and the paragraph after (11.4.4)we obtain the desired diagram.

By (11.2.2) the assertions (i) to (iv) imply that F [m]/Q[m] is either M24 orHe. Since Q[m] is a non-trivial module in which F [m3] stabilizes the 1-dimensionalsubspace Z [3] the latter possibility is excluded, since the index of 26 : 3 ·Sym6 inHe is 29, 155 (cf. Conway et al. (1985) and Section 11.2) and hence (v) follows.The subgroup F [m2] stabilizes in Q[m] the 2-dimensional subspace Z [2] whichcontains the 1-dimensional subspace Z [3] stabilized by F [m3]. In terms of Ivanovand Shpectorov (2002) this means that Q[m] is a quotient of the universal rep-resentation group of M ∼= G(M24), so that (vi) follows from Proposition 4.3.1 inIvanov and Shpectorov (2002). Finally (vii) follows from (i) in view of Gaschutz’stheorem.

It is worth mentioning that the proof of (7.3.3 (v)) is the only place in thepresent volume where we essentially make use of a result (which is (11.2.2))whose proof relies on computer-aided calculations.

7.4 G[4] = G

First we show that G[m4] = G[m] (recall that G[m4] ∼= 211 : M24 by (7.3.3 (v),(vi), (vii)).

Lemma 7.4.1 Suppose that G[m4] ∼= 211 : M24 is a proper subgroup in G[m].Then the coset geometry N corresponding to the embedding into G[m]/Q[m] ofthe amalgam

G[mi]/Q[m] | 1 ≤ i ≤ 4

is described by the rank 4 tilde diagram

T4 :2 ∼

2

2

2.

Proof We claim that under the hypothesis (7.1.2 (ii)) takes place. In fact,otherwise

G[m] ∼= G[m4] ∼= 211 : M24

by (7.2.3), (7.3.3 (v), (vi), (vii)) and the order comparison. Then the structureof the G[mi]’s can be read from (7.2 (i), (ii), (iii) (b)), and (7.3 (v), (vi), (vii)).Calculating the intersections we get the diagram.

Maximal parabolic geometry D 139

Lemma 7.4.2 G[m4] = G[m].

Proof If the claim fails then by (7.4.1) G[m]/Q[m] acts flag-transitively on arank 4 tilde geometry N . In terms of Ivanov and Shpectorov (2002) this geometryis of truncated M24-type and it does not exist by Proposition 12.4.6 and 12.5.1in Ivanov and Shpectorov (2002).

Proof of Proposition 7.2.1 The result is now immediate by (7.3.3) and(7.4.2).

Lemma 7.4.3 The possibilities (7.1.2 (i)) and (7.2.2 (iii) (a)) take place, so that

G[3] ∼= 21+12+ · 3 · Aut (M22) and G[3m]/Q[m] ∼= 26 : 3 · Sym6.

Proof By (7.2.1) G[m]/Q[m] ∼= M24 and the latter group just does not containsubgroups as in (7.2.2 (iii) (b)) already by Lagrange theorem.

We are ready to prove the main result of the section.

Proposition 7.4.4 G[4] = G.

Proof By (7.4.1) G[m] = G[m4] ≤ G[4]. Also G[1] ≤ G[4] since G[1] ≤ G[m] (asremarked before (7.2.2)). But G[0] is generated by G[01] and G[04], and so alsoG[0] ≤ G[4]. This clearly implies G[4] = G.

We refer the reader to sections 9.5, 9.6 in Ivanov (1999) for general discussionabout the existence/non-existence of geometric subgraphs.

Lemma 7.4.5 Let Y be a Sylow 3-subgroup in O2,3(G[3]). Then

(i) CG[3](Y ) ∼= 6 · M22 is a non-split central extension of a cyclic group oforder 6 by M22;

(ii) G[3] does not split over Q[3] = O2(G[3]) ∼= 21+12+ .

Proof Since Y is a Sylow 3-subgroup of N [2] the result is by (4.9.1 (iii)).

By (7.4.5) the Schur multiplier of M22 possesses the cyclic group of order 6as a factor-group. In 1976, when (Janko 1976) was published this cyclic groupwas believed to be the whole Schur multiplier of M22. In (Mazet 1979) themultiplier of M22 was proved to be the cyclic group of order 12.

7.5 Maximal parabolic geometry DWe start this section by summarizing the information about the action of G onΓ we have obtained so far.


Proposition 7.5.1 Let G be a completion of the amalgam G which is constrainedat level 2 and let Γ be the coset graph associated with this completion. Then

(i) Γ is connected of valency 31 and the action of G on Γ is locally projectiveof type (5, 2);

(ii) G(x) = G[0] ∼= 210 : L5(2);(iii) Gx, y = G[1] ∼= 26+4+4 · (L4(2) × 2) ∼= 211 : 24 : L4(2);(iv) the geometric cubic subgraph Γ[2] is isomorphic to the Petersen graph

and

GΓ[2] = G[2] ∼= 23+12 · (L3(2) × Sym5);

(v) the geometric subgraph Γ[3] of valency 7 is isomorphic to octet graph and

GΓ[3] = G[3] ∼= 21+12+ · 3 · Aut (M22);

(v) there are no geometric subgraphs of valency 15 and G[4] := 〈G[04], G[14]〉is the whole of G;

(vi) if Φ = ΦΓ is the local antipodality graph of Γ and Φc is the connectedcomponent of Φ containing x, y then Φc is isomorphic to the octadgraph Γ(M24) and

GΦc = G[m] ∼= 211 : M24.

Proof (i) and (ii) are already in (4.1.1), (iii) is by (7.1.1), (iv) is by (7.4.3), (v)is by (7.4.4). Finally (vi) is by (7.2.3) since (7.1.2 (i)) takes place by (7.4.3).

Let F(G) be a geometry such that

(0) the elements of type 0 are the vertices of Γ;(1) the elements of type 1 are the edges of Γ;(2) the elements of type 2 are the geometric cubic subgraphs;(3) the elements of type 3 are the geometric cubic subgraphs of valency 7;

(i) the incidence relation is via inclusion.

Then it is immediate from (7.5.1) that F(G) belongs to the locally truncatedPetersen diagram

P t5 :

1 P

2

2

2 .

By the Main Theorem G ∼= J4 so F(G) is another geometry for J4 constructedin (Ivanov 1987).

More fruitful for our current purposes is the geometry D = D(G) whoseelements are as in F(G), only instead of the elements of type 1 (which are theedges of Γ) we take elements of type m which are the connected components ofthe local antipodality graph Φ of Γ. The incidence relation between the elementsof type 0, 2, and 3 is as in F(G). A connected component of Φ (an element oftype m) is adjacent to an element f ∈ F(G) if f is incident in F(G) to an edgeof Γ contained in that connected component.

Residues in D 141

Since G is generated by G[0] and G[1] it is a standard result that both F(G)and D(G) are connected.

7.6 Residues in DLet D = D(G) be the geometry defined in Section 7.5. Recall that the set oftypes of D is m, 0, 2, 3. For i ∈ m, 0, 2, 3 the set of elements of type i onD will be denoted by D[i]. Often we will write the type of an element above its

name, for instance we write3d for an element d of type 3. The stabilizer in G of

this element will be denoted by G[3]d . The residue in D of an element a (whose

type will be clear from the context) will be denoted by Da.Recall that a path in D is a sequence π = (a0, a1, a2, a3, . . . , as) of its elements

such that ai is incident to ai+1 but neither equal nor incident to ai+2 for every1 ≤ i ≤ s − 1. In this case s is the length of π.

With every elementia∈ D we associate a certain combinatorial/geometrical

structure (whose isomorphism type depends on i only). Then the residue Da ofa in D and the stabilizer G

[i]a of a in G possess natural descriptions in terms of

this structure. This works in the following way:Type m: If

ma is an element of type m then there is a Witt design W

[24]a of type

S(5, 8, 24). If Ba, Ta and Sa are the octads, trios and sextets of W[24]a , then

D[0]a = Ba × GF (2), D[2]

a = Ta, D[3]a = Sa.

The incidence relation in Da is via the refinement relation on the correspondingpartitions of the element set of W

[24]a . For instance suppose that (B, α) ∈ D[0]

a

and S ∈ D[3]a , where B is an octad of W

[24]a (identified with the partition of

the set of 24 elements into the octad B and its complement), α ∈ GF (2) andS is a sextet. Then (B, α) and S are incident if and only if B is the union oftwo tetrads from S. In particular, α does not effect the incidence. The stabilizerG

[m]a is the semidirect product of the automorphism group M

[m]a

∼= M24 of W[24]a

and the irreducible Todd module Q[m]a

∼= C11. The module Q[m]a is considered

as a section of the GF (2)-permutation module of M[m]a on the set of elements

of W[24]a . In particular M

[m]a has two orbits on the set of non-zero vectors in

Q[m]a ; the elements in one of the orbits are indexed by pairs of elements of W

[24]a ,

while those from the other orbit are indexed by the sextets from Sa. Dually, thehyperplanes in Q

[m]a are indexed by the octads from Ba and by the complementary

pairs of dodecads. The subgroup Q[m]a = O2(G

[m]a ) is the kernel of the action of

G[m]a on D[2]

a ∪ D[3]a . Every orbit of Q

[m]a on D[0]

a is of the form (B, 0), (B, 1),where B ∈ Ba. An element q ∈ Q

[m]a fixes this orbit elementwise if and only if

q is in the hyperplane corresponding to B. For every α ∈ GF (2) the complementM

[m]a stabilizes (B, α) | B ∈ Ba as a whole and acts on it as it acts on Ba.


Type 0: If0b is an element of type 0 then there is a 5-dimensional vector space Vb

over GF (2) such that

D[3]b =

[Vb

2

], D[2]

b =[Vb

3

], D[m]

b =[Vb

4

],

(where[Vb

i

]stands for the set of i-dimensional subspaces in Vb). The incidence

relation in Db is by inclusion. The subspace corresponding to an element x inDb will be denoted by Vb(x). The stabilizer G

[0]b is the semidirect product with

respect to the natural action of the general linear group L[0]b

∼= L5(2) and theexterior square Q

[0]b

∼= 210. The latter is the kernel of the action of G[0]b on Db,

while G[0]b /Q

[0]b

∼= L[0]b acts in the natural way. If b is the vertex x of Γ as in (7.5.1

(ii)) then Vb = U5, G[0]b = G[0] etc.

Type 2: If2c is an element of type 2 then there is a Petersen graph Θc and a

3-dimensional GF (2)-vector space Z[2]c such that

D[3]c =

[Z

[2]c

1

], D[0]

c = V (Θc),

and D[m]c is the set of antipodal triples of edges of Θc (considered also as 6-element

subsets of V (Θc)). Every element from D[3]c is incident to every element from

D[0]c ∪D[m]

c while the incidence between the elements from D[0]c and the elements

from D[m]c is via inclusion. The stabilizer

G[2]c

∼= 23+12 · (L3(2) × Sym5)

is isomorphic to the pentad group. Furthermore, Q[2]c = O2(G

[2]c ) is the kernel of

the action of G[2]c on Dc; N

[2]c

∼= 23+12 : L3(2) is the kernel of the action of G[2]c

on Θc. Let L[2]c

∼= L3(2) be a complement to Q[2]c in N

[2]c and let S

[2]c

∼= Sym5

be a complement to N[2]c in G

[2]c (recall that Q

[2]c is not complemented in G

[2]c ).

If c is the geometric cubic subgraph Γ[2] in Γ as in (7.5.1 (iv)), then G[2]c = G[2],

Q[2]c = Q[2] etc.

Type 3: If3d is an element of type 3 then there is a Witt design W

[22]d of type

S(3, 6, 22) associated with d. If Od is the set of octets, Hd is the set of hexadsand Pd is the set of pairs in W

[22]d then

D[0]d = Od, D[m]

d = Hd, D[2]d = Pd

with the incidence relation as in the geometry H(M22). The stabilizer G[3]d is of

the form

G[3]d

∼= 21+12+ · 3 · Aut (M22)

Intersections of maximal parabolics 143

and N[3]d = O2,3(G

[3]d ) is the kernel of the action of G

[3]d on the residue Dd. Let

M[3]d

∼= 6 · Aut (M22) be the normalizer in G[3]d of a Sylow 3-subgroup Y

[3]d in

N[3]d (compare (7.4.5)). Then M

[3]d ∩ Q

[3]d = Z

[3]d and Q

[3]d M

[3]d = G

[3]d (where

Q[3]d = O2(G

[3]d ) and Z

[3]d = Z(Q[3]

d )). If d is the geometric subgraph Γ[3] ofvalency 7 in Γ as in (7.5.1 (v)) then G

[3]d = G[3], N

[3]d = N [3] etc.

It is immediate from the above that D(G) belongs to the following diagram(cf. Section 10.4 for the definitions of the relevant rank 2 residues). Instead oftypes next to every node we indicate the structure of the corresponding stabilizerin G.

2 2

5

d t

3 2 m

6

21+12+ · 3 · Aut (M22) 23+12 · (L3(2) × Sym5) 211 : M24

0

210 : L5(2)

7.7 Intersections of maximal parabolics

Suppose thatix and

jy are incident elements in D. We require a clear understand-

ing of the structure of the intersection G[i]x ∩ G

[j]y in terms of the chief factors

of G[i]x . This information, as summarized in lemmas below, is not so difficult to

deduce, keeping in mind that

M = G[m], G[0], G[2], G[3]

is the amalgam of maximal parabolic subgroups associated with the action of Gon D.

The action of G[m]a on Da follows from the results in Sections 7.2 and 7.3,

particularly from (7.2.2) and (7.3.3).

Lemma 7.7.1 Let a ∈ D[m] and let G[m]a

∼= 211 : M24 be the stabilizer of a inG. Then

(i) if b ∈ D[0]a then

(1) b = (B, α), where B is an octad from Ba and α ∈ 0, 1;


(2) the subgroup M[m]a (b) ∼= 24 : L4(2) stabilizes a unique hyperplane

Pa(B) in Q[m]a ;

(3) G[m]a ∩ G

[0]b = Pa(B) : M

[m]a (b);

(4) Q[0]b = CPa(B)(O2(M

[m]a (b)))O2(M

[m]a (b)) ∼= 210;

(ii) if c ∈ D[2]a then

(5) c is a trio from Ta;(6) the subgroup M

[m]a (c) ∼= 26 : (L3(2) × Sym3) stabilises in Q

[m]a a

unique subgroup Ra(c) of index 4;(7) G

[m]a ∩ G

[2]c = Q

[m]a : M

[m]a (c);

(8) Q[2]c = Ra(c)O2(M

[m]a (c)) ∼= 23+12.

(iii) if d ∈ D[3]a then

(9) d is a sextet from Sa;(10) if Y is a Sylow 3-subgroup of O2,3(M

[m]a (d)), where M

[m]a (d) ∼= 26 :

3 · Sym6, then Q[m]a /[Q[m]

a , Y ] ∼= 25;(11) G

[m]a ∩ G

[3]b = Qa : M

[m]a (d);

(12) Q[3]d = [Q[m]

a , Y ]O2(M[m]a (d)) ∼= 21+12

+ ;(13) Y is a Sylow 3-subgroup of O2,3(G

[3]d ).

An element b of type 0 in D(G) is a vertex of the locally projective graph Γ.The edges containing b are in the natural bijection with the elements of type m

incident to b in D(G). Therefore the action of G[0]b on Db is isomorphic to the

action of H [0] on the corresponding residue in the dual polar space O+(10, 2)(cf. (2.1.2), (2.1.3)).

Lemma 7.7.2 Let b ∈ D[0] and G[0]b

∼= 210 : L5(2) be the stabilizer of bin G.

(i) If x ∈ D[i]a for i = m, 2, or 3 then

(1) x is a subspace in Vb of dimension 4, 3, or 2, respectively;(2) G

[0]b ∩ G

[i]x = Q

[0]b : L

[0]b (x), where L

[0]b (x) is isomorphic to

24 : L4(2), 26 : (L3(2) × Sym3) and 26 : (L3(2) × Sym3)

in the respective three cases;(3) Q

[i]x ∩ G

[0]b = [Q[0]

b , O2(L[0]b (x))]O2(L

[0]b (x));

(4) if x is of type m then O2(G[m]x ) intersects G

[0]b in a subgroup of index

2 in O2(G[m]x ) and Q

[0]b /[Q[0]

b , O2(L[0]b (x))] ∼= 24;

(5) the subgroup O2(G[i]x ) is contained in G

[0]b for i = 2 and 3, while

Q[0]b /[Q[0]

b , O2(L[0]b (x))] is isomorphic to 2 and 23 in the respective

cases.

The next result follows from the properties of the pentad group establishedin Sections 4.8 and 4.9.

Intersections of maximal parabolics 145

Lemma 7.7.3 Let c ∈ D[2] and let G[2]c

∼= 23+12 · (L3(2) · Sym5) be the stabilizerof c in G. Then

(i) if a ∈ D[m]c then

(1) a is an antipodal triple in the Petersen graph Θc;(2) S

[2]c (a) ∼= Sym4;

(3) G[2]c ∩ G

[m]a = N

[2]c S

[2]c (a);

(4) Q[m]a = C

Q[2]c

(O2(S[2]c (a)))O2(S

[2]c (a));

(5) Z[2]c ≤ Q

[m]a ;

(ii) if b ∈ D[0]c then

(6) b is a vertex of Θc;(7) S

[2]c (b) ∼= Sym3 × 2;

(8) G[2]c ∩ G

[0]b = N

[2]c S

[2]c (b);

(9) Q[0]b = C

Q[2]c

(O2(S[2]c (b)))O2(S

[2]c (b));

(iii) if d ∈ D[3]c then

(10) d is a 1-dimensional subspace in Z[2]c ;

(11) L[2]c (d) ∼= Sym4;

(12) G[2]c ∩ G

[3]d = Q

[2]c S

[2]c L

[2]c (d);

(13) Q[3]d = C

Q[2]c

(O2(L[2]c (d)))O2(L

[2]c (d)).

The structure of G[3]d and its action on Dd follows from results in Sections 5.2,

7.3, and 7.4.

Lemma 7.7.4 Let d ∈ D[3] and let G[3]d

∼= 21+12+ · 3 · Aut (M22) be the stabilizer

of d in G. Then

(i) if a ∈ D[m]d then

(1) a is a hexad from Hd;(2) M

[3]d (a) ∼= 25 : 3 · Sym6;

(3) G[3]d ∩ G

[m]a = Q

[3]d M

[3]d (a);

(4) Q[m]a = C

Q[3]d

(O2(M[3]d (a)))O2(M

[3]d (a)).

(ii) if b ∈ D[0]d then

(5) b is an octet from Od;(6) M

[3]d (b) ∼= 2 × Sym3 × 23 : L3(2);

(8) G[3]d ∩ G

[0]b = Q

[3]d M

[3]d (b);

(9) Q[0]b = C

Q[3]d

(O2(M[3]d (b)))O2(M

[3]d (b)) ∼= 210;

(iii) if c ∈ D[2]d then

(10) c is a pair from Pd;(11) M

[3]d (c) ∼= 26 : 3 : Sym5;


(12) G[3]d ∩ G

[2]c = Q

[3]d M

[3]d (c);

(13) Q[2]c = [O2(G

[3]d ), O2(M

[3]d (c))]O2(M

[3]d (c)) ∼= 23+12.

Exercises

1. Let J = G[0], G[1], G[2] be the amalgam defined in Section 7.1. Show thatthe actions of NG[1](Q[m]) and NG[2](Q[m]) on Q[m] generate the Mathieugroup M24.

2. Show directly that the amalgam F [m1]/Q[m], F [m2]/Q[m], F [m3]/Q[m] as in(7.3.3) is isomorphic to A(M24).

3. Give a computer-free proof of the simple connectedness of the rank 3 tildegeometry G(M24).

8

173,067,389-VERTEX GRAPH ∆

In this chapter we study the graph ∆ on the set of elements of type m inD(G) (with stabilizers 211 : M24) in which two vertices are adjacent wheneverin D(G) they are incident to a common element of type 2 (whose stabilizer isthe pentad group 23+12 · (L3(2) × Sym5)). The ultimate goal is to show thatthe number of vertices in ∆ is as given in the title of the chapter. The numberof vertices gives the order of G; already half way through we establish the sim-plicity of G. Thus the chapter completes the proof of the Main Theorem. Wefollow section 8 in (Ivanov and Meierfrankenfeld 1999) which is mainly due toU. Meierfrankenfeld.

8.1 Defining the graph

The graph ∆ we will study in this chapter is defined as follows: the vertices are theelements of type m in the geometry D(G); two such distinct elements are adjacentwhenever in D(G) they are incident to a common element of type 2. Here G isan arbitrary faithful completion group of the rank 2 amalgam G = G[0], G[1]which is constrained at level 2 (equivalently G is a faithful completion of therank 3 amalgam J = G[0], G[1], G[2]); D(G) is the coset geometry of G definedas in Section 7.5.

We can define ∆ in terms of the locally projective graph Γ of valency 31 asin (7.5.1). Let Σ be a (geometric) Petersen subgraph in Γ. Let e and f be twodistinct edges of Σ which are not antipodal in the line graph of Σ. Then theconnected components Φe and Φf of the local antipodality graph Φ of Γ (thesecomponents are elements of type m in D(G) and therefore they are verticesof ∆) are adjacent in ∆. Unfortunately our knowledge of the structure of Γ isnot sufficient already for accomplishing the next obvious step: calculating thevalency of ∆. We have to show that Σ is the only geometric Petersen subgraphin Γ whose edge-set intersects both Φe and Φf . Within our current knowledgethis appears not at all obvious. This forces us to give up on Γ (at least for a timebeing) and make use of the structure of maximal parabolics in G as reviewed inSection 7.7. Notice that ∆ is connected since Γ is such.

If X is a subset of elements of D consisting, say of a, b, . . . then the vertexwisestabilizer of X in G will be denoted by Gab..., while the set of elements of type i

in D incident to every element in X will be denoted by D[i]ab....

Proposition 8.1.1 Let a ∈ V (∆) and e ∈ ∆(a). Then

(i) the set D[2]ae contains a unique element, say c;

(ii) N[2]c ≤ Gae and Gae/N

[2]c

∼= Sym3;

147

148 173,067,389-Vertex graph ∆

(iii) Gac = Q[m]a Gace;

(iv) Q[m]a ∩ Q

[m]e = Z

[2]c ;

(v) Q[m]a Q

[m]e = Q

[m]a O2(M

[m]a (c));

(vi) Q[m]a acts on ∆(a) with orbits of length 4 indexed by the trios in Ta;

(vii) the valency of ∆ is

∆(a) = |Ta| · 4 = 3795 · 4 = 15,180 = 22 · 3 · 5 · 11 · 23.

Proof Since a and e are adjacent vertices of ∆, by the definition there is atleast one c ∈ D[2]

ae . The set D[m]c consists of five vertices (including a and e). The

group G[2]c induces on D[m]

c the natural action of Sym5 with kernel N[2]c . Since

Q[m]a stabilizes c, it stabilizes D[m]

c as a whole. By (7.7.3 (i) (4)) Q[m]a induces

on D[m]c an elementary abelian group of order 4 which permutes regularly the

vertices in D[m]c \a with kernel Ra(c). Thus Q

[m]a acts on ∆(a) with orbits of

length 4 and Ra(c) is the kernel at one of the orbits. The subgroup Ra(c) isnormalized by

M [m]a (c) ∼= 26 : (L3(2) × Sym3)

and the latter is a maximal subgroup in M[m]a

∼= M24 by (11.2.3 (i)). ThereforeGac = Q

[m]a M

[m]a (c) is the normalizer of Ra(c) in G

[m]a . Since Q

[m]a ∩ G

[m]e =

Q[m]a ∩ Q

[2]c we have

Gae ≤ NG

[m]a

(Ra(c)) = Gac.

Therefore Gae stabilizes c. Since G[2]c acts on D[m]

c doubly transitively, the actionof Gae on D[2]

a ∩D[2]e is transitive. Hence c is unique as claimed in (i). Statements

(ii) and (iii) follow from the above discussions.The subgroup Q

[m]a ∩Q

[m]e is normalized by both Q

[m]a and Q

[m]e . On the other

hand, the actions of Q[m]a and Q

[m]e on D[m]

c generate Alt5. It follows from thestructure of the pentad group (4.7.4) that Alt5 permutes transitively the dents inQ

[2]c , hence the subgroup Q

[m]a ∩Q

[m]e is contained in Z

[2]c . By (7.7.3 (i) (5)) Z

[2]c is

contained in Q[2]c ∩ Q

[m]x for x = a and e. Therefore (iv) follows and immediately

implies (v). Now (vi) and (vii) are rather straightforward.

The next easy lemma describes the subgraph in ∆ induced by the verticesincident to a given element of D.

Lemma 8.1.2 Let c, b, and d be elements in D of type 2, 0, and 3, respectively.Then

(i) the subgraph in ∆ induced by D[m]c is complete on 5 vertices;

(ii) the subgraph in ∆ induced by D[m]b is complete on 31 vertices;

(iii) the subgraph in ∆ induced by D[m]d is isomorphic to the hexad graph on

77 vertices.

Defining the graph 149

Proof Statement (i) follows directly from the definition. The element b isa vertex of Γ. It is incident to 31 edges. Any pair of these edges is containedin a geometric Petersen subgraph. Therefore the connected components of thelocal antipodality graph containing these edges as vertices are pairwise adjacentas vertices of ∆ which gives (ii).

By (7.7.4 (ii)) the set D[m]d can be identified with the vertex-set of the hexad

graph Ξ as defined before (11.4.5). The group G[3]d induces the natural action of

Aut (M22) ∼= G[3]d /N

[3]d

∼= M[3]d /O2,3(M

[3]d )

on Ξ. It is immediate from the diagram of D(G) that, under the above identifi-cation, two vertices from D[m]

d are adjacent in ∆ whenever they are adjacentin Ξ. Thus it only remains to show that two vertices from D[m]

d are not adja-cent in ∆ if they are at distance 2 in Ξ. Let a ∈ D[m]

d . Then by (7.7.4 (2))M

[3]d (a) ∼= 25 : 3 · Sym6 induces on D[m]

d an action isomorphic to 24 : Sym6. By(11.4.5) O2(M

[3]d (a)) permutes transitively the 16 vertices in Ξ2(a). By (7.7.4

(4)) Q[m]a N

[3]d = N

[3]d O2(M

[3]d (a)). Therefore Q

[m]a acts transitively on Ξ2(a) of

size 16. By (8.1.1) (vi)) this implies that Ξ2(a) ⊆ ∆(a) and (iii) follows.

In terms of the proof of (8.1.2 (iii)) Ξ2(a) ⊆ ∆2(a) and as a byproduct weobtain the following.

Proposition 8.1.3 Let ∆12(a) be the set of vertices in ∆ which are incident with

a to a common element of type 3 but not to a common element of type 2 and leth ∈ ∆1

2(a). Then

(i) ∆12(a) is a G

[m]a -orbit on the set of vertices at distance 2 in ∆ from the

vertex a;(ii) the set D[3]

ah contains a unique element, say d;(iii) Q

[m]a acts on ∆1

2(a) with orbits of length 16 indexed by the sextets in Sa;(iv) |∆1

2(a)| = |Sa| · 16 = 1771 · 16 = 28, 336 = 24 · 7 · 11 · 23;(v) Gah

∼= 21+12+ . 3 · Sym6;

(vi) Q[m]a ∩ Q

[m]h = Z(Gah) ∼= 2.

Proof Statement (i) follows directly from (8.1.2 (iii)). Since M[m]a (d) ∼= 26 :

3 · Sym6 is maximal in M[m]a

∼= M24 by (11.2.3 (i)) and M[m]d

∼= 6 · Aut (M22)acts transitively on the set of ordered pairs of non-adjacent vertices in the hexadgraph Ξ, one can argue as in the proof of (8.1.1 (i)) to establish (ii). Since Q

[m]a

stabilizes d, (iii) follows from (7.7.4 (4)). Now (iv) and (v) are direct consequencesof the previous assertions. In order to establish (vi) we apply (7.7.1 (12)) and(7.7.4 (4)).

150 173,067,389-Vertex graph ∆

8.2 The local graph of ∆

In this section we identify the subgraph in ∆ induced by the set ∆(a) of verticesadjacent to a given vertex a. We start by describing the action of G

[m]a

∼= 211 :M24 on ∆(a).

Lemma 8.2.1 Let c ∈ D[2]a . Then for some e ∈ D[m]

c \a we have the equalityGae = Ra(c)M [m]

a (c)

(here Ra(c) is the only subgroup of index 4 in Q[m]a normalized by M

[m]a (c) ∼= 26 :

(L3(2) × Sym3)).

Proof By (7.7.1 (7)) Gac = Q[m]a M

[m]a (c). It is easy to deduce from the shape of

the chief factors of Gac (cf. (11.3.1 (ii))) that Ra(c)M [m]a (c) represents the only

conjugacy class of subgroups of index 4 in Gac. Since the latter group permutestransitively the four vertices in D[m]

c \a, the assertion follows.

Let Λ(a) be the set of pairs (T, β), where T = B1, B2, B3 is a trio from Ta

and β is GF (2)-valued function on T , whose support is even. Notice that if T istreated as the set of non-zero vectors of a non-singular symplectic 2-space, thenβ is an associated quadratic form.

Define an action of G[m]a on Λ(a) by the following rule. If g ∈ M

[m]a and

T g = Bg1 , Bg

2 , Bg3 is the image of T under g then

g : (T, β) → (T g, βg),

where βg(Bgi ) = β(Bi) for 1 ≤ i ≤ 3. If q ∈ Q

[m]a then

q : (T, β) → (T, βq),

where βq(Bi) = β(Bi) if and only if q ∈ Pa(Bi) (where Pa(Bi) is the onlyhyperplane in Q

[m]a stabilized by M

[m]a (Bi) ∼= 24 : L4(2)). Since

Ra(T ) = Pa(B1) ∩ Pa(B2) ∩ Pa(B3),

it is straightforward to check that in this way we obtain a faithful transitiveaction of G

[m]a on Λ(a).

Lemma 8.2.2 The actions of G[m]a on ∆(a) and Λ(a) are permutation iso-

morphic.

Proof It is clear that |Λ(a)| is the number of trios in Ta times the number ofquadratic forms in a 2-space associated with a given (non-singular) symplecticform, which is 4. By (8.1.1 (vii)) this means that

|∆(a)| = |Λ(a)|.Thus it only remains to show that the subgroup Gae as in (8.2.1) stabilizes anelement from Λ(a). Let c be the unique element in D[2]

a ∩ D[2]e treated as a trio

The local graph of ∆ 151

from Ta and let β1 be the function taking value 1 on every octad from c. Thenit is straightforward that (c, β1) is an element of Λ(a) stabilized by Gae.

In what follows λa : ∆(a) → Λ(a) denotes the bijection commuting with theaction of G

[m]a whose existence is guaranteed by (8.2.2).

Lemma 8.2.3 Let a ∈ D[m] and let B be an octad from Ba. Let b(ε) = (B, ε),ε = 0, 1 be the corresponding elements from D[2]

a . Then (up to transposing b(0)

and b(1)) the following holds:

D[m]b(ε) = a ∪ λ−1

a (T, β) | B ∈ T, β(B) = ε.

Proof Let e ∈ D[m]b(ε) for ε = 0 or 1, let c be the unique element from D[2]

a ∩ D[2]e

and let λa(e) = (T, β). Then B ∈ T , Pa(B) contains Ra(c) with index 2 andhence Pa(B) acts on D[m]

c \a with two orbits of length 2 and these orbitsare distinguished by the value of β(B). Let L ∼= L4(2) be a complement toO2(M

[m]a (B)) in M

[m]a (B) ∼= 24 : L4(2). Then L permutes transitively the trios

containing B. On the other hand, if l ∈ L then βl(Bl) = β(B) since L ≤M

[m]a . The subgroup Pa(B)L permutes transitively the vertices in D[m]

b(ε)\a andtherefore the result follows.

Lemma 8.2.4 Let h ∈ D[m]b(0)

\a, k ∈ D[m]b(1)

\a. Suppose that h and k areadjacent in ∆. Then there is a trio T in Ta such that

λa(h) = (T, β(h)), λa(k) = (T, β(k)),

(this is equivalent to the claim that the unique element from D[2]hk is incident

to a).

Proof Let c = D[2]ah and let f = D[2]

ak. Clearly h and k are adjacent wheneverc = f , so suppose that c = f . In this case c and f are distinct trios in Ta sharingthe octad B. Therefore these trios are refined by a common sextet which meansthat there is d ∈ D[3]

c ∩ D[3]f . By (11.4.6) and (8.1.2 (iii)) h is adjacent to only

two vertices from D[m]f \a and these two vertices are in D[m]

b(0), so k is not among

them.

It appears useful to interpret the situation in terms of the locally projectivegraph Γ of valency 31. The element a is a connected component of the localantipodality graph Φ on the edge-set of Γ. The edges of Γ in this connectedcomponent are indexed by the octads from Ba. Thus B is an edge in Γ, sayx, y. Then (up to renaming) we have b(0) = x and b(1) = y, while D[m]

b(0)and

D[m]b(1)

are the connected components of the local antipodality graph containing theedges incident to x and y, respectively. Therefore h is the connected componentof the local antipodality graph containing x, u for some u ∈ Γ(x)\y and kis a similar component containing y, v for some v ∈ Γ(y)\x. If h and k areadjacent in ∆ then there is a Petersen subgraph Σ whose edge-set intersects

152 173,067,389-Vertex graph ∆

both h and k. If Σ contain x, u and y, v then (because Γ does not containtriangles) Σ is forced to contain x, y as on the figure below.

x

y u w

v

t

By (8.2.4) whenever Σ intersects h and k it necessarily contains x, u and y, v,which is not at all obvious at first glance.

From now on we start using the notation for paths in D as introduced inSection 7.6. Consider a path

(me ,

2c,

ma,

2f,

m

h)

in D. Then c and f are distinct elements from D[2]a , e ∈ D[m]

c \a, and h ∈D[m]

f \a. Let us explain the way we are going to use the diagrams for H(M24)and H(M22). The elements c and f are trios in Ta. By the diagram Dt(M24) wesee that the subgroup

M [m]a (c) ∼= 26 : (L3(2) × Sym3)

has four orbits on Ta\c with lengths

2 · 21, 22 · 14, 24 · 63, and 26 · 42.

Since the orbit lengths are pairwise different we can and will refer to an orbitvia its length. When the orbit length is given in the form 2l · k, the subgroupO2(M

[m]a (c)) ∼= 26 acts on this orbit with k orbits of length 2l each. Depending

on the (length of the) M[m]a (c)-orbit containing f we write that

∠caf = 2 · 21, 22 · 14, 24 · 63, and 26 · 42

in the respective cases.

Lemma 8.2.5 Let (me ,

2c,

ma,

2f,

m

h) be a path in D. Then Q[m]a (c) = Ra(c) acts

transitively on D[m]f \a unless ∠caf = 2 · 21, in which case there are two orbits

The local graph of ∆ 153

of length 2. Furthermore, if λa(e) = (T (e), β(e)) and λa(h) = (T (h), β(h)) thenthe following assertions hold:

(i) if ∠caf = 24 · 63 or 26 · 42 then e and h are not adjacent in ∆;(ii) if ∠caf = 2 · 21 then e and h are adjacent in ∆ if and only if β(e)(B) =

β(h)(B), where B = T (e) ∩ T (h);(iii) if ∠caf = 22 · 14 then e and h are adjacent in ∆.

Proof Since NG

[m]a

(Ra(c)) = G[m]a (c) (because of the maximality of M

[m]a (c) ∼=

26 : (L3(2) × Sym3) in M[m]a

∼= M24 by (11.2.3 (i)), we have Ra(c) = Ra(f).Therefore the action of Ra(c) on D[m]

f \a is non-trivial. The three hyperplanes

in Q[m]a containing Ra(c) are Pa(B1), Pa(B2) and Pa(B3) where c = B1, B2, B3

(a trio). The subgroup Ra(f) is contained in one of these hyperplanes if and onlyif the trios c and f intersect. Since the latter takes place only if ∠caf = 2 · 21(compare the diagram Dt(M24)) the first assertion follows.

By (8.1.1 (vi)) h is adjacent to e in ∆ only if the orbit of h under Q[m]e has

length 4. On the other hand, Q[m]a Q

[m]e = Q

[m]a O2(M

[m]a (c)) by (7.7.1 (7)). Since

f is the only element in D[2]ah, the orbit of h under Q

[m]e is at least as long as the

orbit of f under Q[m]e . By Dt(M24) the orbit of f under O2(M

[m]a (c)) has length

2, 22, 24, and 26 when ∠caf = 2 · 21, 22 · 14, 24 · 63, and 26 · 42, respectively.This immediately implies (i).

If ∠caf = 2 · 21 then the trios T (e) and T (h) share an octad B, say. Now (ii)is immediate from (8.2.3).

If ∠caf = 22 · 14 then T (e) and T (h) are disjoint but contained in a commonquad in the octad graph on Oa (equivalently there is a sextet which refines bothT (e) and T (h)). Therefore there is an element d ∈ D[3] incident to both c and f .On the diagram of D the nodes 3 and m are disjoint, therefore e and h are alsocontained in D[m]

d . By (8.1.2 (iii)) the latter subgraph is isomorphic to the hexadgraph Ξ as in (11.4.5). In terms of Ξ the subgraphs D[m]

c and D[m]f correspond to

intersecting pairs in the hexad corresponding to a. Therefore (iii) follows from(11.4.6).

We summarize (8.2.3) and (8.2.5) in the following.

Corollary 8.2.6 Let Λ(a) be the set of pairs (T, β), where T is a trio from Ta

and β is a GF (2)-valued function on T whose support is even. Let G[m]a act on

Λ(a) as described before (8.2.2). Then there is a bijection λa : ∆(a) → Λ(a)which commutes with the action of G

[m]a . Let e and h be distinct vertices from

∆(a), let λa(e) = (T (e), β(e)) and let λa(h) = (T (h), β(h)). Then e and h areadjacent in ∆ if and only if one of the following holds:

(i) T (e) = T (h);(ii) T (e) and T (h) share an octad B and β(e)(B) = β(h)(B);(iii) T (e) and T (h) are disjoint but are refined by a common sextet.

154 173,067,389-Vertex graph ∆

8.3 Distance two neighbourhood

In this section we study the set of vertices at distance 2 in ∆ from a given vertex.

Lemma 8.3.1 Let π = (ma,

2c,

me ,

2f,

m

h) be a path in D such that d∆(a, h) = 2.Then the G-orbit containing π is uniquely determined by ∠cef which is 2 · 21,24 · 63, or 26 · 42.

Proof The assertion is a direct consequence of (8.2.5).

Lemma 8.3.2 Let ∆12(a) be defined as in (8.1.3). Let ∆2

2(a) and ∆32(a) be the

sets of vertices h in D joint with a by paths (ma,

2c,

me ,

2f,

m

h) with ∠cef = 24 · 63and 26 · 42, respectively. Then

(i) G[m]a acts transitively on ∆2

2(a) and on ∆32(a);

(ii) the orbits of Q[m]a on ∆3

2(a) are of length 210 while those on ∆22(a) are of

length at most 28;(iii) ∆1

2(a), ∆22(a) and ∆3

2(a) are the (pairwise distinct) orbits of G[m]a on

∆2(a).

Proof It follows from (8.1.3), (8.2.5), and (8.2.6) that every vertex at distance2 from a in ∆ is contained in ∆i

2(a) for i = 1, 2, or 3. By (8.3.1) G[m]a acts

transitively on each of the ∆i2(a)’s. We have to show that the three orbits are

pairwise different. We accomplish this by estimating the lengths of the orbits ofQ

[m]a on each of the ∆i

2(a)’s. By (8.1.3 (iii)) the orbits of Q[m]a on ∆1

2(a) are oflength 16. Let 2l be the length of the orbit of f under O2(M

[m]a (c)). Then 2l = 24

and 2l = 26 for h ∈ ∆22(a) and h ∈ ∆3

2(a), respectively. By (8.1.1 (v)) the lengthof the orbit of h under Q

[m]a is at least 2l and at most

2l · |D[m]c \ a| · |D[m]

f \ e| = 2l · 4 · 4 = 2l+4.

This immediately shows that ∆12(a) is different from the remaining two orbits

while the upper bounds for the lengths are 28 and 210 for ∆22(a) and ∆3

2(a). Weclaim that in the latter case the upper bound is attained (later we will see thatin the former case the upper bound is not attained (8.3.15 (ii)).

By (8.1.1 (iv)) Q[m]a ∩Q

[m]e = Z

[2]c . We are going to show that the latter group

acts transitively on D[m]f \e provided that ∠cef = 26 · 42. Let B1, B2, and B3

be the trios constituting f (here f is considered as a trio from Te). We haveto show that Z

[2]c ≤ Pe(Bi) for i = 1, 2, and 3. Since Z

[2]c is of order 23, it is

contained in 28 − 1 hyperplanes of Q[m]e . On the other hand, we can see from

the diagram Dt(M24) that every octad Bi is from an orbit of length 25 · 21 ofM

[m]e (c) on Be. Since 25 ·21 > 28−1 we obtain the required transitivity. Therefore

the second factor 4 in the above upper bound is attained.

Distance two neighbourhood 155

By (8.2.6) e is the only vertex in D[m]c adjacent to h. Since Q

[m]a acts transit-

ively on D[m]c \a, the first factor 4 in the upper bound is also attained and the

result follows.

The following lemma is a direct consequence of (8.3.1) and (8.3.2).

Lemma 8.3.3 If h ∈ ∆2(a) then Gah acts transitively on ∆(a) ∩ ∆(h).

For h ∈ ∆i2(a) put µi(a, h) = ∆(a) ∩ ∆(h) and

µi = |µi(a, h)|.Then µi is precisely the number of paths π as in (8.3.1) for a fixed h ∈ ∆i

2(a).Thus, as a direct consequence of (8.2.5) and (8.3.2) we obtain the following.

Lemma 8.3.4 Let h ∈ ∆i2(a) for i = 1, 2 or 3. Let (

ma,

2c,

me ,

2f,

m

h) be a path inD and let δ = 2 if i = 1 and δ = 4 if i = 2 or 3. Then

|∆i2(a)| =

1µi

(|∆(a)| · ∠cef · δ),

in particular

|∆22(a)| =

1µ2

(28 · 33 · 5 · 7 · 11 · 23);

|∆32(a)| =

1µ3

(211 · 32 · 5 · 7 · 11 · 23).

Proof By (8.2.5) δ is the length of the orbit of h under Re(c) on D[m]f \e. The

rest is easy counting.

Since |∆12(a)| is given by (8.1.3 (iv)), in view of (8.3.4) the value of µ1 can

easily be calculated.

Lemma 8.3.5 Let h ∈ ∆12(a) and let d ∈ D[3]

ah. Then

(i) ∆(a) ∩ ∆(h) ⊆ D[m]d ;

(ii) µ1 := |∆(a) ∩ ∆(h)| = 45.

Proof By (8.1.3 (ii)) d is uniquely determined and hence it is stabilized byGah. Since ∆(a) ∩ ∆(h) intersects D[m]

d and by (8.3.3) Gah acts transitively on∆(a) ∩ ∆(h), (i) follows. By (8.1.2 (iii)) the assertion (ii) is immediate from theintersection diagram of the hexad graph (cf. the paragraph before (11.4.5)).

We will calculate µ2 and µ3 by showing that for i = 2 and 3 a vertex h ∈ ∆i2(a)

can be reached from a along a path in D, satisfying a uniqueness criteria in thesense that for every h there is a unique path of that type. In order to accomplishthis we start by analysing various 2-paths in D.

156 173,067,389-Vertex graph ∆

8.3.1 Analysing 2-paths

Let (ix,

kz,

jy) be a 2-path in D. This means that both x and y are incident to z,

but x and y are not incident. We would like to analyse the structure of Gxy. Sinceboth x and y are in the residue of z in D the success of the analysis depends onhow easily we find our way in the residues of D.

The residue ofma in D is closely related to the locally truncated geometry

H(M24) of the Mathieu group M24. Only for every octad in Ba we have got twoelements in D[0]

a , which are (B, 0) and (B, 1) and they are transposed by elementsQ

[m]a . Keeping in mind this remark we can use the diagrams for H(M24) presented

in Section 11.5. These diagrams show the orbits of the stabilizer M[m]a (x) of an

element x in H(M24) on the remaining elements of H(M24). The length of everyorbit is given in the form 2l · k, which means that O2(M

[m]a (x)) has k orbits of

length 2l each (if l = 0 the first factor is dropped). In particular the diagramDb(M24) shows the orbits of M

[m]a (b) ∼= 24 : L4(2) where b ∈ D[0]

a . It is immediatefrom this diagram that the elements in Da fixed by O2(M

[m]a (b)) are exactly those

which are incident to b. Similarly for i, j = 2, 3 and forix∈ D[i]

a the elementsin D[j]

a fixed by O2(M[m]a (x)) are exactly those incident to x. These observations

can be stated in the following two lemmas.

Lemma 8.3.6 Let a ∈ D[m], let b ∈ D[0]a and let

ix∈ D[i]

a , where i = 2 or 3.Then x is incident to b if and only if there is a subgroup R ≤ O2(Gab) such thatR stabilizes x and Q

[m]a R = Q

[m]a O2(M

[m]a (b)).

Lemma 8.3.7 Let a ∈ D[m], let i, j = 2, 3 and let ix,

jy ∈ Da. Then x is

incident to y if and only if there is a subgroup R ≤ O2(Gax) such that R stabilizesy and Q

[m]a R = Q

[m]a O2(M

[m]a (x)).

The next lemma supplies further conditions for elements in Da to be incident.

Lemma 8.3.8 Let a ∈ D[m] and let0b,

2c,

3d ∈ Da. Then the following assertions

hold:

(i) forix=

2c or

3d the elements x and b are incident

(1) if and only if Z[i]x ≤ Q

[0]b ∩ Q

[m]a ;

(2) if and only if Q[i]x ∩ Q

[m]a ≤ G

[0]b ∩ Q

[m]a ;

(ii) the elements c and d are incident(3) if and only if Z

[3]d ≤ Z

[2]c ;

(4) if and only if Q[3]d ∩ Q

[m]a ≤ Q

[2]c ∩ Q

[m]a ;

(iii) Z[3]d ≤ Q

[2]c if and only if ∠cad = 26 · 21.

Proof The only if statements in (i) and (ii) follow from (7.7.1), (7.7.2), (7.7.3),and (7.7.4). With x as in (i) suppose that Z

[i]x ≤ Q

[0]b ∩Q

[m]a . Then Q

[0]b centralizes


Z[i]x . Since Gax is maximal in M

[m]a by (11.2.3 (i)), we have

Q[0]b ≤ N

G[m]a

(Z [i]x ) = Gax

and hence Q[0]b stabilizes x. By (7.7.1 (7))

Q[0]b Q[m]

a = O2(GabQ[m]a )

and hence O2(GabQ[m]a ) stabilizes x. Since

O2(Gab)Q[m]a = Q[m]

a O2(M [m]a (b)),

by (8.3.6) x and b are incident which proves (1).Suppose now that Q

[i]x ∩ Q

[m]a ≤ G

[0]b ∩ Q

[m]a . Since [Q[m]

a , Q[i]x ] ≤ Q

[m]a ∩ Q

[i]x ,

we conclude that Q[i]x normalizes G

[0]2 ∩ Q

[m]a , and so Q

[i]x ≤ GabQ

[m]a . As above

(8.3.6) forces x to be incident to b which gives (2).The proof of statements in (ii) are similar to those in (i).To establish (iii) suppose that ∠cad = 26 · 21. Then by the digram Dt(M24)

in Section 11.5 there is a path

(2c,

3x,

2y,

3d)

in Da. By (i) and (ii) we obtain

Z[3]d ≤ Z [2]

y ≤ Q[3]x ∩ Q[m]

a ≤ Q[2]c ∩ Q[m]

a .

This proves the if statement in (iii). To establish the only if part, observe thatZ

[3]d ≤ Q

[m]a and that Q

[m]a acts faithfully on ∆(a). This means that Z

[3]d acts

non-trivially on D[m]c \a for some c ∈ D[2]

a . In this case Z[3]d ≤ Q

[2]c . By the if

part we necessarily have ∠cad = 26 · 21.

Lemma 8.3.9 If (ma,

3d,

2c) is a path in D with ∠adc = 24 · 6 then

(i) Q[m]a ∩ G

[2]c ≤ Q

[3]d ;

(ii) Q[3]d = (Q[m]

a ∩ Q[3]d )(Q[2]

c ∩ Q[3]d );

(iii) Q[m]a ∩ Z

[2]c = Z

[3]d ;

(iv) Gac = Gadc;(v) |(Q[m]

a ∩ Q[2]c )Z [2]

c /Z[2]c | = 24.

Proof By (7.7.4 (4)) Q[3]d Q

[m]a = Q

[3]d O2(M

[3]d (a)) while by the diagram Dh(M22)

we observe that O2(M[3]d (a)) acts fixed-point freely on the M

[3]a (a)-orbit of length

24 · 6, which gives (i). By the above sentence we also see that Q[m]a does not fix c

and therefore Q[m]a does not normalize Q

[3]d ∩Q

[2]c . Hence Q

[m]a ∩Q

[3]d ≤ Q

[3]d ∩Q

[2]c .

Since N[3]d

∼= 21+12+ . 3 acts irreducibly on Q

[3]d /(Q[3]

d ∩ Q[2]c ) ∼= 22, we obtain (ii).

158 173,067,389-Vertex graph ∆

In particular

|Q[m]a ∩ Q[2]

c | =|Q[m]

a ∩ Q[3]d |

|Q[3]d /(Q[3]

d ∩ Q[2]c )|

=27

22 = 25.

Suppose that Q[m]a ∩Z

[2]c = Z

[3]d . Since N

[3]d acts irreducibly on Z

[2]c /Z

[3]d

∼= 22, weconclude that Z

[2]c ≤ Q

[m]a . But in that case Q

[m]a centralizes Z

[2]c and Q

[m]a ≤ G

[2]c ,

contrary to (i). This gives (iii) and the latter immediately implies (iv) and (v).

Lemma 8.3.10 The group G permutes transitively the paths (ma,

2c,

0b) in D. For

such a path the following assertions hold:

(i) Gacb/Q[2]c

∼= L3(2) × Sym3;(ii) Q

[m]a ∩ Q

[0]b = Z

[2]c ;

(iii) Q[2]c = (Q[2]

c ∩ Q[m]a )(Q[2]

c ∩ Q[0]b );

(iv) Gab = Gacb;(v) GacbQ

[m]a = Gac and GacbQ

[0]b = Gcb;

(vi) Q[m]a ∩ G

[0]b = Q

[m]a ∩ Q

[2]c and Q

[0]b ∩ G

[m]a = Q

[0]b ∩ Q

[2]c ;

(vii) (Q[m]a ∩ G

[0]b )Q[0]

b = O2(Gcb) and (Q[0]b ∩ G

[m]a )Q[m]

a = O2(Gac).

Proof If Θc is the Petersen subgraph associated with c then b is a vertex of Θc

disjoint from the antipodal triple a of edges. It is clear that S[2]c

∼= Sym5 actstransitively on the set of such pairs (a, b) with stabilizer isomorphic to Sym3.This gives the transitivity assertion along with (i). In terms of the figure afterthe proof of (8.2.4) b = u, c = Σ, and a is the connected component of the localantipodality graph containing the edge x, y.

Let us turn to (ii). From (7.7.3 (1), (9)) we conclude that both Q[m]a and

Q[0]b contain Z

[2]c and also that Q

[m]a and Q

[0]b induce on D[2]

c , respectively, anelementary abelian group of order 4 and a group generated by a transpositionwhich does not normalize the action of Q

[m]a . The latter implies that

〈Q[m]a , Q

[0]b 〉Q[2]

c /Q[2]c

∼= Sym5.

By (4.7.4) the latter group permutes transitively the dents in Q[2]c . On the other

hand, since both Q[m]a and Q

[0]b are abelian, they centralize their intersection,

which gives (ii). Now (iii) is immediate by the order consideration.By (ii) we obtain

Gab ≤ NG

[m]a

(Q[m]a ∩ Q

[0]b ) = N

G[m]a

(Z [2]c ) = Gac,

which gives (iv). It is immediate from the elementary properties of the Petersengraph that Q

[m]a permutes transitively the four elements in D[0]

c \D[0]a , while

Q[0]b permutes transitively the two elements in D[m]

c \D[m]b , we obtain (v). The

remaining statements (vi) and (vii) are now easy to verify.


Recall that if b ∈ D[0] then Vb is a 5-dimensional GF (2)-space associatedwith b and for x ∈ Db by Vb(x) we denote the subspace in Vb associated with x.

Lemma 8.3.11 Let (ix,

0b,

2c) be a path in D, where i = 2 or 3. Then Z

[i]x ≤ Q

[2]c

if and only if 〈Vb(x), Vb(c)〉 = Vb.

Proof By (7.7.2) we observe that Q[0]b ≤ Q

[2]c , that

Q[0]b = 〈Z [i]

y | y ∈ D[i]b 〉,

and that Gbc acts transitively on the set

y ∈ D[i]b | 〈Vb(y), Vb(c)〉 = Vb.

Therefore, it is sufficient to show that Z[i]y ≤ Q

[2]c whenever y ∈ D[i]

b and〈Vb(y), Vb(c)〉 = Vb. Take a ∈ D[m] such that 〈Vb(y), Vb(c)〉 ≥ Vb(a). In thiscase a ∈ D[m]

ybc . Then by (8.3.8 (i)) we have Z[i]y ≤ Q

[m]a ∩ Q

[0]b ≤ Q

[2]c ∩ Q

[0]b .

Lemma 8.3.12 Let (3e,

2c,

3d) be a path in D. Then

(i) N[3]e acts transitively on D[3]

c \e;(ii) GecdN

[3]d = Gcd;

(iii) Q[3]e ∩ Q

[3]d ≤ Q

[2]c ;

(iv) |(Q[3]e ∩ Q

[3]d )/Z [2]

c | = 24;(v) Q

[2]c = (Q[3]

e ∩ Q[2]c )(Q[3]

d ∩ Q[2]c );

(vi) (Q[3]e ∩ G

[3]d )N [3]

d /N[3]d = O2(Gcd/N

[3]d ),

(vii) Gecd = CGe(Z [3]

d ) = Ged.

Proof Since Z[3]e ≤ Z

[2]c ≤ Q

[3]e and N

[3]e acts fixed-point freely on Q

[3]e /Z

[3]e , we

conclude that N[3]e acts transitively on Z

[2]c /Z

[3]e . Now [Z [2]

c , Q[3]e ] = Z

[3]e , and so

N[3]e acts transitively on Z

[2]c \Z

[3]e and also on D[3]

c \e. So (i) is established andimplies the equality GecdN

[3]e = Gec which by symmetry gives (ii). Since

|(Q[3]e ∩ Q[2]

c )/Z [2]c | =

213

22+3 = 28

and |Q[2]c /Z

[2]c | = 212, we have the following lower and upper bounds:

24 ≤ |(Q[3]e ∩ Q[2]

c ∩ Q[3]d )/Z [2]

c | ≤ 28

The lower bound is attained if and only if (v) holds. Since Gec is a maximalsubgroup in G

[2]c , we have Q

[3]e ∩ Q

[2]c = Q

[3]d ∩ Q

[2]c , therefore the upper bound

is not attained. Moreover, by the properties of the pentad group (4.9.1 (i)) weknow that the elements of order 5 in Gecd act fixed-point freely on Q

[2]c /Z

[2]c

and hence also on (Q[3]e ∩ Q

[2]c ∩ Q

[3]d )/Z [2]

c . This means that the lower bound is

160 173,067,389-Vertex graph ∆

attained and (v) holds. By the order consideration we also obtain (iii) and (iv).By (7.7.3 (13)) Q

[2]c Q

[3]d = O2(Gcd) and therefore (Q[3]

e ∩ Q[2]c )Q[3]

d = O2(Gcd). Itis clear that (Q[3]

e ∩ G[3]d )N [3]

d /N[3]d ≤ O2(Gcd/N

[3]d ) and hence (vi) follows. Since

Q[3]e is extraspecial,

[Z [2]c , Q[3]

e ∩ G[3]d ] = [Z [2]

c , CQ

[3]e

(Z [3]d )] = Z [3]

e ≤ Z[3]d

and so Q[3]e ∩Gd inverts N

[3]d /Q

[3]d . Since the conjugates of Z

[3]d under N

[3]e

generate Z[2]c , we have CGe(Z

[3]d ) ≤ NGe(Z

[2]c ) = Gec and hence

Ged ≤ CGe(Z [3]

d ) ≤ CGec(Z [3]

d ) = Gecd,

proving (vii).

8.3.2 Calculating µ2

Recall that our strategy is to reach h ∈ ∆22(a) from a by a path in D which

satisfies a uniqueness condition.

Lemma 8.3.13 Let Φ be the set of paths

ϕ = (ma,

2c,

0b,

2e,

m

h)

in D subject to the condition that 〈Vb(c), Vb(e)〉 = Vb. Then the action of G onΦ is transitive and the following assertions hold for a path ϕ from Φ:

(i) |Z [2]c ∩ G

[m]h | = 22;

(ii) |Gacbeh| = 214 · 32;(iii) Gcbe = GacbehQ

[0]b ;

(iv) GacbehQ[m]a /Q

[m]a = NGac

(Z [2]c ∩ G

[m]h )/Q

[m]a is of order 210 · 32;

(v) Gah = Gacbeh;(vi) |Gah| = 214 · 32 and |Gah ∩ Q

[m]a | = 24;

(vii) ϕ is the unique path in Φ which joins a with h;(viii) GahQ

[0]b /Q

[0]b

∼= Sym4 × Sym4;(ix) Q

[m]a ∩ Q

[m]h = 1.

Proof By (8.3.10 (v)) GacbQ[0]b = Gcb and so there exists a unique G-orbit on

paths (ma,

2c,

0b,

2e) with 〈Vb(c), Vb(e)〉 = Vb. By (8.3.11) Z

[2]c ≤ Q

[2]e and so Z

[2]c

permutes transitively the pair of elements in D[m]e \D[m]

b . Thus the uniquenessassertion is proved along with (i). Since

|Gacbeh| =|Gacb|

∠cbe · ∠beh=

219 · 32 · 724 · 7 · 2

= 214 · 32,


(ii) holds. Moreover, by (8.3.11) Z[2]c is not contained in Q

[2]e . We claim that

Z[2]c is not contained in N

[2]e . In fact, otherwise Z

[2]c would act non-trivially on

D[m]b \D[m]

e contrary to the inclusion Z[2]c ≤ Q

[0]b . Hence Z

[2]c permutes the pair of

vertices in D[m]e \D[0]

b . This gives (iii) and also proves the equality GacbehZ[2]c =

Gacbe. By (8.3.10 (vii)) (Q[m]a ∩ G

[0]b )Q[0]

b = O2(Gcb) and therefore (Q[m]a ∩ G

[0]b )

acts transitively on the set of 3-dimensional subspaces U in Vb such that U ∩Vb(c) = Vb(e) ∩ Vb(c) (there are exactly 16 such subspaces and each of them canbe taken as Vb(e) for a suitable path in Φ). From this conclusion it is easy todeduce (iii). Furthermore, the established transitivity implies the equalities

|(Q[m]a ∩ G

[0]b )/(Q[m]

a ∩ Gbe)| = 24, |Q[m]a ∩ Gbe| = 29/24 = 25

and

|Q[m]a ∩ Gbeh| = 24.

Since Q[m]a fixes a and c, easy counting gives |GacbehQa/Qa| = 210 · 32. Since

|Gac/Q[m]a | = |26 : (L3(2) × Sym3)| = 210 · 32 · 7

and Gab acts transitively on the set of 7 subgroups of order 4 in Z[2]c , we conclude

that |NGac(Z[2]c ∩ G

[m]e )| is also 210 · 32 and (v) follows. Notice that NGac(Z

[2]c ∩

G[m]e )Q[m]

a /Q[m]a is a conjugate of the subgroup NMt(R) of M24 ∼= G

[m]a /Q

[m]a as

in (11.2.3 (iii)).Since 〈Vb(c), Vb(e)〉 = Vb, D[3]

cbe = ∅ and hence Z[2]c ∩ Z

[2]e = 1. By (8.3.10

(ii)) Q[0]b ∩ Q

[m]h = Z

[2]e , and so Z

[2]c ∩ Q

[m]h = Z

[2]c ∩ Z

[2]e = 1. In particular,

Q[m]a ∩ G

[m]h Q

[m]h , and since Gah normalizes Q

[m]a ∩ G

[m]h , we conclude that

GahQ[m]h = G

[m]h . By symmetry GahQ

[m]a = G

[m]a . By (11.2.3 (iii)) the only group

between NGac(Z [2]

c ∩ G[m]h ) and G

[m]a is Gac. Hence Gah ≤ G

[2]c . By symmetry

Gah ≤ G[2]e . Since Z

[2]c Q

[2]e = Q

[0]b Q

[2]e , we have

Gce ≤ NG

[2]e

(Q[0]b Q[2]

e ) = Gbe,

which gives Gce = Gcbe and also Gah = Gacbeh, which is (vi). Comparing (ii),(v), and (vi) we obtain (vii). The transitivity of the action of G on Φ and (vii)imply the uniqueness condition (viii). Hence

GahQ[0]b /Q

[0]b = GacbehQ

[0]b /Q

[0]b = Gcbe/Q

[0]b

∼= Sym4 × Sym4

and (ix) follows. It only remains to prove (x). As 〈Vb(c), Vb(e)〉 = Vb, we haveQ

[2]c ∩ Q

[2]e ≤ Q

[0]b . By (8.3.10 (vi)) Q

[m]a ∩ G

[0]b ≤ Q

[2]c and so

Q[m]a ∩ Q

[m]h ≤ Q[m]

a ∩ Q[2]c ∩ Q[2]

e ∩ Q[m]h

≤ (Q[m]a ∩ Q

[0]b ) ∩ (Q[m]

h ∩ Q[0]b ) = Z [2]

c ∩ Z [2]e = 1.

162 173,067,389-Vertex graph ∆

Lemma 8.3.14 If h ∈ ∆22(a) then there is a path π = (

ma,

2p,

0q,

2r,

m

h) in D, suchthat 〈Vq(p), Vq(r)〉 = Vq.

Proof Let (ma,

2c,

me ,

2f,

m

h) be a path joining a and h as in (8.3.4) such that ∠cef =24 · 63. We are going to ‘shift’ the path π to obtain a path from the orbit Φ asin (8.3.13) which still joins a with h. The shifting process can be seen on thefollowing diagram where edges indicate incidence in D.

mi

0q

m

k

2p

0x

2y

0z

2r

ma

2c

me

2f

m

h

By the diagram Dt(M24) in the residue De there is a path (2c,

0x,

2y,

0z,

2f) such that

∠cex = 3, ∠cey = 2 · 21, ∠cez = 22 · 21.

Recall that for every B ∈ Be there are two elements in D[0]e forming an orbit under

Q[m]e . In particular the elements x and z are contained in different Q

[m]e -orbits

on D[0]e . Therefore we can independently choose x and z in their Q

[m]e -orbits so

that a is incident to x and h is incident to z.Let us take a closer look at the residue of y. The elements x and z are non-

adjacent vertices of the Petersen graph Θy, while e is an antipodal triple of edges,containing both x and z. Since the diameter of Θy is two, the vertices x and z,being non-adjacent, are joint by a (unique) 2-path (x, q, z) in Θy. Therefore there

is a path (0x,

mi ,

0q,

m

k,0z) in Dy, where i and k are the connected components of the

local antipodality graph of Γ containing the edges x, q and q, z respectively(now treated as edges of Γ).

Treating x as a vertex of Γ the elements a and i are the connected componentsof two distinct edges incident to x in the local antipodality graph. Let p be theelement of type 2 corresponding to the unique geometric Petersen subgraphcontaining these edges. Let r be a similar element defined with respect to z, h


and k. Then

p = D[2]axi, r = D[2]

hzk.

Since q is adjacent in Γ to both x and z, both p and r are incident to q. Con-sidering the Petersen graph Θp we observe that a and q are not incident in D.Similarly in Θr we observe that h and q are not incident. Therefore

(ma,

2p,

0q,

2r,

m

h)

is a path in D.Now it only remains to show that 〈Vq(p), Vq(r)〉 = Vq. Suppose the contrary.

Then Vq(p) ∩ Vq(r) contains a 2-dimensional subspace Vq(w) for an elementw ∈ D[3]

pqr. Since the nodes 3 and m are disjoint on the diagram of D, theelement w is incident to i and k. In the residue of q we observe that w is alsoincident to y. Again directly from the diagram of D we see that w further isincident to x and e. Now a and w are both incident to p and x and hence theyare incident to each other. Next we observe that

Vx(c) = Vx(a) ∩ Vx(c) ≥ Vx(w)

and c is incident to w. By symmetry w is incident to f . Then w ∈ D[3]cef contrary

to the assumption that ∠cef = 24 · 63 and the diagram Dt(M24). Hence no

such w exists, (ma,

2p,

0q,

2r,

m

h) belongs to the orbit Φ as in (8.3.13) and the resultfollows.


(i) |∆22(a)| = 27 · 3 · 5 · 7 · 11 · 23 = 3, 400, 320;

(ii) Q[m]a acts on ∆2

2(a) with orbits of length 27;(iii) if h ∈ ∆2

2(a) then GahQ[m]a = NMt

(R)Q[m]a where NMt

(R) ∼= 26 : (Sym4×Sym3) is the stabilizer in M

[m]a

∼= M24 of an element of the intermediatetype from the rank 3 tilde geometry G(M24);

(iv) µ2 = 18.

Proof Since ∆22(a) is a G

[m]a -orbit, |∆2

2(a)| = [G[m]a : Gah] for h ∈ ∆2

2(a),therefore (8.3.13 (vii)) and (8.3.14) give (i) and (ii). Statement (iii) is by (8.3.13(v), (vi)) and the remark made at the end of the first paragraph of the proof of(8.3.13). Finally (iv) is by (i) and (8.3.4).

8.3.3 Calculating µ3

Similarly to what has been accomplished in the previous section we will reachh ∈ ∆3

2(a) along a path in D satisfying a uniqueness condition.We need an extra piece of notation. If a, b, c, d are elements of D such that

a, d ∈ Dbc (which means each of a and d is incident to both b and c) then ∠a bcd

denotes the length of the orbit of d under Gabc.

164 173,067,389-Vertex graph ∆

Lemma 8.3.16 The set of paths (ma,

3b,

2c,

3d) in D with ∠abc = 24 · 6 is non-empty

andG acts on this set transitively. For such a path the following assertions hold:

(i) |Gabcd| = 215 · 3 · 5;(ii) GabcdN

[3]d = Gcd;

(iii) Z[3]d Q

[m]a /Q

[m]a is generated by a non-2-central involution of G

[m]a /Q

[m]a

∼=M24;

(iv) GabcdQ[m]a /Q

[m]a is the centralizer of Z

[3]d Q

[m]a /Q

[m]a in G

[m]a /Q

[m]a ;

(v) Gabcd = CG

[m]a

(Z [3]d ) = Gad.

Proof Let (ma,

3b,

2c) be a path in D such that ∠abc = 24 · 6. Then by (8.3.12 (i))

N[3]b acts transitively on D[3]

c \b. Therefore the existence and the transitivitystatements hold with

|Gabcd| =|Gab|

∠abc × ∠bcd=

221 · 33 · 524 · 6 · 6

= 215 · 3 · 5,

which gives (i). By (8.3.12 (vi)) (Q[3]b ∩ G

[3]d )N [3]

d /N[3]d = O2(Gdc/N

[3]d ) and by

Dp(M22) the group Q[3]b Q

[2]c /Q

[3]b acts regularly on the set

a | a ∈ D[m]b ,∠abc = 24 · 6.

Hence (ii) holds. Note that Z[3]d ≤ Z

[2]c and that Z

[3]d = Z

[3]b . By (8.3.9 (iii)) Q

[m]a ∩

Z[2]c = Z

[3]b . Thus Z

[3]d ≤ Q

[m]a and d is not incident to a. Since Gabcd centralizes

Z[3]d and has order divisible by 5, we obtain (iii) from the information about

centralizers of involutions in M24. Therefore the centralizer of Z[3]d Q

[m]a /Q

[m]a

in G[m]a /Q

[m]a has order 29 · 3 · 5. Since Q

[m]a ∩ G

[2]c ≤ Q

[3]b by (8.3.9 (i)) and

CG

[3]b

(Z [3]d ) = Gbcd by (8.3.12 (vii)), we obtain

CQ

[m]a

(Z [3]d ) = Q[m]

a ∩ Gbcd = CQ

[m]a ∩Q

[3]b

(Z [3]d ).

Furthermore, since Q[3]b is extraspecial and Q

[m]a ∩ Q

[m]b is a maximal subgroup

of Q[3]b , we have |Q[m]

a ∩ Gbcd| = |Q[m]a ∩ Q

[3]b |/2 = 26. Thus

|GabcdQ[m]a /Q[m]

a | =|Gabcd|

26 = 29 · 3 · 5,

which gives (iv) and (v).

Lemma 8.3.17 The group G acts transitively on the set of paths

(ma,

3b,

2c,

3d,

me ) with ∠abc = ∠edc = 24 · 6. For such a path the following assertions

hold:

(i) |Gabcde| = 210 · 3 · 5;(ii) Z

[2]c ≤ Q

[3]b ∩ Q

[3]d and |(Q[3]

b ∩ Q[3]d )/Z [2]

c | = 24;


(iii) Gae = Gabcde;(iv) Q

[m]a ∩ Q

[m]e = 1;

(v) Gae/(Q[3]b ∩ Q

[3]d ) ∼= Sym5.

Proof The transitivity assertion and (i) follow from (8.3.16 (i), (ii)). By (8.3.12(iv)) |(Q[3]

b ∩ Q[3]d )/Z [2]

c | = 24 which is (ii). By (8.3.9 (v)) we have |(Q[m]a ∩

Q[2]c )Z [2]

c /Z[2]c | = 24. By (8.3.12 (vi)) (Q[3]

d ∩G[3]b )N [3]

b /N[3]b = O2(Gbc/N

[3]b ). Also

by the diagram Dp(M22) the group Q[3]d ∩ G

[3]b acts transitively on the set of 32

elements x ∈ D[m]b with ∠xbc = 24 · 6. Thus Gbc = Gabc(Q

[3]d ∩ G

[3]b ). Suppose

that (Q[m]a ∩ Q

[2]c )Z [2]

c = Q[3]b ∩ Q

[3]d . Then Gbc = Gabc(Q

[3]d ∩ G

[3]b ) normalizes

(Q[m]a ∩Q

[2]c )Z [2]

c /Z[2]c . But this is impossible, since by the structure of the pentad

group (4.9.1) it follows that

Gbc∼= 23+12 · (Sym4 × Sym5)

does not normalize subgroups of order 24 in Q[2]c /Z

[2]c . So (Q[m]

a ∩Q[2]c )Z [2]

c =Q

[3]b ∩Q

[3]d . Since 5 divides |Gabcd|, we have

Q[m]a ∩Q[2]

c ∩ Q[3]d ≤ Z [2]

c .

Since O2(Gcd/N[3]d ) ∩Gde/N

[3]d = 1, Q

[m]a ∩Gcde ≤ Q

[d]d . Similarly

Q[m]a ∩G

[2]c ≤ Q

[3]b . By (8.3.12 (iii)) Q

[3]b ∩ Q

[3]d ≤ Q

[2]c . Hence

Q[m]a ∩ Gcde ≤ Q[m]

a ∩ Q[2]c ∩ Q

[3]d ≤ Z [2]

c .

By (8.3.9 (iii)) we have Q[m]a ∩ Z

[2]c ≤ Z

[3]b and thus Q

[m]a ∩ Gcde = Z

[3]b . By

symmetry Q[m]e ∩ Gabc = Z

[3]d . By (8.3.16 (v)) Gabcd = Gad, and so Q

[m]e ∩

G[m]a = Q

[m]e ∩ Gabc = Z

[3]d . Thus Gae = C

G[m]e

(Z [3]d ) = Ged and Gae = Gabcde

which is (iii). By symmetry Q[m]a ∩ Ge = Z

[3]b and so Q

[m]a ∩ Q

[m]e = 1, which

is (iv). Finally since Gae/O2(Gae) ∼= Sym5 and O2(Gae) ≤ Q[3]b ∩ Q

[3]d we have

Gae/(Q[3]b ∩ Q

[3]d ) ∼= Sym5, completing the proof of (v).

Lemma 8.3.18 Let h ∈ ∆32(a) and let (

ma,

2c,

me ,

2f,

m

h) be a path as in (8.3.4)

joining a and h, such that ∠cef = 26 · 42. Then there is a path (ma,

3l,

2g,

3k,

m

h) inD, such that l, g, k ∈ De and ∠alg = ∠hkg = 24 · 6.

Proof Since ∠cef = 26 · 42 by Dt(M24) there is a path (2c,

3l,

2g,

3k,

2f) in De with

∠c leg = 8 = ∠ck

e g. Since the nodes m and 3 are disjoint on the diagram of D,a is incident to l and h is incident to k. Since ∠f l

eg = 8 we conclude from thediagram Dh(M22) that ∠alg = 24 · 6, and by symmetry ∠hkg = 24 · 6. Thereforethe result follows.

166 173,067,389-Vertex graph ∆


(i) |∆32(a)| = 211 · 32 · 7 · 11 · 23 = 32, 643, 072;

(ii) if h ∈ ∆32(a) then the subgroup G

[m]h ∩ Q

[m]a is of order 2 and it is a

conjugate in G[m]a of Z

[3]c for some c ∈ D[3]

a ;(iii) let Ms

∼= 26 : 3 ·Sym6 be the stabilizer in M[m]a

∼= M24 of a sextet S fromSa, let F be a subgroup in Ms such that

(1) F ∩ O2,3(Ms) = O2(Ms);(2) F/O2(Ms) ∼= Sym5;(3) F acts on the set of 15 trios from Ta refined by the sextet S with two

orbits of lengths 5 and 10;

then Q[m]a Gah and Q

[m]a F are conjugate in G

[m]a ;

(iv) µ3 = 5.

Proof Because of the transitivity of the action of G[m]a on ∆3

2(a), since the orderof G

[m]a is known (8.3.18) and (8.3.17 (i)) imply (i). By (8.3.2 (ii)) the orbit of h

under Q[m]a has length 210. Therefore Q

[m]a ∩G

[m]h is of order 2 and clearly Gah ≤

CG

[m]a

(Q[m]a ∩ G

[m]h ). The group Q

[m]a is a Todd module for M

[m]a

∼= G[m]a

∼= M24

and the latter has two orbits on the set of subgroups of order 2 in Q[m]a with

lengths 1771 and 276. The corresponding stabilisers are

26 : 3 · Sym6 of order 210 · 32 · 5

and

Aut (M22) of order 28 · 3 · 5 · 7 · 11.

By the order consideration G[m]h ∩ Q

[m]a is from the former orbit which gives

(ii) and (iii). Finally (iv) is by (i) and (8.3.4).

8.3.4 The µ-subgraphsLet a be a vertex of ∆ and let h ∈ ∆2(a). In this section we describe the subgraphin the local graph ∆(a) as in (8.2.6) induced by the set µ(a, h) = ∆(a) ∩ ∆(h).This subgraph is commonly known as a µ-subgraph in ∆. If h ∈ ∆α

2 (a) then by(8.3.5), (8.3.15), and (8.3.19) the number of vertices in the µ-subgraph µ(a, h)is 45, 18, and 5 for α = 1, 2, and 3, respectively. Therefore we have establishedthe following fragment of the suborbit diagram of ∆:

In order to deal with µ-subgraphs we first introduce some further termino-logy concerning the locally truncated geometry H(M24) and the tilde geometryG(M24) of the Mathieu group M24 (cf. Section 11.2). Let Σ be a sextet, let BΣ andTΣ be respectively the octads and trios refined by Σ. By the definition BΣ and TΣare the octads and trios incident to Σ in H(M24). Clearly BΣ corresponds to thepairs of the tetrads in Σ while TΣ corresponds to the partitions of the tetrads intothree disjoint pairs. Thus by Section 1.8 there is a non-singular 4-dimensionalsymplectic space VΣ such that TΣ are the non-zero vectors in VΣ; BΣ correspondsto the totally singular 2-dimensional subspaces; the incidence (in H(M24)) is via


inclusion. The group Ms = Ms/O2,3(Ms) ∼= Sym6 (where Ms∼= 26 : 3 · Sym6 is

the stabilizer of Σ in M ∼= M24) induces the full symplectic group of the space VΣ.

a

∆1(a)

∆12(a) ∆2

2(a) ∆32(a)

15,180

1

84

4,032

10,752

3+84+224

45 185

By (1.8.5) Ms contains up to conjugation two Sym5-subgroups. One of them actstransitively on TΣ while the other one stabilizes an associated quadratic form ofminus type on VΣ and has two orbits on TΣ with lengths 5 and 10 (isotropic andnon-isotropic vectors, respectively).

Let T be a trio and let ST be the set of sextets which refine T . Then |ST | = 7and Mt

∼= 26 : (L3(2) × Sym3) (the stabilizer of T in M) induces on ST thenatural action of L3(2). This action preserves on ST a unique structure π of theprojective plane of order 2. A line

l = Σ1,Σ2,Σ3

of π is called a sextet-line . The stabilizer of l in M is contained in Mt,

Mt(l) ∼= 26 : (Sym4 × Sym3)

and Mt(l) is the stabilizer of an element from G(M24) of the intermediate type.

Lemma 8.3.20 Let h ∈ ∆12(a). Then there is a unique Σ ∈ Sa such that

(i) if T ∈ Ta, then (T, β) = λa(e) for some e ∈ µ(a, h) and for aGF (2)-valued function β on T if and only if T ∈ TΣ (equivalently ifT is refined by Σ);

(ii) if T ∈ TΣ then there are exactly three functions β on T such thatλ−1

a (T, β) ∈ µ(a, h).

168 173,067,389-Vertex graph ∆

Proof Let d be the unique element from D[3]ah. Then d is a sextet from Sa and

the elements b ∈ D[2]a such that D[m]

b ⊆ D[m]d are the trios in Ta refined by d. In

view of (8.3.3) the result now follows from (8.3.5).

Lemma 8.3.21 Let h ∈ ∆22(a). Then there is a unique trio Th ∈ Ta and a

sextet-line l = Σ1,Σ2,Σ3 ⊆ ST h such that

(i) if T ∈ Ta, then (T, β) = λa(e) for some e ∈ µ(a, h) and for a GF (2)-valued function β on T if and only if T ∈ TΣi for i = 1, 2, or 3 and thereis a unique octad in Ba which is contained in both T and Th;

(ii) if (T, β) = λa(e) for e ∈ µ(a, h) then β is uniquely determined by T .

Proof By (8.3.15) GahQ[m]a = NMt

(R)Q[m]a , where NMt

(R) is the stabilizer ofan element of the intermediate type from the tilde geometry G(M24). We claimthat NMt

(R) has a unique orbit X on Ta of length 18 and no orbits of length 9.In fact, NMt

(R) contains O2(Mt), therefore by the diagram Dt(M24) an orbit oflength 18 or 9 must be contained in the orbit Y of Mt on Ta of length 42 = 2 ·21.Notice that if Th is the trio stabilized by Mt then Y are the trios intersectingTh in one octad. It is now elementary to check that NMt(R) acts on Y with twoorbits of lengths 18 and 24. This proves the claim. Now the result is rather easyto deduce.

Lemma 8.3.22 Let h ∈ ∆32(a). There there is a unique Σ ∈ Sa and a unique

associated quadratic form q of minus type on the symplectic space VΣ such that

(i) if T ∈ Ta, then (T, β) = λa(e) for some e ∈ µ(a, h) and for a GF (2)-valued function β on T if and only if T ∈ TΣ (equivalently if T is refinedby Σ) and q(T ) = 0;

(ii) if T ∈ TΣ and q(T ) = 0 then there is a unique function β on T such thatλ−1

a (T, β) ∈ µ(a, h).

Proof The assertion is a reformulation of (8.3.19 (iii)).

8.4 Earthing up ∆13(a)

Lemma 8.4.1 If (ma,

3b,

mc ,

2d,

me ) is an arc in D with d∆(a, e) = 3 then ∠abc = 24

and ∠bcd = 26 · 45.

Proof Clearly d∆(a, c) = 2, and so ∠abc = 24. If d is incident to b, then b isincident to e in which case the diagram Dh(M22) shows that d∆(a, e) = 2. Thisis a contradiction.

Let x ∈ D[2]bc and suppose that ∠xcd = 22 · 14. Then by (8.2.5 (iii)) the set

D[m]x is contained in e∪∆(e). On the other hand, by Dt(M24) we observe that

∆(a) ∩ D[m]x = ∅. Therefore d∆(e, a) ≤ 2, which is again a contradiction.

Earthing up ∆13(a) 169

Thus ∠xcd = 22 · 14 for all x ∈ D[2]bc . Suppose that ∠bcd = 23 · 45. Then by

Ds(M24) there exists a path (3b,

2x,

3y,

2d) in the residue Dc with ∠x c

y d = 8. Thusby Dt(M24) we observe that ∠xcd must be 22 · 14, which is a contradiction.

Suppose that ∠bcd = 22 · 45. Then the residue Dc contains a path (3b,

0x,

2d).

Since ∠abc = 24 we can choose x in its Q[m]c -orbit in such a way that ∠abx =

22 · 15 (compare the diagram Dh(M22)), in which case there exists y ∈ D[2]abx.

Thus we have found a path

(ma,

2y,

0x,

2d,

me ).

Since d∆(a, e) = 3, D[m]yxd = ∅ and hence 〈Vx(y), Vx(d)〉 = Vx. By (8.3.13) and

(8.3.14) once again we have d∆(a, e) = 2, which is a contradiction.Thus ∠bcd = 26 · 45 and the lemma is proved.

Lemma 8.4.2 There exists a unique G-orbit on the set of paths

(ma,

3b,

mc ,

2d,

me )

in D with ∠abc = 24 and ∠bcd = 26 ·45. Moreover for such a path there is a path

(ma,

2h,

0f,

3g,

me )

with 〈Vf (h), Vf (g)〉 = Vf and ∠fge = 23 · 7.

Proof By the properties of the hexad graph (11.4.5) we have GacQ[m]c = Gbc

and so there exists a unique G-orbit on the set of paths

(ma,

3b,

mc ,

2d)

with ∠abc = 24 and ∠bcd = 26 · 45. Moreover, D[0]bcd = ∅ and so by (8.2.5)

(Q[3]b ∩ Q[m]

c )(Q[2]d ∩ Q[m]

c ) = Q[m]c .

Since Q[m]c acts transitively on D[m]

d \c, the uniqueness assertion holds.

By the diagram Ds(M24) the residue Dc contains a path (3b,

0f,

3g,

2d). Replacing,

if necessary, f by the other element in its Q[m]c -orbit, we assume that ∠abf =

22 · 45 (compare the diagram Dh(M22)), in which case there exists h ∈ D[2]abf .

Note that d is incident to e and g, and so e and g are incident because of thediagram of D. Since ∠bcd = 26 · 15, we see from the diagram Ds(M24) that D[2]

bcg

is empty and so 〈Vf (b), Vf (g)〉 = Vf (c). Moreover, since ∠abc = 24, c and h arenot incident. So Vf (b) ≤ Vf (h) ≤ Vf (c) and therefore 〈Vf (h), Vf (g)〉 = Vf .

Since ∠bcd = 26 · 45, d is not incident to f . On the other hand, both d andf are incident to c, and we see from Do(M22) that ∠fgd = 22 · 21 and hence∠fge = 22 · 14.

170 173,067,389-Vertex graph ∆

Lemma 8.4.3 There is a unique G-orbit on the set Ψ of paths

ψ = (ma,

2b,

0c,

3d,

me )

with Vc(b)∩Vc(d) = 0 and ∠cde = 23 · 7. Moreover, for such a path the followingassertions hold:

(i) |Gabcde| = 210 · 32;(ii) there is a unique path

χ = (ma,

3i,

0g,

2h,

me )

with Vg(h) ∩ Vg(i) = 0 and ∠gia = 22 · 14;(iii) Q

[m]a ∩ G

[m]e = 1;

(iv) d∆(a, e) = 3;(v) Gae = Gabcde;(vi) Q

[0]g ∩ Gae is elementary abelian of order 26 normal in Gae.

Proof By (8.3.10 (vii)) we have (Q[m]a ∩ G

[0]c )Q[0]

c = O2(Gcb) and so Q[m]a ∩ G

[0]c

acts transitively on the 64-element set

x ∈ D[3]c | Vc(b) ∩ Vc(x) = 0

Hence there is a unique G-orbit on the set of paths (ma,

2b,

0c,

3d) with Vc(b)∩

Vc(d) = 0. Moreover, in G[0]c /Q

[0]c

∼= L5(2) we observe that Gbcd/Q[0]c

∼= L3(2) ×Sym3 is a complement to O2(Gcd/Q

[0]c ) in Gcd/Q

[0]c . Thus Gcd = GbcdQd. By

(8.3.10 (v)) we have Gbc = GabcQ[0]c and so

Gcd = GbcdQ[3]d = ((GabcQ

[0]c ) ∩ G

[3]d )Q[3]

d = GabcdQ[0]c Q

[3]d .

We claim that Zb(Q[0]c ∩ Q

[3]d ) = Q

[0]c . In fact, let us identify the orbit of length

155 of G[0]c /Q

[0]c

∼= L5(2) on Q[0]c with the set of 2-dimensional subspaces in

Q[0]c

∼=∧2

Vc. Then by (7.7.2 (3)) the intersection Q[0]c ∩ Q

[3]d is generated by the

2-dimensional subspaces having non-zero intersection with Vc(d), while Z[2]b is

generated by the 2-dimensional subspaces contained in Vc(b). Since

Vc = Vc(d) ⊕ Vc(b)

the claim follows. Since Z[2]b ≤ Gabcd, we have the equality

Gcd = GabcdQ[0]c Q

[3]d = GabcdQ

[3]d ,

which particularly proves the transitivity of the G-action on the set ofpaths in Ψ.


By the transitivity assertion |Gabcde| is the quotient of |G[m]a | over the number

of paths in Ψ starting with a. Therefore

|Gabcde| =|G[m]

a ||D[2]

a | · ∠abc · ∠bcd · ∠cde= 210 · 32,

which gives (i).The shift from ψ to χ which constitutes the proof of (ii) can be illustrated

by the following diagram: 0g

3i

m

f2h

ma

2b

0c

3d

me

Since ∠cde = 23 · 7, ∠edc = 24 · 15, if follows form Dh(M22) that there is aunique f ∈ D[m]

cd with ∠edf = 24. Moreover, if g, c is the orbit of c under Q[m]f ,

then there exists h ∈ D[2]gde. Since Vc(d) ≤ Vc(f), Vc(b) ≤ Vc(f), there is a unique

element i ∈ D[3]bcf . In fact i is determined by the equality Vc(i) = Vc(f) ∩ Vc(b).

Then because of the shape of the diagram of D i is adjacent to a and to g aswell. Since Vc(b) ∩ Vc(d) = 0 and Vc(i) ≤ Vc(b), we have 〈Vc(i), Vc(d)〉 = Vc(f).Conjugation under Q

[m]f yields 〈Vg(i), Vg(d)〉 = Vg(f). Since ∠fde = 24, f is not

incident to h, and so since Vg(h) ≥ Vg(d), the equality Vg(i) ∩ Vg(h) = 0 holds.

Consider the path (0g,

m

f ,0c,

2b,

ma) in Di. By the diagram Do(M22) and since g, c

is a Q[m]f -orbit, we have ∠gic = 7 and ∠gib = 22 · 7. Now it is immediate to

observe from the diagram Do(M22) that D[m]ic and D[2]

ic are the point-set and theline-set of the projective plane of order 2 with the natural incidence relation. Inview of this observation and the indexes in Do(M22), every element from

x | x ∈ D[m]ib ,∠gix = 2 · 7

is incident to c. Hence ∠gia = 23 · 7, which proves (ii).Since

CQ

[3]i ∩G

[2]h

(Vg) = CQ

[3]i ∩G

[2]h

(Vg(i) ∩ Vg(h)) ≤ 〈Vg(i), Vg(h)〉 = Vg,

we have

Q[3]i ∩ G

[2]h ≤ Q[0]

g ,

172 173,067,389-Vertex graph ∆

and so Z[2]b ∩G

[2]h ≤ Q

[0]c ∩Q

[0]g . Assuming as above that 2-dimensional subspaces

in Vc are considered as elements of Q[0]c

∼=∧2

Vc we obtain

Z[2]b ∩ Q[0]

c ∩ Q[0]g = Z

[3]i .

Hence Z[2]b ∩ G

[2]h = Z

[3]i . Since Vg(i) ∩ Vg(h) = 0, Z

[3]i ≤ Q

[0]g ∩ Q

[2]h . Moreover,

Q[0]g ∩ G

[m]e = Q

[0]g ∩ Q

[2]h by (8.3.10 (vi)), and thus Z

[3]i ≤ G

[m]e .

Put K = Gabcde. Since each of f , g, h, and i is uniquely determined by(a, b, c, d, e), K ≤ Gfghi, and in particular

Z[2]b ∩ G[m]

e = Z[2]b ∩ K ≤ Z

[3]i ∩ G[m]

e = 1.

From the equalities Q[m]a ∩ G

[0]c ≤ Q

[2]b , Q

[2]b ∩ G

[3]d ≤ Q

[0]c , and Q

[m]a ∩ Q

[0]c = Z

[2]b

(cf. (8.3.10 (ii)) for the latter equality) we conclude that Q[m]a ∩ K = 1. By (i)

|K| = 210 · 32. Since K ≤ Gabi and

|Gabi/Q[m]a | =

|Gab/Q[m]a |

7= 210 · 32,

we have KQ[m]a = Gabi.

Suppose that Q[m]a ∩ G

[m]e = 1 and let R be a Sylow 2-subgroup of K. Then

CQ

[m]a ∩G

[e]e

(R) = 1, and since CQ

[m]a

(R) = CQ

[m]a

(RQ[m]a ) = Z

[3]i , we get Z

[3]i ≤

G[m]e , a contradiction. Hence Q

[m]a ∩ G

[m]e = 1 which gives (iii).

By (8.1.3 (iii)) and (8.3.2 (ii)) we know that for every y ∈ ∆2(a) there is anon-identity element in Q

[m]a which stabilizes y. Therefore by (iii) we conclude

that d∆(a, e) = 3 which is (iv).To establish (v) suppose to the contrary that K = Gae. Then by (11.2.3 (ii))

the subgroup GaeQ[m]a is one of G

[m]a , Gab and Gai. Suppose that Gabe = K. Then

GabeQ[m]a = Gab. In particular |Gabe/K| = 7 and so Gabc = O2,3(Gabe)K. From

the equality ∠abc = 4 we conclude that O2,3(G[2]b ) ≤ G

[0]c . Now also K ≤ G

[0]c

and so Gabe ≤ G[0]c . Note that

Q[0]c ∩ G[m]

e = Z[2]b (Q[0]

c ∩ Q[3]d ) ∩ G[m]

e = Q[0]c ∩ Q

[3]d ,

and so Gce ≤ NG

[0]c

(Q[0]c ∩ Q

[3]d ) = Gcd. Thus Gce = Gcde and Gabe = Gabce =

Gabcde = K, a contradiction.Therefore Gabe = K and GaeQ

[m]a = Gai. On the other hand, by (8.3.10 (iv))

Gge ≤ Gghe, and as seen above Q[3]i ∩ G

[2]h ≤ Q

[0]g . In particular Q

[3]i ∩ G

[m]e ≤

Q[3]i ∩G

[2]h = Q

[3]i ∩Q

[0]g . Since Z

[3]i permutes transitively the set of two elements in

D[m]h \D[m]

gh , Q[3]i ∩Q

[0]g = Z

[3]i (Q[3]

i ∩G[m]e ). Thus Gae = Gaie ≤ N

G[3]i

(Q[3]i ∩Q

[0]g ) =

Gig. This is a contradiction, since 33 divides |Gae| = |Gai/Q[m]a | but not |Gig|.

Thus Gae = Gabcde = Gabcde = Gabcdefghi completing the proof of (v).Now it only remains to prove (vi). By (v) and the transitivity assertion

ψ is uniquely determined by a and e. By (ii) χ is uniquely determined by ψ.


Therefore Gae stabilizes g and hence normalizes Q[0]g ∩ Gae. Let us identify the

latter intersection. We identify the 2-dimensional subspaces in Vg with elementsin Q

[0]g

∼=∧2

Vg. Since Vg(h) ∩ Vg(i) = 0 we have Vg = Vg(h) ⊕ Vg(i). ThereforeQ

[0]g ≤ Gigh,

F := Gigh/Q[0]g

∼= Sym3 × L3(2)

and as an F -module Q[0]g is the direct sum of three irreducible submodules Wi,

Wh, and Wih of dimensions 3, 1 and 6, respectively. Here Wi is generated bythe 2-dimensional subspace of Vg contained in Vg(i), Wh is generated by Vg(h)(which is itself a 2-dimensional subspace) and Wih is generated by the subspaceswhich intersect both Vg(i) and Vg(h). By (7.7.3 (8)) Q

[0]g induces an action of

order 2 on D[m]i and by (7.7.4 (9)) Q

[0]g induces an action of order 24 on D[m]

h .Clearly Q

[0]g permutes transitively the pair of elements in D[m]

h \ D[m]g (notice

that a is one of them), while by Do(M22) the orbit of e under Q[0]g has length

23. In the above terms this means that Gae ∩ Q[0]g = Wih completing the proof

of (vi).

Define ∆13(a) to be the set of vertices e ∈ ∆ such that there is a path ψ ∈ Ψ

as in (8.4.3) which joins a and e.


(i) ∆13(a) is an orbit of G

[m]a on ∆3(a);

(ii) |∆13(a)| = 211 · 3 · 5 · 7 · 11 · 23 = 54, 405, 120;

(iii) Q[m]a acts fixed-point freely on ∆1

3(a);(iv) if e ∈ ∆1

3(a) then the set ∆(e) ∩ ∆12(a) is of size 6 and the action of Gae

on this set is transitive.

Proof Statement (i) follows from the transitivity assertion in (8.4.3) and thedefinition of ∆1

3(a). By (i) |∆13(a)| is |G[m]

a | divided by |Gae|. Therefore (8.4.3 (i))and (8.4.3 (v)) give (ii). Statement (iii) is by (8.4.3 (iii)). By (8.4.1), (8.4.2), and(8.4.3 (iv)) for every vertex c ∈ ∆1

2(a) the set ∆(c)∩∆13(a) is of size 2880 = 26 ·45

and Gae acts transitively on this set. Comparing the size of ∆13(a) in (ii) with

the size of ∆12(a) in (8.1.3) we obtain (iv).

A direct consequence of (8.4.1), (8.4.2), and (8.4.3) is the following result.

Corollary 8.4.5 Let (ma,

3j,

m

k,2l,

me ) be an arc in D. Then either d∆(a, e) ≤ 2 or

e ∈ ∆13(a).

Now we are ready to prove the final result of the section.

Lemma 8.4.6 Let (ma,

3j,

m

k,3l,

me ) be a path in D. Then either d∆(a, e) ≤ 2 or

e ∈ ∆13(a).

174 173,067,389-Vertex graph ∆

Proof First consider an arc (ma,

0b,

3c,

0d,

me ). Then by Do(M22) there is an arc

(0b,

2f,

mg ,

0d) in Dc. Take i ∈ D

[3]abf and h ∈ D

[2]gde. Then by the diagram of D

the element i is incident to g while h is incident to e. Thus we have an arc

(ma,

3i,

mg ,

2h,

me ). By (8.4.5) the assertion holds.

Now consider a path (ma,

3j,

m

k,3l,

me ) as in the hypothesis of lemma. By the dia-

gram Ds(M24) there is a path (3j,

2f,

3g,

2h,

3l) in Dk. If d∆(a, k) ≤ 1 or d∆(k, e) ≤ 1

then we are done by (8.4.5). So suppose that ∠ajk = 24 = ∠kle. Then by thediagram Dh(M22) applied to the residues of j and l we conclude that

∠ajf = 24 · 6, ∠elh = 24 · 6

and there exist b ∈ D[0]ajf and d ∈ D[0]

elh. By the diagram of D both b and d areincident to g. Hence we are done by the first paragraph of the proof.

8.5 Earthing up ∆23(a)

Lemma 8.5.1 Let ϕ = (3b,

2c,

3d,

2e,

3f) be a path in D such that [Z [3]

b , Z[3]f ] = 1.

Then

(i) ∠cde = 24 · 10;(ii) Q

[3]b ∩ Q

[3]f = Z

[3]d ;

(iii) Q[3]d ≤ (Q[3]

b ∩ Q[3]d )Q[2]

e ;(iv) Q

[3]b ∩ G

[2]e acts transitively on the set of four elements in

α ∈ D[3]e | [Z [3]

b , Z [3]α ] = 1;

(v) GbdfQ[3]d /Q

[3]d

∼= Sym3 × 2;(vi) (Q[3]

b ∩ Q[3]d ∩ G

[3]f )N [3]

f /N[3]f = (Q[3]

d ∩ G[3]f )N [3]

f /N[3]f = O2(Gef/N

[3]f ).

Proof By the diagram Dp(M22) if ∠cde = 24 · 10 then there exists x ∈ D[m]cde , in

which case Z[2]c Z

[2]e ≤ Q

[m]c and [Z [2]

c , Z[2]e ] = 1, which contradicts the hypothesis,

since Z[3]b ≤ Z

[2]c and Z

[3]f ≤ Z

[2]e . Hence (i) follows.

We are going to construct some bypasses of the path ϕ as shown on thefollowing diagram:

Since ∠cde = 24 · 10, by the diagram Dp(M22) there exits a unique path

(2c,

0g,

m

h,0i,

2e) in Dd such that i, g is a Q

[m]h -orbit. Then h is not incident to c.

On the other hand, by the diagram of D the element b is incident to g, while theelement f is incident to i.


3δ

2γ

0g

m

h0i

3b

2c

3d

2e

3f

Notice that 〈Vg(b), Vg(d)〉 = Vg(c), that Vg(c) ≤ Vg(h) and thatVg(h) ≥ Vg(d). Thus Vg(b) ≤ Vg(h) and so h is not incident to b. By symmetryh is not incident to f . Hence

Z[3]f ≤ Q

[m]h ∩ Q

[0]i = Q

[m]h ∩ Q[0]

g .

Put R = Q[0]g ∩ Q

[m]h . Then R ∩ Q

[3]b = (Q[3]

b ∩ Q[0]g ) ∩ (Q[m]

h ∩ Q[0]g ). On the other

hand, identifying as usual the 2-dimensional subspaces in Vg with the corres-ponding elements in Q

[0]g we observe the following. The intersection Q

[3]b ∩ Q

[0]g

is generated by the 2-dimensional subspaces having non-zero intersection withVg(b), while Q

[m]h ∩Q

[0]g is generated by the 2-dimensional subspaces with non-zero

intersection with Vg(h). Since Vg(b) is itself 2-dimensional and Vg(b) intersectsVg(h) in a 1-dimensional subspace, we conclude that the subgroups of order 2 inR ∩ Q

[3]b are all of the form Z

[3]δ for δ ∈ D[3]

gh = D[3]ghi with

Vg(b) ∩ Vg(h) ≤ Vg(δ) ≤ Vg(h).

Notice that for any such δ there exists γ ∈ D[2]bgδ and so Z

[3]b ≤ Z

[2]γ ≤ Q

[3]δ .

Suppose that R ∩ Q[3]b ∩ Q

[2]f = Z

[3]d and pick δ ∈ Dghi\d with Z

[3]δ ≤

R ∩ Q[3]b ∩ Q

[3]f . Then Z

[3]b ≤ Q

[3]δ , and similarly Z

[3]f ≤ Q

[3]δ . Thus both Z

[3]b and

Z[3]f are contained in the elementary abelian group Q

[3]d ∩ Q

[3]δ , a contradiction.

Thus R ∩ Q[3]b ∩ Q

[3]f = Z

[3]d . Since |R| = 26 and

|R ∩ Q[3]b | = 23 = |R ∩ Q

[3]f |,

we get |R/(R ∩ Q[3]b )(R ∩ Q

[3]f )| = 2. Since

[R ∩ Q[3]b , Q

[3]b ∩ Q

[3]f ] ≤ Z

[3]b ∩ R = 1,

176 173,067,389-Vertex graph ∆

we have [(R ∩ Q[3]b )(R ∩ Q

[3]f ), Q[3]

b ∩ Q[3]f ] = 1. Now R is a natural orthogonal

module for Gig/Q[0]i Q

[0]g

∼= Ω+6 (2), and so no element of Gig acts as a transvection

on R. Thus Q[3]b ∩ Q

[3]f ≤ CGig

(R) = Q[0]i Q

[0]g . By (7.7.2) we have

CVi(Q[0]

i Q[0]g ) ≥ Vi(h), CVi

(Q[3]f ) ≥ Vi(f),

CVi(Q[0]

i Q[0]g ∩ Q

[3]f ) ≥ 〈Vi(f), Vi(h)〉 = Vi,

and so Q[0]g Q

[0]i ∩ Q

[3]f ≤ Q

[0]i . By symmetry Q

[0]g Q

[0]i ∩ Q

[3]b ≤ Q

[0]g and thus

Z[3]d ≤ Q

[3]b ∩ Q

[3]f ≤ (Q[0]

i Q[0]g ∩ Q

[3]b ) ∩ (Q[0]

i Q[0]g ∩ Q

[3]f )

= Q[3]b ∩ Q[0]

g ∩ Q[0]i ∩ Q

[3]f = Q

[3]b ∩ R ∩ Q

[3]f = Z

[3]d ,

which gives (ii).Since |Q[3]

b ∩ Q[3]d | = 27 = |Q[3]

f ∩ Q[3]d | and |Q[3]

d | = 213, we conclude that

Q[3]d = (Q[3]

b ∩ Q[3]d )(Q[3]

f ∩ Q[3]d ) ≤ (Q[3]

b ∩ Q[3]d )Q[2]

e ,

which is (iii).By (8.3.12 (vi)) (Q[3]

b ∩ G[3]d )N [3]

d /N[3]d = O2(Gcd/N

[3]d ) and by (8.3.12 (ii))

GbcdN[3]d = Gcd. Thus GbcdeN

[3]d = Gcde. Also

(Q[3]b ∩ G

[3]d )N [3]

d ∩ GbcdeN[3]d = ((Q[3]

b ∩ G[3]d ) ∩ (GbcdeN

[3]d ))/N [3]

d

= (Q[3]b ∩ G[2]

e )N [3]d

and

(Q[3]b ∩ G[2]

e )N [3]d /N

[3]d = O2(Gcd/N

[3]d ) ∩ Gcde/N

[3]d .

It is easy to deduce from the diagram Dp(M22) and the structure of the stabilizer25 : Sym5 of a pair in Aut (M22) that GbdeQ

[3]d /(Q[3]

b ∩ G[2]e )Q[3]

d∼= Sym3 × 2,

while (Q[3]b ∩ G

[2]e )Q[3]

d /Q[3]d has order two and inverts N

[3]d /Q

[3]d

∼= 3. SinceN

[3]d Q

[2]e /Q

[2]e

∼= Alt4 and since by (iii) the equality Q[3]d Q

[2]e /Q

[2]e = (Q[3]

b ∩Q

[3]d )Q[2]

e /Q[2]e holds, we conclude that Q

[3]b ∩ G

[2]e acts as the dihedral group

D8 of order eight on Z[2]e with Z

[3]b mapping onto the centre of D8. Hence (iv)

follows. Furthermore,

[Q[3]b ∩ G

[3]f , Z [2]

e ] = [Q[3]b ∩ G

[3]f , C

Z[2]e

(Z [3]b )Z [3]

f ] ≤ Z[3]d ,

Q[3]b ∩ G

[3]f ≤ Q

[3]d and GbdfQ

[3]d /Q

[3]d

∼= Sym3 × 2,

which is (v). Since by (iii)

(Q[3]b ∩ G

[3]f )N [3]

f = (Q[3]b ∩ Q

[3]d ∩ G

[3]f )N [3]

f = (Q[3]d ∩ Gf )N [3]

f ,

(vi) follows from (8.3.12 (vi)).



ρ = (ma,

3b,

2c,

3d,

2e,

3f,

mg )

in D such that ∠abc = 24 · 6 = ∠gfe and [Z [3]b , Z

[3]f ] = 1. Moreover, for such a

path ρ the following assertions hold:

(i) CGag (Zd) = Gadg = Gabcdefg;(ii) GadgQ

[3]d /Q

[3]d

∼= Sym3 × 2;(iii) Q

[3]d ∩ Gadg = Z

[3]d ;

(iv) |Gadg| = 24;(v) Q

[m]a ∩ G

[m]g = 1;

(vi) d∆(a, g) ≥ 3 and g ∈ ∆13(a);

(vii) there exists a path

π = (ma,

2l,

0k,

2p,

mi ,

2n,

mg )

in D such that(1) |Gadg/(G[3]

d ∩ G(π))| = 3;(2) i ∈ D[m]

d ;(3) Z

[3]d ≤ Q

[m]i ∩ G

[3]d (π).

Proof By (8.3.16) there is a unique G-orbit of the set of paths (ma,

3b,

2c,

3d) with

∠abc = 24 · 6. Furthermore, by (8.3.16 (ii), (iv), (v)) we have respectively

GabcdN[3]d = Gcd,

GabcdQ[m]a /Q[m]

a is the centralizer of Z[3]d in G[m]

a /Q[m]a and

Gabcd = CG

[m]a

(Z [3]d ) = Gad.

The latter equality implies that in (i).From (8.5.1 (i), (ii), (iv)) we conclude that ∠cde = 24 · 10, Q

[m]b ∩ Q

[3]f = Z

[3]d

and that Q[m]b ∩ G

[2]e acts transitively on Z

[2]e \ Z

[3]b . Thus the G-orbit of the

subpath in ρ which joins a and f is uniquely determined. Next by (8.5.1 (v))we have GbdfQ

[3]d /Q

[3]d

∼= Sym3 × 2. It follows from the diagram Dp(M22) thatO2(Gef/N

[3]f ) acts regularly on X = α ∈ D[m]

f | ∠efα = 25. Finally by(8.5.1 (vi))

O2(Gef/N[3]f ) = (Q[3]

b ∩ Q[3]d ∩ Gf )N [3]

f /N[3]f = (Q[3]

d ∩ G[3]f )N [3]

f /N[3]f ,

Q[3]b ∩Q

[3]d ∩G

[3]f acts transitively on X and Q

[3]d ∩G

[3]f ≤ Q

[3]f . Similarly, Q

[3]d ∩Q

[3]f ∩

G[3]b acts transitively on Y = β ∈ D[m]

b | ∠cbβ = 25, and Qd ∩Gab ≤ Q[3]b . Since

(Q[3]b ∩ Q

[3]d ) ∩ (Q[3]

f ∩ Q[3]d ) = Z

[3]d

178 173,067,389-Vertex graph ∆

the transitivity assertion follows along with (ii).Notice that

CQ

[3]b ∩G

[m]g

(Z [3]d ) ≤ Q

[3]b ∩ Ggf ≤ Q

[3]b ∩ Q

[3]f = Z

[3]d ,

and so Q[3]b ∩ G

[m]g = Z

[3]d . Thus

Q[3]d ∩ Gadg = Q

[3]b ∩ Q

[3]d ∩ Q

[3]f = Z

[3]d ,

which is (iii). Furthermore, since

|Gadg| =|Gab|

∠abc · ∠bcd · ∠cde · ∠def · ∠efg=

|Gab|96 · 6 · 160 · 4 · 32

= 24

(which is (iv)) and Q[m]a ∩ G

[2]c ≤ Q

[3]b , we obtain the equalities

Q[m]a ∩ Gdg = Q[m]

a ∩ Q[3]b ∩ G[m]

g = Q[m]a ∩ Z

[3]d = 1.

Hence CQ

[m]a ∩G

[m]g

(Z [3]d ) = 1 and Q

[m]a ∩ G

[m]g = Q

[m]a ∩ Gag = 1 and (v) follows.

By (v) d∆(a, g) ≥ 3. Suppose that g ∈ ∆13(a). Then by (8.4.3 (vi)) Gag

contains a normal elementary abelian subgroup A of order 26. If Z[3]d is in A,

then 26 divides |CGag(Z [3]

d )|, and if Z[3]d ∈ A, then 23 divides |CA(Zd)| and so 24

divides CA(Zd)Zd|, and in any case we reach a contradiction with (iv) (which is|Gadg| = 24) and hence (vi) follows.

The proof of (vii) is illustrated by the following diagram:

2p

2l

0k

mj

0h

mi

ma

3b

2c

3d

2e

2n

3f

mg

By the diagram Dp(M22) in Dd there are three paths (2c,

0h,

mi ,

2e). Since an

element of order 3 in N[3]d acts fixed-point freely on Q

[3]d /Z

[3]d , since Z

[3]b ≤ Q

[3]d

and since N[3]d ∩G

[3]b ≤ Q

[3]d , these three paths are transitively permuted by Gcde.

We choose one of these paths. Then h is incident to b, and since ∠abc = 24 · 6,

it follows from the diagram Dh(M22) there is a unique path (ma,

2l,

0k,

mj ,

0h) in Db

such that k, h is a Q[m]j -orbit. Let p be the unique vertex in D[2]

jhi. Since k, h is

a Q[m]h -orbit, p is incident to k. Since ∠gfe = 24 ·6, i is incident to both e and f .

Now Dh(M22) shows that there is a unique n ∈ D[2]gfi. Then by the construction


π = (ma,

2l,

0k,

2p,

mi ,

2n,

mg ) possesses the properties (1) and (2). Since h and i are

incident to d, Z[3]d ≤ Gadghi = G

[3]d (π) and Z

[3]d ≤ Q

[m]i , which gives (3).


π = (ma,

2b,

0c,

2d,

me ,

2f,

mg )

satisfying d∆(a, g) ≥ 3 and g ∈ ∆13(a). Moreover, for such a path π

|G(π)| = 24 and G(π)/O2(G(π)) ∼= Sym3.

Proof The existence of π has been established in (8.5.2). Suppose there isx ∈ D[3]

bcd. Then by the diagram of D the element x is incident to both a ande, which contradicts (8.4.6), so 〈Vc(b), Vc(d)〉 = Vc and the subpath in π joininga and e is as in (8.3.13).

Suppose that Z[2]d ∩G

[m]a ∩Q

[2]f = 1. We will reach a contradiction by extending

the path π as shown on the following diagram:

0v

my 3

x2z

ma

2b

0c

2d

me

3u

2f

mg

3s

2w

mt

Under the above assumption pick x ∈ D[3]dc with Z

[3]x ≤ Z

[2]d ∩G

[m]a ∩Q

[2]f . Then

x is incident to c and Z[3]x ≤ Q

[0]c . Since Q

[0]c ∩ G

[m]a ≤ Q

[2]b , we get Z

[3]x ≤ Q

[2]b .

Thus by (8.3.11) Vc(x) ∩ Vc(b) = 0. In particular there exists y ∈ D[m]bcx. Since

Z[3]x ≤ Q

[2]f , (8.3.8 (iii)) implies that ∠fex = 26 · 21 and so by Ds(M24) there

exists a path (3x,

2z,

3u,

3f) in De. Then g is incident to u. Let c, v be the Q

[m]y -

orbit containing c. Then in the residue of b we observe that a is incident tov, since it is not incident to c. Clearly v is also incident to x. By the diagramDp(M22) there exists a path (

0v,

2w,

mt ,

2z) in the residue Dx. Then u is incident to

t and g. Pick s ∈ D[3]avw. Then s is incident to t and we have constructed a path

(ma,

3s,

mt ,

3u,

mg ) contrary to (8.5.1).

180 173,067,389-Vertex graph ∆

Hence Z[2]d ∩ G

[m]a ∩ Q

[2]f = 1. If ∠def = 26 · 42 then the diagram Ds(M24)

and (8.3.8 (iii)) imply that |Z [2]d ∩ Q

[2]f | ≥ 4. By (8.3.13 (i)) |Z [2]

d ∩ G[m]a | = 22.

Since Z[2]d has order 8, the latter means that Z

[2]d ∩ Q

[2]f ∩ G

[m]a = 1. So ∠def =

26 · 42, |Z [2]d ∩ Q

[2]f | = 2 and by (8.3.13 (v)) GaeQ

[m]e = NGde

(Z [2]d ∩ G

[m]a ).

Put X = CGde(Zd). Now it is easy to deduce from the diagram Dt(M24) that

GdefX = NGde(Z [2]

d ∩ Q[2]f ) and Gdef/Q

[m]e

∼= Sym4. It follows that Gdef actstransitively on the set

A ≤ Z[2]d

∣∣∣|A| = 4, A ∩ Q[2]f = 1.

Moreover, NGdef(A)/Q

[m]e

∼= Sym3 for any such A. Thus both NGde(Z [2]

d ∩ G[m]a )

and Gae act transitively on

f ∈ D[2]e | ∠def = 26 · 42, Z

[2]d ∩ G[m]

a ∩ Q[2]f = 1.

Also the following isomorphisms take place

GaefQ[m]e /Q[m]

e∼= Sym3

∼= Gaef/O2(Gaef ).

Moreover, Q[m]e = (Z [2]

d ∩ G[m]a )(Q[m]

e ∩ Q[2]f ) and so Z

[2]d ∩ G

[m]a acts transitively

on D[m]f \ e. Thus the paths as in the hypothesis constitute a single G-orbit

and

G(π)Q[m]e /Q[m]

e∼= Sym3

∼= G(π)/O2(G(π)).

Finally

|Gabcdefg| =|Gae|

47 · (26 · 42) · 4

= 24

and the lemma is proved.

Let g be a vertex of ∆ such that the path π as in (8.5.2 (vii)) and (8.5.3)joins a with g. Let ∆2

3(a) be the set of images of g under the elements of G[m]a .

Lemma 8.5.4 The set ∆23(a) is an orbit of G

[m]a on ∆3(a).

Proof We assume that π is a path which joins a and g as in (8.5.3). In view of thedefinition we only have to show that d∆(a, g) = 3. By (8.5.2 (vi)) d∆(a, g) ≥ 3.By the first paragraph of the proof of (8.5.3) along with (8.3.13), (8.3.14), and(8.3.15) it follows that d∆(a, e) = 2. Clearly d∆(e, g) = 1 by the shape of π, theresult follows.

Lemma 8.5.5 Let g ∈ ∆23(a). Then

(i) |Gag| = 23 · 3 · 11 · 23;(ii) Gag has two orbits on D[0]

g and permutes transitively the Q[m]g -orbits

on D[0]g .


Proof Let the paths

ρ = (ma,

3b,

2c,

3d,

2e,

3f,

mg )

and

π = (ma,

3l,

0k,

2p,

mi ,

2n,

mg )

be as in (8.5.2). Then by (8.5.2) and (8.5.3) we have the following:

|Gadg| = 24 = |G(π)|, |Gd ∩ G(π)| = 8,

CGag(Z [3]

d ) = Gadg, Gag ∩ Q[m]g = 1,

Gadg/Z[3]d

∼= Sym3 × 2, G(π)/O2(G(π)) ∼= Sym3.

Put A = Q[m]i (π). Then by (8.5.2 (3)) Z

[3]d ≤ A. Hence A is a non-trivial

normal 2-subgroup of G(π), and CG(π)(A) ≤ G[3]d ∩ G(π) is a 2-group. Since

|O2(G(π))| = 4, we get that A is elementary abelian of order 4 and thatG(π) ∼= Sym4. Thus Gd(π) is a dihedral group of order 8, and so NGag

(G[3]d (π)) ≤

CMag(Z [3]

d ). In particular, G[3]d (π) is a Sylow 2-subgroup of Gag. Moreover, there

exists an element t ∈ G such that

(t(a), t(b), t(c), t(d), t(e), t(f), t(g)) = (g, f, e, d, c, b, a).

Notice t ∈ G[3]d , and so t normalizes Gadg. Thus we may assume that t normalizes

G[3]d (π).

We claim that A ∩ At = Z[3]d . Clearly, Z

[3]d ≤ A ∩ At. By (8.5.2) i is incident

to d and since t ∈ G[3]d , t(i) is also incident to d. Since d∆(a, g) > 2, we have

i = t(i). We claim that Q[m]i ∩ (Q[m]

i )t ≤ G[3]d . In fact, if D[2]

idt(i) = ∅ (i.e. if

∠idt(i) = 24), then Q[m]i ∩ (Q[m]

i )t ≤ O2(Gidt(i)) ≤ Q[3]d , and if δ ∈ D[2]

idt(i), then

by (8.2.5) Q[m]i ∩ (Q[m]

i )t = Z[2]δ ≤ Q

[3]d . By (8.5.2 (iii)) Q

[3]d ∩ Gag = Z

[3]d , and so

A ∩ At ≤ Gag ∩ Q[m]i ∩ (Q[m]

i )t ≤ Gag ∩ Qd ≤ Z[3]d .

In particular, A = At. Put E = O2(Gadg). Since Gadg/Z[3]d

∼= Sym3 × 2, |E| = 4.Moreover, t normalizes E, and so A = E = At, E is cyclic of order 4 and Gadg is adihedral group of order 24. Let D = O3(Gadg) and note that ED = CGag (Z [3]

d D).Since D centralizes Z

[3]d , the information on involution centralizers in M24 implies

that

CG

[m]g /Q

[m]g

(D) ∼= 3 × L3(2).

Now a subgroup of L3(2) with a centralizer of an involution isomorphic to a cyclicgroup of order four clearly is a cyclic group of order four, and so CGag

(D) = DE.In particular, D is a Sylow 3-subgroup of Gag. Note that all involutions in

182 173,067,389-Vertex graph ∆

G[3]d (π) are contained in A ∪ At and so conjugate into Z

[3]d under G(π) and

G(π)t, respectively. Thus Gag has a unique class of involutions. Let z be anyinvolution in Gag and put

C(z) = Gadg ∩ CGag(z).

If 3 divides |C(z)|, z ∈ CGag(D) = DE and z ∈ Z

[3]d . Hence exactly one of the

following holds: z ∈ Gadg, |C(z)| = 2, or C(z) = 1. Moreover, if C(z) = 〈y〉for one of the 12 involutions y ∈ Gadg \ Z

[3]d , then z is one of the 10 involu-

tions in CGag(y) \ Gadg. Thus, if r is the number of involutions in Gag, that is,

r = |Gag/Gadg|, then

r = 13 + 12 · 10 + 24s = 133 + 24s

for some non-negative integer s. On the other hand, since

|G[m]g /Q[m]

g | = 210 · 33 · 5 · 7 · 11 · 23,

r divides 5 · 7 · 11 · 23, and we conclude that r = 11 · 23 or r = 5 · 7 · 23. Thelatter case is impossible by Burnside’s p-complement theorem for p = 23, and sor = 11 · 23.

Hence |Gag| = 23 · 3 · 11 · 23. In particular, Gadg and G(π) are maximal2, 3, 5, 7-subgroups of Gag. Since both Gfg and Gng are 2, 3, 5, 7-groups, weconclude that Gafg = Gadg and Gang = G(π).

Since |D[0]ing| = 3, we can choose x ∈ D[0]

nig with G[3]d (π) ≤ G

[0]x . Since the non-

trivial elements of odd order in Gag act fixed-point freely on D[2]g , we conclude

that Gagx = G[3]d (π) = Gagy = Gagx, y, where x, y is a Q

[m]g -orbit. In

particular, |Gag/Gagx| = 759, and the lemma is proved.

It can be shown using the list maximal subgroups in M24 and/or the classi-fication of groups with dihedral Sylow 2-subgroup that in the above lemma Gag

is isomorphic to L2(23).As a consequence of (8.5.5 (i)) and direct calculations we have the following.

Corollary 8.5.6 |∆23(a)| = 218 · 32 · 5 · 7 = 82, 575, 360.

We have constructed seven orbits of G[m]a (including a itself) on the vertex-set

of ∆. The information about these orbits from (8.1.1 (vii)), (8.1.3 (iv)), (8.3.15(i)), (8.3.19 (i)), (8.4.4 (ii)), and (8.5.6) is summarized in Table 3 below. Thistable shows that we already see the required number of vertices. In the nextsection we show that all the vertices of ∆ are in the accounted orbits.

In the last column of Table 3 the stabilizers Gae are given in the form(Gae ∩ Q

[m]a ).(GaeQ

[m]a /Q

[m]a ).

8.6 | G | = 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43

Lemma 8.6.1 Let g ∈ V (∆) and d∆(a, g) = 3 then g ∈ ∆13(a) ∪ ∆2

3(a).

| G | = 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43 183

Table 3. G[m]a -orbits on ∆

Orbit Size Prime decomposition Stabilizer

a 1 1 (211) : (M24)∆(a) 15,180 22 · 3 · 5 · 11 · 23 (29) · (26.(L3(2) × Sym3))∆1

2(a) 28,336 24 · 7 · 11 · 23 (27) · (26 : 3 · Sym6)∆2

2(a) 3,400,320 27 · 3 · 5 · 7 · 11 · 23 (24) · (26 · (Sym4 × Sym3))∆3

2(a) 32,643,072 211 · 32 · 7 · 11 · 23 (2) · (26.Sym5)∆1

3(a) 54,405,120 211 · 3 · 5 · 7 · 11 · 23 (1) · (26 · (Sym4 × Sym3))∆2

3(a) 82,575,360 218 · 32 · 5 · 7 (1) · (L2(23))Total 173,067,389 112 · 29 · 31 · 37 · 43

Proof Let e be a vertex of ∆ adjacent to g which is at distance 2 from a. Ife ∈ ∆1

2(a) then by (8.4.5) g ∈ ∆13(a). If e ∈ ∆2

2(a) then g ∈ ∆23(a) by (8.5.3),

(8.5.4), (8.4.3), and (8.3.13).So we assume that e ∈ ∆3

2(a). Then by (8.3.18) there exists a path

(ma,

3b,

2c,

3d,

me ,

2f,

mg )

with ∠abc = 24 · 6 = ∠edc. We are going to produce an alternative path in Djoining a and g as on the below diagram.

2n

m

l2k

0h

ma 3

b2c

3d

me

0j

2f

mg

2i

By the diagram Ds(M24) there is an arc (3d,

0h,

2i,

0j,

2f) in De. Note that

∠cde = 25 and hence there is k ∈ D[2]deh with ∠cdk = 23 · 5. Thus by the dia-

gram Dp(M22) there exists l ∈ D[m]cdk. Then by the diagram of D the elements l

and b are incident. Next, by the diagram Dh(M22) and since ∠abc = 24 · 6, there

exists n ∈ D[2]abl, and so a and l are incident. Considering the arc (

2k,

0h,

2i,

0j,

2f)

contained in De, we see by the diagram Ds(M24) that ∠kef = 26 · 42. Thus by

(8.3.18) applied to the arc (mg ,

2f,

me ,

2k,

m

l ) we observe that

d∆(l, g) ≤ 3 and l ∈ ∆32(g).

Thus by the first paragraph of the proof applied to (g, l, a) in place of (a, e, g),we obtain the inclusion a ∈ ∆1

3(g) ∪ ∆23(g). In view of the obvious symmetry

between a and g the result is established.

184 173,067,389-Vertex graph ∆

Proposition 8.6.2 If z is any vertex of ∆ then d∆(a, z) ≤ 3.

Proof Suppose the contrary, then by the connectivity of ∆ we may assume thatd∆(a, z) = 4.

We are going to establish the following claim:

there does not exist a path π = (ma,

3b,

mc ,

2d,

me ,

2f,

mz ).

Suppose to the contrary that such apathπ exists. Thend∆(a, e) = 3, d∆(c, z) =2, by (8.3.2 (iii)) and (8.4.1) this means that ∠abc = 24, ∠bcd = 26 · 45 and

c ∈ ∆α2 (z) for α = 1, 2, or 3.

We are going to consider the three possible values for α successively.α = 1: In this case there exists ρ ∈ D[3]

cz and we obtain a path

(ma,

3b,

mc ,

3ρ,

mz ).

Hence by (8.4.6) d∆(a, z) ≤ 3, a contradiction.α = 2: We are going to reduce the path π to a path with α = 1. This

reduction process is illustrated on the following diagram.

0j

2g 0

h2i

ma

3b

mc

2d

me

2f

mz

2y

m

k3l

mn 2

o

mp

0q

2r

We start by choosing a path (mc ,

2g,

0h,

2i,

mz ) with 〈Vh(g), Vh(i)〉 = Vh. By the

diagram Ds(M24) there exists j ∈ D[0]cg with ∠bcj = 26 ·6. Replacing, if necessary

j by the other vertex from the same Q[m]c -orbit we assume that j, h is a Q

[m]k -

orbit for some k ∈ D[m]gh .

| G | = 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43 185

Next take l ∈ D[3]khi. Then l is incident to j and z. Since c ∈ ∆2

2(z), the setD[3]

cz is empty and hence l is not incident to c. Put

Λ = Vj(c)/Vj(x) | x ∈ D[2]cj , Vj(x) ≥ Vj(c) ∩ Vj(l)

and

Θ = Vj(c)/Vj(x) | x ∈ D[2]cj ,∠bcx = 26 · 45.

Then Λ is the set of 1-dimensional factor-spaces of a 4-dimensional space Vj(c).Moreover, since ∠bij = 26 · 6, we deduce from the diagram Ds(M24) that Θ isthe set of 1-dimensional factor-spaces of Vj(c) over subspaces containing a given1- or 2-dimensional subspace. Thus

|Θ ∩ Λ| ≥ 3

and there exists y ∈ D[2]cj with y = g, ∠bcy = 26 · 45 and Vj(y) ≥ Vj(c) ∩ Vj(l).

In particular 〈Vj(y), Vj(l)〉 = Vj(n) for some n ∈ D[m]yjl . If n = k then Vj(g) =

Vj(k) ∩ Vj(c) = Vj(y) contrary to the inequality m = g. Thus n = k and thereexists a unique o ∈ D[2]

kjn. Since Vj(o) = Vj(k)∩Vj(n) ≥ Vj(l), we have o ∈ D[2]kjn.

Since Vj(o) = Vj(k)∩Vj(n) ≥ Vj(l), the element o is incident to l. Since h, j isa Q

[m]k -orbit, o and i are both incident to l and h. Hence there exists p ∈ D[m]

ihol.Let q, p be the Q

[m]h -orbit containing p. Since n is incident to j and o, we

conclude from some elementary properties of the Petersen graph Θo, that n isnot incident to h. Since p ∈ D[m]

o , we conclude that n is incident to q. Similarly,since z is not incident to h, calculating in the residue Di we observe that z isincident to q. Hence there exists r ∈ D[2]

nqz and thus we have found a path

(ma,

3b,

mc ,

2y,

mn,

2r,

mz )

with ∠bcy = 26 · 45 contrary to the case α = 1 of the proof, which is alreadysettled.

α = 3: Finally we assume that c ∈ ∆32(z) and by (8.3.17) we choose

a path (mc ,

3g ,

2h,

3i,

mz ) with ∠cgh = 24 · 6 = ∠zih. Regard D[2]

cg and D[0]cg

as the 1-dimensional and 2-dimensional isotropic subspaces of a non-singular4-dimensional symplectic space S over GF (2). Using the diagram Ds(M24) wewill show that there exits y ∈ D[0]

cg such that ∠bcx = 26 · 42 for all x ∈ D[2]cgy.

Indeed, if ∠bcg = 25 · 45, we choose y so that x is incident to b. If ∠bcg = 25 · 45,there exists u ∈ D[0]

cg such that v ∈ D[2]cg is perpendicular to u in S if and only if

∠bcv = 26 · 45. Choose y = u in this case. By the diagram Dh(M22) there existsx ∈ D[2]

cgy with ∠ghx = 23 · 5. Hence by Dp(M22) there exists n ∈ D[m]xgh. Then n

is incident to h and i, and since ∠zih = 24 · 6, there exists r ∈ D[2]inz, and again

we obtain a path

(ma,

3b,

mc ,

2x,

mn,

2r,

mz )

186 173,067,389-Vertex graph ∆

with ∠bcx = 26 · 45 which brings us back to the α = 1 case. The path extensionwe have just performed is illustrated on the following diagram:

2x

0y 3

g2h

ma

3b

mc

2d

me

3i

2f

mz

mn 2

r

This completes the proof of the claim. We continue to assume thatd∆(a, z) = 4. Let g be a vertex adjacent to z such that d∆(g, a) = 3. By (8.6.1)g ∈ ∆1

3(a) ∪ ∆23(a). By the above claim we have g ∈ ∆1

3(a) and therefore

g ∈ ∆23(a). Pick a path (

me ,

2f,

mg ,

3h,

mz ) with d∆(e, a) = 2. Let j ∈ D[0]

ghz. Thenby (8.5.5 (ii)) and its proof there exists t ∈ Gag such that t(j) is incident tof , and so t−1(f) is incident to j. Hence, replacing (e, f) by (t−1(e), t−1(f)), wemay assume that f is incident to j. Since Vj(g) ≥ 〈Vj(f), Vj(h)〉, there existsk ∈ D[3]

fjh. Then k is adjacent to e and z, and we get a contradiction to the claimapplied, with the roles of a and z interchanged.

By (8.6.1), (8.6.2), and the paragraph after (8.5.6) we deduce that

V (∆) = a ∪ ∆(a) ∪ ∆12(a) ∪ ∆2

2(a) ∪ ∆32(a) ∪ ∆1

3(a) ∪ ∆23(a).

In view of Table 3 this equality gives the following:

Proposition 8.6.3

|V (∆)| = 173, 067, 389;

|G| = |G[m]a | · |V (∆)| = 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43.

8.7 The simplicity of G

First we show that the amalgam J = G[0], G[1], G[2] is simple in the sense thatit does not possess non-trivial factor-amalgams.

Proposition 8.7.1 Let J = G[0], G[1], G[2] be an amalgam and

ν : J → Jbe a surjective homomorphism of amalgams. This means that ν maps (theelement-set of) J onto (that of) J and for each i = 0, 1, and 2 the restriction

The simplicity of G 187

of ν to G[i] is a homomorphism onto G[i]. Then unless J contains only oneelement, ν is an isomorphism.

Proof Let K = g ∈ J | ν(g) = 1 be the kernel of ν. Suppose first thatK ∩ G[i] is non-identity for some i = 0, 1, or 2. Then K ∩ G[i] contains theminimal normal subgroup of G[i], which is

Q[0] ∼= 210, Z [1] ∼= 26, and Z [2] ∼= 23

for i = 0, 1, and 2, respectively. Since

Z [2] < Z [1] < Q[0],

we conclude that K contains Z [2]. The normal closure of Z [2] in G[0] is Q[0], thenormal closure of Q[0] in G[1] is Q[1] and the normal closure of Q[1] in G[0] is thewhole G[0]. Finally the normal closure of G[0] ∩ G[i] in G[i] is the whole G[i] fori = 1 and 2. Therefore in the considered case K = J and |J | = 1.

Next assume that K ∩G[i] = 1 for i = 0, 1, and 2 (equivalently the restrictionof ν to each of the G[i]’s is an isomorphism). In this case, unless ν is an isomorph-ism, there are elements g ∈ G[i], h ∈ G[j] for 0 ≤ i < j ≤ 2 with g, h ∈ G[i] ∩G[j]

and

ν(g) = ν(h).

This means that the image of G[i]∩G[j] under ν is a proper subgroup in G[i]∩G[j].On the other hand G[0] ∩ G[1] and G[0] ∩ G[2] are maximal subgroups in G[0],while G[1] ∩ G[2] is maximal in G[1]. Since the G[i]’s are pairwise non-isomorphicthis means that their images cannot have larger intersections than they alreadyhave in J . Hence ν is an isomorphism.

As a direct consequence of (8.7.1) we obtain the following.

Lemma 8.7.2 Let G be a faithful generating completion group of the amal-gam J = G[0], G[1], G[2] and let N be a proper normal subgroup of G. Then(identifying J with its image in G) the amalgam

G[0]N/N, G[1]N/N, G[2]N/N

is isomorphic to J and G/N is also a faithful completion of J .

Proposition 8.7.3 Let G be a faithful generating completion of the amalgamJ . Then

(i) G is a finite simple group;(ii) G is the universal completion of J ;(iii) G is the only faithful generating completion of J .

Proof Suppose that N is a proper normal subgroup of N and let G = G/N .Then by (8.7.2) G is also a faithful generating completion group of J and (8.6.3)

188 173,067,389-Vertex graph ∆

applies to G as good as to G. This is a contradiction and hence (i) is estab-lished. By (i) the universal completion is simple and hence there are no furthercompletions which gives (ii) and (iii).

8.8 The involution centralizer

Now we can show that G[3] is the full centralizer of Z [3] in G.

Proposition 8.8.1 Let z be the generator of Z [3]. Then

G[3] ∼= 21+12+ · 3 · Aut (M22)

is the full centralizer of z in G.

Proof Let d be the element from D[3] such that G[3]d = G[3]. Then z is the

generator of Z[3]d . It is clear that G

[3]d is the setwise stabilizer of D[m]

d (the latterset being of size 77 by (8.1.2 (iii))). On the other hand, CG(z) clearly stabilizesas a whole the set

Θ(z) = x ∈ V (∆) | z ∈ Q[m]x .

By (8.3.13 (x)), (8.3.17 (vi)), (8.4.3 (iii)), and (8.5.2)

Θ(z) ⊆ a ∪ ∆(a) ∪ ∆12(a)

Now we apply (8.1.1 (vi)), (8.1.3 (vi)) and perform easy calculations in the Toddmodule Q

[m]a to show that |Θ(z)| = 77. Thus the sets D[m]

d and Θ(z) coincideand the result follows.

Now the proof of the Main Theorem is complete.

Exercises

1. Show that the subgroup Gag in (8.5.5) is isomorphic to L2(23).2. Calculate the orbits of G[3] ∼= 21+12

+ · 3 · Aut (M22) on the vertex-set of ∆.3. Calculate the automorphism group of the local graph of ∆ described in (8.2.6).4. Suppose that Σ is a graph whose local graph is isomorphic to that of ∆. Is

there a geometry S whose diagram coincides with that of D such that Σ isthe graph on the set of elements of type m in S in which two of them areadjacent whenever they are incident to a common element of type 2?

5. Classify the graphs whose local graphs are isomorphic to the local graphof ∆.

6. Classify the geometries having the same diagram as D.

The involution centralizer 189

7. Calculate the suborbit diagram of the graph Γ of valency 31 associated withJ4. The permutation character χ of J4 on the vertex-set of Γ (which is thecharacter on the cosets of G[0] ∼= 210 : L5(2)) as calculated by Jurger Mullerfrom Aachen is the following:

χ = 1 + 8 + 11 + 14 + 2 · 19 + 2 · 20 + 2 · 21 + 22 + 23 + 24

+ 29 + 30 + 32 + 36 + 37 + 38 + 39 + 51

(we follow the Conway et al. (1985) notation). Thus there are 27 suborbits.

9

HISTORY AND BEYOND

In this chapter we survey the history of discovery, construction andcharacterization of J4. This shows when and how the crucial ingredients of theproof of the Main Theorem emerged.

9.1 Janko’s discovery

The first evidence for the existence of the group now known as the fourth Jankogroup and denoted by J4 was given by Zvonimir Janko in his remarkable paper(Janko 1976). The main result of Janko (1976) is Theorem A reproduced below(we keep the original notation of Janko, particularly the symmetric group ofdegree n is denoted by Σn unlike Symn as in the rest of the book).

Theorem A Let G be a nonabelian finite simple group which possesses aninvolution z such that H = CG(z) satisfies the following conditions.

(i) The subgroup E = O2(H) is an extraspecial 2-group of order 213 andCH(E) ⊆ E.

(ii) An S3-subgroup P of O2,3(H) has the order 3 and CE(P ) = Z(E) = 〈z〉.(iii) We have H/O2,3(H) ∼= Aut(M22), NH(P ) = CH(P ), and P ⊆ (CH(P ))′.

Then G has the following properties.

(1) The order of G is 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 43.(2) The subgroup F0 = CH(P ) is isomorphic to the full covering group of

M22 (i.e. the perfect central extension of a cyclic group of order 6 byM22). Moreover, F = NH(P ) = NG(P ) and the group G has exactly oneconjugacy class of elements of order 3.

(3) Let R be an S3-subgroup of H. Then R is an extraspecial group oforder 27 and exponent 3 and we have NG(R) = NH(R) = R(〈z〉 × D),where D is a semidihedral group of order 16.

(4) Let T be an S2-subgroup of G. Then T possesses exactly one ele-mentary abelian subgroup V of order 211. We have CG(V ) ⊆ V andNG(V ) = V K, where K is isomorphic to M24. The orbits of K on V #

have lengths 7 · 11 · 23 (with representative z′) and 4 · 3 · 23 (with repres-entative t). Here z′ is conjugate in G to z and t is not conjugate to z inG. Moreover, we have CG(t) = CG(t) ∩ NG(V ).

(5) The group G has precisely two conjugacy classes of involutions withrepresentatives z and t. Hence CG(t) is a splitting extension of anelementary abelian group of order 211 by Aut (M22).

190

Janko’s discovery 191

(6) The group G possesses exactly one conjugacy class of selfcentralizingelementary abelian subgroups of order 210 and if A is one of them, thenwe have NG(A) = AB, where B ∼= L5(2) and B acts irreducibly on A.The orbits of B on A# have lengths 5 · 31 and 4 · 7 · 31.

(7) Let Q be an S5-subgroup of H. Then Q is also an S5-subgroup of G,CG(Q) = Q × J , where J is isomorphic to the non-splitting extension ofan elementary abelian group of order 8 by L3(2). Also NG(Q) containsa Frobenius subgroup of order 20. Hence a Sylow 5-normalizer in G hasorder 28 · 3 · 5 · 7.

(8) Let S be an S7-subgroup of H. Then S is also an S7-subgroup of G,CG(S) =S ×I, where I is isomorphic with Σ5 and |NG(S)|= 3 · |CG(S)|.Hence a Sylow 7-normalizer in G has the order 23 · 32 · 5 · 7.

(9) The group G possesses a special 2-group L of order 215 with |Z(L)| = 23,so that NG(L)/L ∼= Σ5 × L3(2). Also, NG(L) does not split over L andnevertheless NG(L) contains subgroups isomorphic to Σ5 and to L3(2).The subgroup NG(L) contains both a Sylow 5-normalizer in G and aSylow 7-normalizer in G.

(10) A Sylow 11-normalizer in G has order 24 · 3 · 5 · 113 and contains asubgroup isomorphic to GL2(3). An S11-subgroup of G is extraspecialof order 113 and exponent 11. The group G has exactly two conjugacyclasses of elements of order 11.

(11) An Sp-subgroup is self-centralizing in G for p = 23, 29, 31, 37, and 43.A Sylow p-normalizer in G has order 23 · 22, 29 · 28, 31 · 10, 37 · 12, and43 · 14, respectively.

(12) The group G possesses PGL2(23) as a subgroup.(13) The group G has exactly 62 conjugacy classes of elements. The charac-

ter table of G is unique and was computed by J. Conway, S. Norton,J. G. Thompson, and D. Hunt.

In 1976, Janko deduced almost the whole p-local structure of G from thestructure of the involution centralizer. Once in Gaeta in 1990 he mentionedthat this was the easier part of the job; the hard part was to eliminate manyother possibilities for the structure of centralizer which did not lead to consistentconfigurations.

At the time of (Janko 1976) the cyclic group Z(F0) of order 6 in (2) wasbelieved to be the full Schur multiplier of the Mathieu group M22 (in (Mazet1979) the multiplier was proved to be cyclic of order 12).

As pointed out on p. 494 in Kleidman and Wilson (1988) the structure of theSylow 3-normalizer is slightly mis-stated in (3); the correct structure is (2×31+2

+ :8) : 2, where the outer involution conjugates the elements of order 8 to its cubetimes the central involution.

Let us relate the present work with Janko’s theorem. The structure of involu-tion centralizer in (i) to (iii) which is the starting point of Janko’s work appearsin our treatment at the very last stage (8.8.1). Our penultimate result (8.6.3) isthe order of G which coincides with that in (1).

192 History and beyond

The subgroup NG(V ) = V K in (4) is the subgroup G[m] ∼= 211 : M24 in ourterms; the subgroup NG(A) = AB in (6) is our subgroup G[0] ∼= 210 : L5(2); thesubgroup NG(L) in (9) is our pentad group G[3] ∼= 23+12 · (L3(2) × Sym5). Theinformation on 5- and 7-normalizers in (7) and (8) corresponds to the propertiesof the pentad group in (4.9.1).

The subgroup PGL2(23) in (12) is the subgroup Ga, g = 〈Gag, t〉 in theproof of (8.5.5).

The character table mentioned in (13) can be found in (Conway et al. 1985)(under the name J4 of course). The number 1333 which first appeared in (6.2.1)within our project is the degree of the minimal faithful character of J4.

Janko left unsettled the questions about existence and uniqueness of thegroup he had discovered in (Janko 1976).

9.2 Characterizations

Immediately after Janko’s discovery the existence of further simple groups withsimilar 2- and 3-local structure was questioned. A group G is said to be ofJ4-type if it satisfies the conditions (i) to (iii) in Janko’s Theorem A. Soon afterpublication of (Janko 1976) it was proved by various authors that each of theproperties (2), (4), (5), (6), and (7) is characteristic for groups of J4-type.

Theorem 9.2.1 (Stroth 1978; Stafford 1979) Let G be a finite group whichis not 3-normal. Furthermore, let x be an element of order 3 in G and F0 =CG(x). Then the following assertions hold:

(i) if G is simple and F0 is isomorphic to 3 · M22 or 6 · M22, then G is ofJ4-type;

(ii) if G is of characteristic 2-type and F0 is isomorphic to 6 · M22, then G ofJ4-type.

Theorem 9.2.2 (Reifart 1978) Let G be a finite group containing anelementary abelian subgroup V of order 211 such that

(i) CG(V ) = V ;(ii) NG(V )/V is isomorphic to M24;(iii) O(CG(z)) = 1 for an involution z in the centre of a Sylow 2-subgroup

of NG(V ).

Then one of the following holds:

(a) V is normal in G;(b) G is of J4-type;(c) G is simple, |G| = |M(24)′|, and the centralizer of a 2-central involution

of G is isomorphic to the centralizer of a 2-central involution of M(24)′;(d) G is isomorphic to the first Conway group Co1.

It is known by now (cf. (Parrott 1981) or theorem 34.1 in Aschbacher 1997)that M(24)′ is the only group which satisfies (9.2.2 (c)).

Ronan–Smith geometry 193

Theorem 9.2.3 (Reifart 1977a, 1977b) Let G be a finite group containingan involution f such that V := O2(CG(f)) is elementary abelian of order 211

and CG(f)/V ∼= Aut(M22). Then one of the following holds:

(i) G = O(G) : CG(f);(ii) V is normal in G and G/V ∼= M24;(iii) G is of J4-type.

Theorem 9.2.4 (Lempken 1978a) Let G be a finite group containing anelementary abelian subgroup A of order 210 such that NG(A)/A ∼= L5(2) actsirreducibly on A and O(CG(z)) = 1 for a 2-central involution z of NG(A). Thenone of the following holds:

(i) G = NG(A) ∼= 210 : L5(2);(ii) G is isomorphic to Ω+

10(2) or to O+10(2);

(iii) G is of J4-type.

Theorem 9.2.5 (Van Trung 1980) Let G be a finite simple group ofcharacteristic 2-type and let K be a 2-local subgroup of G satisfying thefollowing:

(i) L := O2(K) is special of order 215, Z := Z(K) ∼= 23; K = NG(Z) andK/R ∼= Sym5 × L3(2);

(ii) if x and y are two commuting elements of K of orders 3 and 7,respectively, then CL(x) = Z and CL(y) = 1.

Then G is of J4-type.

9.3 Ronan–Smith geometry

It was not stated in Janko’s Theorem A but can be deduced from its proofthat the maximal 2-local subgroups can be chosen to have ‘large’ pairwise inter-sections. The following explicit form of this observation led to discovery of themaximal 2-local parabolic geometry of J4.

Proposition 9.3.1 (Ronan and Smith 1980) The subgroups

G[0] = NG(A) ∼= 210 : L5(2), G[2] = NG(L) ∼= 23+12 · (L3(2) × Sym5),

G[3] = CG(z) ∼= 21+12+ · 3 · Aut (M22), G[m] = NG(V ) ∼= 211 : M24.

in Janko’s Theorem A can be chosen in such a way that the coset geometry D(G)associated with the completion in G of the amalgam G[m], G[0], G[2], G[3] is


described by the diagram

2 2

5

d t

3 2 m

6

21+12+ · 3 · Aut (M22) 23+12 · (L3(2) × Sym5) 211 : M24

0

210 : L5(2)

The existence of the geometry D(J4) was stated in Ronan and Smith(1980) subject to the existence of the group J4. In fact these two exist-ences are equivalent and here we have constructed the geometry before thegroup herself was constructed. The main question about D(J4) (as well asabout other sporadic geometries) posed in Ronan and Smith (1980) wasthe question about the simple connectedness. The affirmative answer tothis question (originally given in Ivanov (1992)) also follows from our MainTheorem.

9.4 Cambridge five

The group J4 was constructed in Cambridge in 1980 by D. J. Benson,J. H. Conway, S. P. Norton, R. A. Parker, and J. G. Thackrey as a group of112 × 112 matrices over GF (2) and the construction made a heavy use of acomputer.

The construction process was very dramatical. This is reflected in themathematical (Norton 1980) and popular (Conway 1981) summaries. Thesesummaries give the exact date when the last step was accomplished: February20, 1980.

It would not be wise to reproduce the content of Norton (1980) here ‘in myown words’. Although it should be mentioned that a lot of essential informationabout the action of J4 on the cosets of 211 : M24 can be found in Norton (1980)(more details can be found in Benson(1980)). For instance the structure of the2-point stabilizers from Table 3 at the end of Section 8.5 are already given inNorton (1980).

9.5 P -geometry

In Ivanov (1987) it was shown that there exists a graph Γ = Γ(J4) of valency 31on which J4 acts locally projectively. The graph Γ contains geometric Petersen

Uniqueness of J4 195

subgraphs and geometric octet subgraphs. Hence J4 acts flag-transitively on alocally truncated geometry F(J4) with the diagram

P t5 :

1 P

2

2

2 .

(cf. Section 7.5). The geometric subgraphs of valency 15 are missing andtherefore F(J4) is not a truncation of a P -geometry of rank 5. It was noticed byS. V. Shpectorov that the non-existence of the valency 15 geometric subgraphsmeans that the corresponding geometric subgroup (which still can be defined)is the whole of J4. Therefore J4 acts flag-transitively on a rank 4 P -geometryG(J4) with the diagram

P4 :1 P

2

2

2.

as in (7.3.2).In Shpectorov (1988) the amalgams of maximal parabolic subgroups asso-

ciated with flag-transitive actions on P -geometries of rank 4 were classified.It turned out to exists just five such amalgams, one of them being thesubamalgam of J4.

In 1988, G. Stroth and R. Weiss have written down generators and relationswhich must hold in any group acting locally projectively on a graph of valency31 and girth 5 with the vertex stabilizer isomorphic to G[0] ∼= 210 : L5(2).This is a presentation for the automorphism group of the universal cover of thegeometry F(J4). The presentation of the automorphism group of the universalcover of G(J4) was also explicitly written down. It was asked in Stroth and Weiss(1988) whether the groups defined by the presentations are isomorphic to J4. Theaffirmative answers to these questions are equivalent to the simple connectednessof F(J4) and G(J4), respectively.

The simple connectedness of G(J4), F(J4), and D(J4) was established in(Ivanov 1992).

Within the theory of P -geometries an important role is played by the so-called universal abelian representations of geometries (cf. Ivanov and Shpectorov(2002) for the definition). It was shown in Ivanov and Shpectorov (1990) thatG(J4) does not possess abelian representations. In a sense this is the reason whyG(J4) does not appear as a residue in P -geometries of higher rank. In Ivanov andShpectorov (1997) it was shown that the universal non-abelian representationgroup of G(J4) is J4 itself.

9.6 Uniqueness of J4

The first uniqueness proof for J4 was announced in (Norton 1980). It was basedon the fact that the amalgam

G[0], G[1] ∼= 210 : L5(2), 211 : 24 : L4(2)


possesses a unique 1333-dimensional representation with a given character(the character being the restriction of a 1333-dimensional character of J4),see (Thompson 1981). This uniqueness ensures the success of our constructionin Chapter 6.

It is a remarkable fact (already mentioned in (Norton 1980)) that the 1333-dimensional representation of a larger amalgam

G[0], G[m] ∼= 210 : L5(2), 211 : M24

is not unique, even when the character is prescribed.Within the classification of the finite simple groups J4 falls into the most

dramatic class of the quasi-thin groups (Mason 1980; Aschbacher and Smith2003). In Berlin in 1990 Geoff Mason suggested me that the simple connectednessproof for G(J4) can be extended to the uniqueness proof for J4. This was fullyaccomplished in (Ivanov 1992). An independent uniqueness proof along similarlines by M. Aschbacher and Y. Segev appeared in 1991.

9.7 Lempken’s construction

In (Lempken 1993) J4 was constructed as a group of 1333 × 1333-matrices overthe field GF (11). The construction is similar to our’s in Chapter 6. First arepresentation of G[0] ∼= 210 : L5(2) was constructed, then it was restricted tothe subgroup G[0] ∩ G[1] ∼= 210 : 24 : L4(2) and finally a matrix t was found,which extends G[0] ∩ G[1] to the whole of G[1]. To justify the fact that the groupgenerated is indeed J4, the existence of J4 was needed.

9.8 Computer-free construction

In July 1990 during the Durham Symposium U. Meierfrankenfeld and the authoragreed that the simple connectedness proof for the 2-local geometries of J4 canbe turned into a computer-free construction of J4. It took almost 10 years beforethe project was completed and published in (Ivanov and Meierfrankenfeld 1999).The present volume is essentially based on this project.

9.9 The maximal subgroups

The maximal p-local subgroups of a group of J4-type were determined in(Lempken 1989) while the non-local subgroups were determined in (Kleidmanand Wilson 1988) with considerable use of computer.

Theorem 9.9.1 The group J4 contains exactly 13 classes of maximal subgroups.If X is a maximal subgroup in J4, then one of the following holds:

(i) X is 2-local and is isomorphic to one of the following: 211 : M24, 210 :L5(2); 21+12

+ · 3 · Aut(M22), 23+12 · (L3(2) × Sym5);(ii) X is a Sylow p-normalizer for some p ∈ 11, 29, 37, 43 and is isomorphic

to the respective groups 111+2+ : (GL2(3) × 5), 29 : 28, 37 : 12, 43 : 14;

Locally projective graphs 197

(iii) X is a non-local subgroup isomorphic to one of the following: U3(11) : 2,M22 : 2, L2(32) : 5, L2(23) : 2, U3(3).

9.10 Rowley–Walker diagram

In (Rowley and Walker 1994) the following suborbrit diagram of the action ofJ4 on the cosets of 211 : M24 was computed.

a

∆1(a)

∆12(a) ∆2

2(a) ∆32(a)

∆13(a) ∆2

3(a)

15,180

1

84

4,032

10,752

311

45 185

3,125375463

3,240 27 3,168 330

11,520 5,360 6,144 4,6807,040

6

359

2,808

7,296 4,807

2532,783

4,711 7,337

9.11 Locally projective graphs

Let n ≥ 2 be an integer and q be a prime power. Recall that a locally finite,vertex-transitive action of a group F on a graph Φ is said to be locally projectiveof type (n, q) if for a vertex u of Φ the subconstituent F (u)Φ(u) contains anormal subgroup isomorphic to the special linear group Ln(q) acting on the set


of 1-dimensional subspaces in its natural module (which is an n-dimensionalGF (q)-space), so that the valency of Φ is (qn − 1)/(q − 1),

Ln(q) F (u)Φ(u) ≤ PΓL(n, q),

and the action of F on Φ is 2-arc-transitive.The most interesting case among the locally projective actions is when q = 2

and the following condition holds: whenever u, v is an edge of Φ, there is anelement t ∈ Fu, v \ F (u, v) which centralizes F (u, v)/O2(F (u, v)).

Under these assumptions the amalgam

F = A(F, Φ) = F [0], F [1],

where F [0] = F (u) and F [1] = Fu, v is said to be the amalgam determined bythe locally projective action or simply a locally projective amalgam. The locallyprojective amalgams can be characterized as abstract rank 2 amalgams subjectto the following conditions:

(A1) F [0]/O2(F [0]) ∼= Ln(2) for some n ≥ 2;(A2) F [01] := F [0]∩F [1] is the stabilizer of a 1-space U1 in the natural module

U of F [0]/O2(F [0]);(A3) [F [1] : F [01]] = 2;(A4) F [1]/O2(F [01]) ∼= Ln−1(2) × 2 or, equivalently [t, F [1]] ≤ O2(F [01]) for

some t ∈ F [1] \ F [0];(A5) no non-identity subgroup in F [01] is normal in both F [0] and F [1].

D. Z. Djokovic and G. L. Miller (1980) used the classical Tutte’s theorem(Tutte 1947), to show that there are exactly seven locally projective amalgams forn = 2. In Ivanov and Shpectorov (2004) we used Trofimov’s theorem (Trofimov2003) to extend the classification to the case n ≥ 3 by proving the following.

Theorem 9.11.1 Let F = F [0], F [1] be a locally projective amalgam, that isan amalgam satisfying (A1) to (A5) for some n ≥ 3. Then exactly one of thefollowing holds:

(i) F ∼= A(AGLn(2), K2n);(ii) F ∼= A(O+

2n(2), D+(2n, 2));(iii) F is isomorphic to one of twelve exceptional amalgams in Table 4.

In order to explain the data in Table 4 we need to recall some definitions. By(A2) the subgroup F [01] is the stabilizer in F [0] of a 1-space U1 in the naturalmodule U of F [0]/O2(F [0]) ∼= Ln(2). Let

0 < U1 < U2 < · · · < Un−1 < U

be a maximal flag in U containing U1. For 2 ≤ i ≤ n−1 let F [0i] be the stabilizerof Ui in F [0], F [01i] = F [0i] ∩F [1] and let F [1i] be the normalizer of F [01i] in F [1].Let K [i] be the largest subgroup in F [01i] which is normal in both F [0i] and F [1i].Let F [i] be the subgroup in the outer automorphism group Out (K [i]) of K [i]

generated by the natural images in Out (K [i]) of F [0i] and F [1i].

Locally projective graphs 199

Table 4. Exceptional amalgams

n F F [0]/O2(F [0])O2(F [0])

F [2]

K[2]F [3]

K[3]F [4]

K[4] Completionsconstrainedat level 2

3 A(1)3

L3(2)23

O+4 (2)24 —

A(2)3

L3(2)23

Sym524 M22

A(3)3

L3(2)23

Sym524 —

A(4)3

L3(2)23×2

24:O+4 (2)

25 (Sym8 2)+

A(5)3

L3(2)23×2

Sym525 Aut (M22)

4 A(1)4

L4(2)1

Sym524:3

11 M23

A(2)4

L4(2)26

Sym5×221+8+ :Sym3

L6(2)26 Alt64

A(3)4

L4(2)21+4+6

Sym524+10·Sym3

Aut (M22)210 Co2

A(4)4

L4(2)24+4+6

Sym523+12+2·Sym3

3·Aut (M22)21+12+

J4

A(5)4

L4(2)24+4+6

Sym5×223+12+2·Sym3

L6(2):221+12+

Alt256

5 A(1)5

L5(2)210

Sym523+12·L3(2)

Aut (M22)21+12+ :3

11 J4

A(2)5

L5(2)25+5+10+10

Sym523·[232]·L3(2)

Aut (M22)22+10+20·Sym3

Co2

21+22+

BM

Let φ : F → F be a faithful generating completion (which might or mightnot be the universal completion). Let us identify F with its image under φ.For 2 ≤ i ≤ n − 1 let F [i] be the subgroup in F generated by F [0i] and F [1i].Then

F [i] = F [i]/K [i]CF [i](K [i])

and the completion φ : F → F is said to be constrained at level i if

CF [i](K [i]) ≤ K [i].

The amalgam H defined in Section 2.3 is A(O+10(2), D+(10, 2)), the amalgam

G defined in Section 3.5 is A(1)5 while the amalgam G[4] = G[04], G[14] as in

Section 5.4 is A(4)4 . The Main Theorem of the present work states that J4 is the

only completion of A(1)5 which is constrained at level 2.

Because of the classification of the flag-transitive Petersen type geometriesaccomplished in (Ivanov 1999) and (Ivanov and Shpectorov 2002) we know thatthe Baby Monster sporadic simple group BM and the non-split extension of an


Table 5. Dimensions of minimal representations

F A(1)3 A(2)

3 A(3)3 A(4)

3 A(5)3 A(1)

4m(F) 7 20 20 14 20 20

F A(2)4 A(3)

4 A(4)4 A(5)

4 A(1)5 A(2)

5m(F) 63 23 1333 255 1333 4371

elementary abelian group of order 34371 by BM are the only completions of A(2)5

which are constrained at level 2. It is desirable to prove this result starting fromthe basic principles (as we did here for J4).

In (Ivanov and Pasechnik 2004) the following result was established.

Proposition 9.11.2 Let F be a locally projective amalgam andm = m(F) be thesmallest positive integer such that F possesses GLm(C) as a faithful completiongroup (which might or might not be generating). Then

(i) if F = A(AGLn(2), K2n), then m(F) = 3, 7 and 2n −2 for n = 3, n = 4and n ≥ 5, respectively;

(ii) if F = A(O+2n, D+(2n, 2)) then m(F) = 7, 28 and (2n − 1)(2n−1 − 1)/3

for n = 3, n = 4 and n ≥ 5, respectively;(iii) if F is an exceptional amalgam then m(F) is given in Table 5.

The fact that m(A(4)4 ) = m(A(1)

5 ) = 1333 is the dimension of the minimalrepresentation of J4 has played a crucial role in the present work. Notice thatm(A(2)

5 ) = 4371 is the dimension of the minimal representation of the BabyMonster.

9.12 On the 112-dimensional module

The information on the 112-dimensional GF (2)-module B for J4 given in(Conway et al. 1985) suggests that B is a quotient of the so-called universalderived module C of the Petersen geometry G(J4) (cf. Ivanov and Shpectorov(2002) for the relevant definitions). The module C can be defined as follows:

(a) take one involutory generator c(e) for every edge e of the locally projectivegraph Γ = Γ(J4);

(b) declare the c(e)’s to commute pairwise;(c) whenever edges e, f and g are contained in a common geometric Petersen

subgraph Σ in Γ and constitute an antipodal triple in Σ, impose therelation c(e)c(f)c(g) = 1.

Conjecture 9.12.1 The above defined J4-module C over GF (2) is112-dimensional.

Miscellaneous 201

We can also define the non-abelian version D of C simply by dropping thecondition (b). Then C is the quotient of D over the commutator subgroup of D.

Conjecture 9.12.2 The group D is abelian.

9.13 Miscellaneous

Further results on construction, characterization and properties of J4 can befound in Cooperman et al. (1997), Finkelstein (1977), Ganief and Moori (1999),Green (1993), Guloglu (1981), Ivanov and Shpectorov (1989), Lempken (1978b),Mason (1977), Michler and Weller (2001), Sitnikov (1990), Smith (1980),Yoshiara (2000).

Exercises

1. Prove (or disprove) that the 112-dimensional GF (2)-module for J4 is aquotient of the module C from (9.12.1).

2. Describe the group G[0] from the amalgam A(2)5 in Table 4 in terms

independent of the Baby Monster group.

10

APPENDIX: TERMINOLOGY AND NOTATION

In this appendix we indicate our main terminology and notation concerninggroups, graphs, amalgams, and diagram geometries. As is common we could saythat our notation and terminology are standard, but this would sound more likea declaration of what ‘standard’ is.

10.1 Groups

Let G be a group, H be a subset of G which might or might not be a subgroupand let g ∈ G and let h ∈ H. Then hg = g−1hg, Hg = hg | h ∈ H and〈H〉 denotes the subgroup in G generated by the elements of H. The subset(subgroup) H is said to be normal if Hg = H for every g ∈ G. The centralizerand the normalizer of H in G are defined as follows:

CG(H) = g | g ∈ G, hg = h for every h ∈ H;

NG(H) = g | g ∈ G, hg ∈ H for every h ∈ H.

The centralizer and normalizer are subgroups in G and

CG(H) ≤ NG(H).

For a group G we write Aut (G) and Inn (G) for the automorphism group ofG and for the group of inner automorphisms, respectively. An automorphismσ ∈ Aut (G) \ Inn (G) is called an outer automorphism of G; it should not beconfused with the coset of Inn (G) in Aut (G) containing σ. The latter coset isan element of the outer automorphism group Out (G) = Aut (G) /Inn (G) of G.

An involution in a group is an element of order 2.

If N and H are groups and σ : H → Aut N is a homomorphism, then(H × N, ∗), where (h1, n1) ∗ (h2, n2) = (h1h2, n

σ(h2)1 n2) is a group called the

semidirect product of N and H with respect to σ and is denoted by S(N, H, σ)(or simply by N : H when σ is clear from the context or irrelevant). If σ is trivial(in the sense that its image is the identity) the product is direct.

For a finite group G the symbol O(G) denotes the largest odd order (solvable)normal subgroup of G. If p is a prime then Op(G) denotes the largest normalp-subgroup in G and Op(G) the smallest normal subgroup of G such that thecorresponding factor group is a p-group. The centre of G is denoted by Z(G)and G′ is the commutator subgroup of G.

We mostly follow the Atlas notation (Conway et al. 1985) for groups. Inparticular pn denotes the elementary abelian group of that order; 21+2n

ε denotes

202

Amalgams 203

the extraspecial group of order 22n+1 of type ε ∈ +,−. The symmetric andalternating groups of degree n are denoted by Symn and Altn, respectively. ByLn(q) we denote the projective special linear group in dimension n over the fieldGF (q) of q elements. By F b

a we denote a Frobenius group with kernel of order aand complement of order b.

If H is a subgroup of a group G then there is a natural homomorphism ofNG(H) into Aut (H); if this homomorphism is surjective then is said that H isfully normalized in G.

Suppose that a group G acts on a set Ω by permutation (which means thereis a homomorphism of G into the symmetric group of Ω). If a is an element of Ωthen the stabilizer of a in G is denoted either by G(a) or by Ga. If X ⊂ Ω thenGX denotes the stabilizer in G of X as a whole and G(X) denotes the elementswise stabilizer (then G(X) is the intersection of the subgroups G(x) = Gx takenfor all x ∈ X). If H ≤ GX then by HX we denote the permutation groupinduced by H on X (in this case clearly HX ∼= H/H(X) and H = HX).

Let G be a group, let H be a subgroup of G and G/H = gH|g ∈ G be thesets of left cosets of H in G. The action of G on G/H via

f : gH fgH

for f ∈ G defines a homomorphism of G into the symmetric group of the setG/H. The image of this homomorphism is a transitive subgroup of the symmetricgroup; if H = 1H is considered as an element of G/H then G(H) = H and thekernel of the homomorphism is the largest normal subgroup of G contained inH. The latter normal subgroup is known as the core of H in G:

core(H, G) =⋂

g∈G

Hg.

10.2 Amalgams

An amalgam of rank m is a collection

A = (G[i], ∗i) | 0 ≤ i ≤ m − 1

of m groups (G[i], ∗i), 0 ≤ i ≤ m − 1 such that for all 0 ≤ i < j ≤ m − 1 theintersection G[ij] := G[i] ∩ G[j] of the element sets is non-empty and the groupoperations ∗i and ∗j coincide, when restricted to G[ij]. If (G, ∗) is a group andG[0], G[1], ..., G[m−1] are subgroups in G, then (G[i], ∗|G[i]) | 0 ≤ i ≤ m − 1 isan amalgam.

Let A = (G[i], ∗i) | 0 ≤ 0 ≤ m−1 be an amalgam, let (G, ∗) be a group andlet ϕ be a mapping of the union of the element sets of the groups constitutingA into G such that for every 0 ≤ i ≤ m − 1 and all g, h ∈ G[i] the equality

ϕ(g ∗i h) = ϕ(g) ∗ ϕ(h)

holds (which means that the restriction of ϕ to each G[i] is a homomorphism).Then the pair (G, ϕ) is called a completion of A (here G is the completion group

204 Appendix: Terminology and notation

and ϕ is the completion map). The completion is said to be faithful if ϕ is injectiveand generating if G is generated by the image of ϕ. A completion (G, ϕ) is said tobe universal if for every completion (G, ϕ) there is a homomorphism ψ of G intoG, such that ϕ is the composition of ϕ and ψ. A universal completion is alwaysgenerating and the universal completion group is unique up to isomorphism.Furthermore, an amalgam possesses a faithful completion (which is not alwaysthe case) if and only if its universal completion is faithful. It is natural to definetwo completions (G, ϕ(1)) and (G, ϕ(2)) to be equivalent whenever there is l ∈ G,such that

ϕ(1)(a) = l−1ϕ(2)(a)l

for every a ∈ A =⋃m

i=0 G[i]. Clearly equivalent completions generate in Gconjugate subgroups.

Whenever the group operations are clear from the context or irrelevant, wesimply write

A = G[i] | 0 ≤ i ≤ m − 1for an amalgam of rank m. We will also drop the explicit reference to the comple-tion maps whenever it does not cause confusion (in this case by ‘completion’ wemean the completion group). For an amalgam A = G[0], G[1] of rank 2, the uni-versal completion is faithful and the universal completion group is isomorphic tothe free product of G[0] and G[1] amalgamated over the common subgroup G[01].The free amalgamated product is infinite whenever G[01] is proper in both G[0]

and G[1].

10.3 Graphs

An undirected graph Γ without loops is a pair of sets V (Γ) (the set of vertices orsimply the vertexset) and E(Γ) (the set of edges or simply the edgeset) togetherwith an incidence relation with respect to which every edge is incident to exactlytwo distinct vertices. For such a graph we will write Γ = (V (Γ), E(Γ)) assumingthat the incidence relation is clear from the context. Two vertices are calledadjacent if they are incident to a common edge. A graph is said to contain nomultiple edges if every pair of vertices is incident to at most one common edge.

In this case every edge is identified with the pair of vertices it is incident to,so that the incidence relation is via inclusion. For the remainder of the section weassume that Γ = (V (Γ), E(Γ)) contains no multiple edges, that E(Γ) is simply aset of 2-element subsets of V (Γ) and the incidence relation is via inclusion. Fora vertex x ∈ V (Γ) the number of edges incident to x is called the valency of xin Γ. Since there are no multiple edges the valency of x is equal to the numberof vertices adjacent to x in Γ. If the valency is independent of the choice of thevertex x, it is called the valency of the graph Γ in which case the graph Γ is saidto be regular of that valency.

A sequence π = (x0, x1, ..., xs) of vertices in a graph Γ is said to be an s-arcif xi, xi+1 ∈ E(Γ) for all 0 ≤ i ≤ s−1 and xi = xi+2 for all 0 ≤ i ≤ s−2. This

Graphs 205

arc is said to join x0 with xs. If in addition xi and xi+2 are not adjacent in Γ thenπ is said to be a path (of length s). A graph is said to be connected if any two ofits vertices are joint by a path. The length of a shortest path joining vertices xand y of Γ is called the distance between x and y in Γ and is denoted by dΓ(x, y).The diameter of a connected graph Γ is the maximum among distances betweenits vertices.

If Γ = (V (Γ), E(Γ)) and Σ = (V (Σ), E(Σ)) are graphs then Σ is said to be asubgraph of Γ if

V (Σ) ⊆ V (Γ) and E(Σ) ⊆ E(Γ).

If in addition x, y ∈ E(Σ) whenever both x, y ∈ V (Σ) and x, y ∈ E(Γ), thenΣ is said to be the subgraph of Γ induced by V (Σ). A subgraph Σ in Γ is said tobe convex or geodetically closed if the following condition holds:

whenever x, y ∈ V (Σ), Σ contains every shortest path (that is a path of lengthdΓ(x, y)) which joins x and y in Γ.

Let G be a group of automorphisms of Γ, and suppose that the action ofG on Γ is 2-arc-transitive, that is transitive on the set (y, x, z) | y, x, z ∈V (Γ), y, x, x, z ∈ E(Γ), y = z of 2-arcs in Γ. For x ∈ V (Γ) let

Γ(x) = y | y ∈ V (Γ), x, y ∈ E(Γ)be the set of neighbours of x in Γ and let

G(x) = g | g ∈ G, g(x) = xbe the stabilizer (subgroup) of x in G. We always assume that the action is locallyfinite so that G(x) is of finite order. Then, because of the 2-arc-transitivity, thepermutation group G(x)Γ(x) (known as the subconstituent) is doubly transitive.

The action of a group G ≤ Aut Γ on Γ is distance-transitive if for every0 ≤ i ≤ d (where d is the diameter of Γ) the group G acts transitively on the set

Γi = (x, y) | x, y ∈ Γ, d(x, y) = i.

A graph which possesses a distance-transitive action is called a distance-transitivegraph. If Γ is distance-transitive then for every i, 0 ≤ i ≤ d, the parameters

ci = |Γi−1(y) ∩ Γ(x)|, ai = |Γi(y) ∩ Γ(x)|, bi = |Γi+1(y) ∩ Γ(x)|are independent of the choice of the pair x, y ∈ Γ satisfying d(x, y) = i. Clearlyin this case Γ is regular and ci + ai + bi = |Γ(x)| = k is the valency of Γ. Thesequence

i(Γ) = b0 = k, b1, . . . , bd−1; c1 = 1, c2, . . . , cdis called the intersection array of the distance-transitive graph Γ. If we putki = |Γi(x)| for 1 ≤ i ≤ d then

ki =k · b1 · · · · · bi−1

c1 · c2 · · · · · ci.


To represent the decomposition of a distance-transitive graph with respect to avertex we draw the following distance diagram:

.. . .. .1 k k2 ki kd

a1 a2 ai ad

k 1 b1 c2 b2 ci bi cd

We draw similar diagrams for non-distance-transitive actions. Let G be agroup acting on a graph Γ and x be a (basic) vertex. The suborbit diagram(with respect to x) consists of ovals (or circles) joined by curves (or lines). Theovals represent the orbits of G(x) on the vertex set of Γ. Inside the oval whichrepresents an orbit Σi (the Σi-oval) we show the size of Σi or place its nameexplained in the context. Next to the Σi-oval we show the number ni (if non-zero) of vertices in Σi and adjacent to a given vertex yi ∈ Σi. On the curvejoining the Σi- and Σj-ovals we put the numbers nij and nji (called valencies.)Here nij (appearing closer to the Σi-oval) is the number of vertices in Σj adjacentto yi. Clearly

|Σi| · nij = |Σj | · nji

and we draw no curve if nij = nji = 0. Normally we present the valencies ni

and nij as sums of lengths of orbits of G(x, yi) on the vertices in Σi and Σj

adjacent to yi. When the orbit lengths are unknown or irrelevant, we put thevalencies into square brackets. Generally the suborbit diagram depends on theorbit of G on Γ from which the basic vertex x is taken. Even if a graph is notnecessarily distance-transitive (but still generates an association scheme), we usethe notation ci, ai, bi if the corresponding parameters are independent of thechoice of a pair of vertices at distance i.

If Γ and ∆ are graphs, then a surjective mapping ν : V (Γ) → V (∆) is saidto be a covering if for every x ∈ V (Γ) the restriction of ν to Γ(x) is a bijectiononto ∆(ν(x)).

Let Γ be a graph and G be a group of automorphisms of Γ. Suppose thatthe action is 2-arc transitive and for an integer n ≥ 2 and a prime power q thefollowing holds:

there is a group SL(n, q) ≤ H ≤ ΓL(n, q), such that for every x ∈ V (Γ) the actionof G(x) on Γ(x) is permutation isomorphic to the permutation action of H on the setof 1-dimensional subspaces in its natural module (the latter being an n-dimensionalGF (q)-space).

Then the action of G on Γ is said to be locally projective of type (n, q).A very special role in our project is played by the famous Petersen graph

Π. The standard way to define Π is the following. Take a set Ω of size 5. Thenthe vertices of Π are the 2-element subsets of Ω (so that there are 10 vertices)and two vertices are adjacent if they are disjoint (as subsets of Ω). Taking Ω =1, 2, 3, 4, 5 we obtain the familiar picture of the Petersen graph presented below.

Graphs 207

The automorphism group G of the Petersen graph is the symmetric groupSym5 of Ω. The vertices of Π are naturally identified with the transpositions ofG. In these terms two transpositions are adjacent if and only if they commute.Therefore x is the unique non-identity element of G1(x).

Due to the isomorphism Sym5∼= O−

4 (2) we obtain another description of Π.Let V −

4 = (V, f, q) be a non-singular orthogonal space of minus type of dimension4 over GF (2). Then the vertices of Π are the vectors of V which are non-singularwith respect to the quadratic form q; two such vectors are adjacent if they areperpendicular.

1,2

3,4

3,5

4,51,5

2,5

2,3

1,3

1,4 2,4

Let l(Π) be the line graph of Π. This means that the vertices of l(Π) arethe edges of Π and two of them are adjacent if they are incident to a commonvertex of Π. The graph l(Π) is distance-transitive with the following intersectiondiagram:

1 4 8 24 1 2 1 1 4

1 2

The graph l(Π) is antipodal in the following sense: the relation with respectto which two edges are related if they are either equal or are at distance three inthe line graph is an equivalence relation (with classes of size 3). The classes withrespect of this antipodality relation will be called antipodality classes. When thevertices of Π are treated as transpositions of the underlying 5-element set Ω, twodistinct edges x, y and u, v are antipodal if and only if xy and uv are evenpermutations of Ω fixing the same element. Therefore the antipodality classesare indexed by the elements of Ω and the setwise stabilizer of an antipodalityclass in G is Sym4.


In terms of the orthogonal space the antipodality classes are indexed by thenon-zero singular vectors in V ; a vertex is contained in an antipodality class if andonly if the corresponding non-singular and singular vectors are perpendicular.

10.4 Diagram geometries

A geometry is a quadruple (F , I, t,Ω), where F is a non-empty set (theelement-set of the geometry), I is a symmetric binary relation on F (the incid-ence relation) and t : F → Ω is the type function which assigns to every elementof F its type, the latter being an element of the set Ω of possible types. It isconvenient to treat I as a subset of the set

(F2)

of all 2-element subsets of F .A flag in a geometry (F , I, t,Ω) is a set of pairwise incident elements of the geo-metry. Thus Φ ⊆ F is a flag whenever a, b ∈ I for all distinct a, b ∈ Φ. Clearlya subset of a flag is also a flag. A flag is said to be maximal if it is not a subsetof a larger flag. The following conditions are assumed to hold in a geometry

(G1) t(a) = t(b) for any two distinct elements a, b of a flag Φ;(G2) if Φ is a maximal flag then t(Φ) = t(F) = Ω.

In order to simplify the notation, instead of (F , I, t,Ω) we simply write F .The number of types of elements in F (that is the size of Ω) is called the rankof F . For i ∈ Ω let F [i] denote the set of elements of type i in F (so thatF [i] = t−1(i)). Let Γ = Γ(F) be the incidence graph of F which is a graphon the element-set of F in which I is the set of edges. Then (G1) and (G2)imply that

(G3) Γ is an r-partite graph (where r = |Ω| is the rank of F) with partsF [i], i ∈ Ω and every maximal clique in Γ (that is a maximal completesubgraph) intersects all the parts.

It is clear that every graph which satisfies (G3) is the incidence graph ofa geometry of rank r. Therefore it is common to identify a geometry with itsincidence graph. A path in a geometry is a path in its incidence graph.

Important information about a geometry F is carried by its residual geomet-ries often called residues and defined as follows. Let Φ be a flag in a geometryF . The residue of Φ in F is the quadruple (FΦ, IΦ, tΦ,ΩΦ), where

FΦ = a | a ∈ F \ Φ, a ∪ Φ is a flag in F, IΦ = I ∩(

FΦ2

),

tΦ is the restriction of t to FΦ, ΩΦ = Ω \ t(Φ).

It is immediate that the residue of a non-maximal flag is a geometry. Supposethat Φ is a premaximal flag of cotype i ∈ Ω (this means that t(Φ) = Ω\i). ThenFΦ is a geometry of rank 1 whose set of types is i. This geometry is determinedby the number of its elements. If this number, say qi + 1, is independent on thechoice of the flag Φ of cotype i then qi it is called the ith index of F (in whatfollows we assume that all the indexes exist). If F is the projective plane overGF (q) then the point- and line-indexes are equal to q.

Diagram geometries 209

Suppose now that i and j are two different types in F and Φ is a flag ofcotype i, j in F (this means that t(Φ) = Ω \ i, j). Then the residue FΦ of Φin F is a geometry of rank 2 over the set i, j of types. Every bipartite graphwithout isolated vertices is the incidence graph of a rank 2 geometry. Since theindexes are assumed to exist, the vertices in one part of Γ(FΦ) have valencyqi + 1 while those from the other part have valency qj + 1.

Some particularly nice geometries are characterized by their residual geomet-ries of rank 2. It is common to present information about rank 2 residues in theform of a diagram. The diagram consists of nodes indexed by the types of ele-ments in F (that is by the elements of Ω) and the specific edge which joins thenodes i and j symbolizes the class of rank 2 geometries which arise as residuesin F of flags whose cotype is i, j. The meaning of some (labelled) edges willbe explain below. On a diagram above the node we place the corresponding typei ∈ Ω (sometimes the types are suppressed) and below the node we put the indexqi. Having a diagram of F it is easy to see the diagrams of its residual geometries.In order to obtain the diagram of the residue of an element of type i (which is aflag of size 1) all we have to do is to remove from the diagram the node of typei along with all the edges incident to this node. In order to obtain the diagramsof residues of larger flags we proceed by induction.

The diagram business is probably best seen on the following example. LetV = Vr+1(q) be a vector space of dimension r + 1 over the field GF (q) of qelements. Let

P = (P, I, t,Ω)

be the projective geometry of V . This means that P is the set of proper subspacesof V , I is the set of pairs U, W of such subspaces where either U < W orW < U (in particular U = W ). The type t(U) of a subspace is the dimensionof U , so that Ω = 1, 2, . . . , r. Of course the particular names for the types areirrelevant, for instance, we could have defined t(U) to be the projective dimensionof U (which is dim U − 1). A flag in F is a chain

0 < Vi1 < Vi2 < · · · < Vis< V

and the diagram of P is

Ar :1

q 2

q · · · r−1

q r

q.

This can be illustrated as follows. Suppose that r ≥ 3 and that Φ is a flag

0 < V2 < V3 < · · · < Vr−1 < V

of cotype 1, r, then the collinearity graph of PΦ is the complete bipartite graphKq+1,q+1 whose vertex-set consists of the 1-dimensional subspaces containedin V2 and the r-dimensional subspaces containing Vr−1. Any geometry whosecollinearity graph is complete bipartite is said to be a generalized digon and on


a diagram it is symbolized by the empty edge. On the other hand, if Φ is a flag

0 < V1 < · · · < Vi−2 < Vi+1 < · · · < Vr < V

of cotype i − 1, i then PΦ is the projective plane of order q associated withthe 3-dimensional GF (q)-space Vi+1/Vi−2. On diagrams the projective planesare symbolized by simple edges.

Let F = (F , I, t,Ω) be a geometry. A permutation of the element-set is saidto be an automorphism of the geometry if it preserves both the incidence relationand the type function. A permutation of the elements which preserves the incid-ence relation and permutes the types is called a diagram automorphism of thegeometry. For instance the automorphism group of the above defined projectivegeometry P is the group PΓL(r + 1, q) of projective semilinear transformationsof V . On the other hand, the isomorphism between V and its dual space V ∗

induces a diagram automorphism of P which maps the elements of type i ontothe elements of type r + 1 − i for every 1 ≤ i ≤

[r2

].

Let F = (F , I, t,Ω) be a geometry and F be an automorphism group of F .Then F is said to be flag-transitive if it acts transitively on the set of maximalflags in F . In this case the whole of F can be described in terms of the groupF and certain of its subgroups. This procedure works as follows. Let Φ = ui |i ∈ Ω be a maximal flag in F , where t(ui) = i. Let F [i] = F (ui) be the stabilizerof ui in F and let

A = F [i] | i ∈ Ω

be the subamalgam in F forms by these stabilizers. Then the F [i]’s are the max-imal parabolic subgroups and A is the amalgam of maximal parabolic subgroupsassociated with the action of F on F (a parabolic subgroup in F is the stabilizerof a flag in F). Because of the flag-transitivity the isomorphism type of A isindependent on the choice of the maximal flag Φ. Furthermore for every elementvi of type i in F there is an element in F which maps ui onto vi; the set ofall such elements is a left coset of F [i] in F which determines vi uniquely. Thisobservation enables one to show that the geometry F is isomorphic to the cosetincidence system F(F,A) defined as follows:

(C1) the set of types is Ω;(C2) the set of elements is fF [i] | f ∈ F, i ∈ Ω;(C3) the type function is fF [i] → i;(C4) the incidence relation is fF [i], hF [j] | i = j, fF [i] ∩ hF [j] = ∅.

In the case of the projective geometry P a maximal flag Φ is a chain

0 < V1 < V2 < · · · < Vr < V

the action of the automorphism group L = PΓL(r + 1, q) is flag-transitive (theresult is known as the main theorem of projective geometry). The maximalparabolic subgroup L[i] is the stabilizer in L of the i-dimensional subspace Vi

from V . The subamalgam L = L[i] | 1 ≤ i ≤ r can be specified as the one

Diagram geometries 211

formed by the maximal p-local subgroups in L containing the normalizer of agiven Sylow p-subgroup (here p is the characteristic of GF (q) and the normalizeris the stabilizer of Φ in L). In this case the isomorphism P ∼= F(L,L) providesus with a purely group-theoretical definition of the projective geometry.

An important feature of the above description is that the proper residues inF are described in terms of the amalgam of maximal parabolics only (that isindependently of whole group F ). For instance the residue Fui is isomorphicto F(F [i],A[i]), where

A[i] = F [i] ∩ F [j] | j ∈ Ω \ i.

Now suppose that Ω is a set of size r, let A = F [i] | i ∈ Ω be an abstractamalgam of rank r and ϕ : A → F is a faithful completion of A (we identify Awith its image under ϕ). Then the cosets incidence system F(F,A) might be ageometry but it also might not (in that case it contains so-called non-standardflags). The conditions for F(F,A) to be a geometry are known (cf. sections10.1.3 and 10.1.4 in (Pasini 1994) and Lemma 1.4.1 in (Ivanov 1999)). Theseconditions are formulated in terms of the amalgam A and are independent onthe completion group F . These conditions are satisfied in all the cases whichoccur in the present work. Therefore in the remaining of the section we assumethat F(F,A) is a geometry.

The diagram of F(F,A) carries in a very compact form a lot of informationabout the structure of A (recall that the diagram is independent on the com-pletion F ) and we use this language extensively throughout the volume. Thecompletion F is generating if and only if the geometry F(F,A) is connected(the latter means that the incidence graph of F(F, A) is connected in the usualsense).

Let ϕ(1) : A → F (1) and ϕ(2) : A → F (2) be two completions of A andψ : F (1) → F (2) be a homomorphism of completions (so that ϕ(2)(a) = ψ(ϕ(1)(a))for every a ∈ A). Then the mapping χ of F(F (1),A) onto F(F (2),A) defined by

χ : fϕ(1)(F [i]) → ψ(f)ϕ(2)(F [i])

is a covering of geometries in the sense that the restriction of χ to every properresidue is an isomorphism (in fact, the residues are determined by the amalgamA which stays the same). Certainly we can introduce the notion of a coveringfor any geometry (which might or might not be flag-transitive) and to definethe universal covering in the usual way (cf. Pasini 1994; Ivanov 1999; Ivanovand Shpectorov 2002). A geometry which is isomorphic to its universal cover issaid to be simply connected. The following principle independently establishedby A. Pasini, S.V. Shpectorov and J. Tits is of a fundamental importance forthe theory of flag-transitive diagram geometries.

Theorem 10.4.1 If ϕ : A → F is the universal completion of A then χ :F(F ,A) → F(F,A) is the universal covering.


It is not so difficult to check that the projective geometry P of rank at least3 is simply connected. Therefore L = PΓL(r + 1, q) is the universal completionof the amalgam L of maximal parabolic subgroups.

Below we list the diagrams of some geometries we came across within thepresent volume.

p

q – the generalized digon, whose incidence graph is complete bipartite

Kp+1,q+1;

2

2 – the projective plane of order 2 formed by the 1- and 2-dimensional

subspaces in a 3-dimensional GF (2)-space, where the incidence is via inclusion;

2

2 – the generalized quadrangle of order (2,2) formed by the 1-

and 2-dimensional totally isotropic subspaces in a non-singular 4-dimensionalsymplectic GF (2)-space, the incidence is via inclusion;

2 ∼

2 – the rank 2 tilde geometry which is a triple cover of the generalized

quadrangle of order (2,2) associated with the non-split extension 3 × S4(2) ∼=3 × Sym6;

2 d

5 – this geometry possesses the following description in terms of the

generalized quadrangle of order (2,2): take one clone of every totally isotropic1-space and two clones of every totally isotropic 2-space, a 1-space is incident tothe both clones of a 2-space containing the 1-space;

1

2 – the generalized quadrangle of order (1,2) which is the geometry

of vertices and edges of the complete bipartite graph K3,3 with the naturalincidence relation;

1 P

2 – the geometry of vertices and edges of the Petersen graph with

the natural incidence relation;

5 t

2 – the geometry of vertices and the antipodal triples of edges of the

Petersen graph with the natural incidence relation;

2

2 – the geometry of 1- and 2-dimensional subspaces in a

4-dimensional GF (2)-space (this is a truncation of the rank 3 projective GF (2)-geometry).

A geometry of rank r with the digram

Pr :1 P

2

2 · · ·

2

2

is said to be a P -geometry (or Petersen-type geometry) of rank r.A geometry of rank r with the diagram

Tr :2 ∼

2

2 · · ·

2

2

is said to be a tilde geometry of rank r.

11

APPENDIX: MATHIEU GROUPS AND THEIR GEOMETRIES

In this chapter we summarize some properties of the Mathieu groups M24 andM22, and of the associated combinatorial structures. These properties are usedthroughout the second half of the volume. In order to fully appreciate Chapter8 one should be fluent in Mathieu groups and their Steiner systems. In (Ivanov1999) all the needed properties were deduced from the very basic principles.It would not be appropriate to reproduce all the proofs here. Most of theinformation is well known, not so hard to deduce and can be found in theextensive literature on the Mathieu groups including Aschbacher (1994), Brouweret al. (1989), Conway et al. (1985), Curtis (1976), Griess (1998), James (1973),Mathieu (1860, 1861), Mazet (1979), Syskin (1980), Todd (1966), Witt (1938).Less commonly known (but crucial for our strategy) are the diagrams for H(M24)and H(M22) presented in Sections 11.5 and 11.6. These diagram were first intro-duced in (Ivanov and Meierfrankenfeld 1999) exactly for the purpose they areused here. The proofs for the diagrams are given in sections 3.7 and 3.9 in (Ivanov1999). These diagram extend the calculations of the orbits of one maximal sub-group in a Mathieu group on the coset of another maximal subgroup pioneeredin (Curtis 1976).

11.1 Witt design S(5, 8, 24)

Let P24 be a set of 24 elements. Then there exists a collection B of 8-elementsubsets of P24 such that every 5-element subset of P24 is contained in exactlyone subset from B. The pair (P24,B) is the unique Steiner system S(5, 8, 24)also known as the Witt design (Witt 1938). In (Todd 1966) the name octads forthe subsets in B was introduced. An elementary combinatorial calculation showsthat there are exactly (

245

)/(85

)= 759

octads.The automorphism group M of the Witt design (P24,B) is the Math-

ieu sporadic simple group M24 discovered by E. Mathieu as early as in 1860(cf. Mathieu (1860, 1861)). The order of M24 is

|M24| = 210 · 33 · 5 · 7 · 11 · 23,

and the action of M24 on P24 is 5-fold transitive, that is transitive on the set ofordered 5-element subsets. Let M24−i denote the elementwise stabilizer in M24of an i-element subset of P24 and let M∗

24−i denote the stabilizer of such a subset

213

214 Appendix: Mathieu groups and their geometries

as a whole. Then M23 and M22 are two further Mathieu sporadic simple groupsof orders

|M23| = |M24|/24 = 27 · 32 · 5 · 7 · 11 · 23

and

|M22| = |M23|/23 = 27 · 32 · 5 · 11,

while M21 ∼= L3(4). We have M∗22/M22 ∼= 2, M∗

21/M21 ∼= Sym3, so that

M∗22

∼= Aut (M22) and M∗21

∼= PΓL3(4),

respectively.Let B ∈ B be an octad. The stabilizer Mb of B in M ∼= M24 is the semidirect

product of a subgroup Qb∼= 24 which fixes B elementwise acting regularly on

P24 \ B and a subgroup Kb∼= L4(2) which fixes an element outside B. The

subgroup Kb acts faithfully on B as the alternating group of degree eight andKb

∼= L4(2) acts on Qb by conjugation as on its natural 4-dimensional GF (2)-module. Therefore

Mb = Qb : Kb∼= 24 : L4(2) ∼= AGL4(2).

In order to simplify the terminology we will identify B with the partition of P24into B and the complement of B.

The octad graph Γ(M24) is the graph on the set B of 759 octads in which twooctads are adjacent if they are disjoint. The octad graph is distance-transitivewith respect to the action of M24. The intersection diagram of Γ(M24) is thefollowing

1 30 280 44830 1

128 3

324 15

15

A trio is a partition T = B1, B2, B3 of P24 into three pairwise disjointoctads. By the definition the trios are in a natural correspondence with thetriangles in the octad graph Γ(M24) and it is immediate from the intersectiondiagram of Γ(M24) that every octad appears in exactly 15 trios, so that thetotal number of trios is 3795 = (759 · 15)/3. The group M24 permutes the triostransitively and the stabilizer Mt of a trio is a semidirect product of a subgroupQt

∼= 26 and a subgroup Kt∼= L3(2) × Sym3. The subgroup Kt acts on Qt by

conjugation as it acts on the tensor product of a natural module U3 of L3(2) anda natural module U2 of L2(2) ∼= Sym3. Thus

Mt = Qt : Kt∼= 26 : (L3(2) × Sym3).

A 4-element subset of P24 is called a tetrad. Every tetrad S is contained in aunique sextet Σ = S1 = S, S2, ..., S6 which is a partition of P24 into six tetradssuch that the union of any two distinct ones is an octad. The octads refined by agiven sextet induce in the octad graph Γ(M24) a subgraph known as a quad and

Geometries of M24 215

isomorphic to the collinearity graph of the generalized quadrangle of order (2, 2)(cf. Shult and Yanushka (1980) and Brouwer et al. (1989) for further propertiesof the octad graph, which is also an example of a near hexagon with three pointper a line). There are

1771 =(

244

)/6

sextets transitively permuted by M and the stabilizer Ms of a sextet is a semi-direct product of a subgroup Qs

∼= 26 and a subgroup Ks∼= 3 · Sym6. The latter

group is a non-split non-central extension of a group of order 3 by Sym6. Thegroup Ks acts on Qs as on the hexacode module, in particular O3(Ks) acts onQs fixed-point freely. The group Ms induces the symmetric group on the set oftetrads in the corresponding sextet.

11.2 Geometries of M24

Every octad from B can be identified with the partition of P24 into two sub-sets, one of them being the octad itself. Let T and S denote, respectively theset of trios and sextets in (P24,B). The Ronan–Smith geometry H(M24) is thegeometry on

B ∪ T ∪ S

(so that every element of H(M24) is a partition of P24). Two elements of H(M24)are incident if one of them refines the other one. The geometry H(M24) belongsto the truncated C4-diagram

Ct4 :

2

2

2 .

The natural action of M24 on H(M24) is flag-transitive and

B(M24) = Mb, Mt, Ms

is the amalgam of maximal parabolic subgroups associated with this action (herethe octad, the trio and the sextet stabilized by Mb, Mt, and Ms, respectivelyare assumed to be pairwise incident).

The following is a slightly weakened version of the main result of(Ronan 1982).

Theorem 11.2.1 Let X = Xb, Xt, Xs be an amalgam of rank 3, which cor-responds to the locally truncated diagram Ct

4 and suppose that Xb∼= Mb

∼= 24 :L4(2). Then the following assertions hold:

(i) X ∼= B(M24);(ii) the geometry H(M24) is simply connected;(iii) M24 is the only faithful completion of B(M24).


Proof The assertion (i) is proved in section 12.4 of (Ivanov and Shpectorov2002), the assertion (ii) is proposition 3.2.4 in (Ivanov 1999). Finally (iii) is animmediate consequence of (i), (ii), and the simplicity of M24.

As above let B(M24) denote the amalgam of maximal parabolic subgroupsassociated with the action of M ∼= M24 on H(M24). Then Qb ∩ Qt is elementaryabelian of order 23 and Mb ∩ Mt

∼= 23+3+1 : L3(2) induces on Qb ∩ Qt an actionof L3(2) as on the natural module. Let τ ∈ Qb ∩ Qt, let R ∼= 22 be the subgroupin Qt generated by the images of τ under O2,3(Mt) and consider the amalgam

A(M24) = CMb(τ), NMt

(R), Ms.

Then the isomorphism type of A(M24) is independent on the choice of such τ ;

CMb(τ) = CM (τ) ∼= 21+6

+ : L3(2)

is isomorphic to the centralizer of a central involution in L5(2) and the cosetgeometry corresponding to the embedding of A(M24) into M24 is described bythe tilde diagram

T3 :2 ∼

2

2.

Theorem 11.2.2 Let Y = Yb, Yt, Ys be an amalgam of rank 3 whichcorresponds to the diagram T3 and suppose that

|Ys| = |Ms| = |26 : 3 · Sym6|.Then

(i) Y is isomorphic either to A(M24) or to one further amalgam A(He);(ii) M24 is the only faithful completion of A(M24);(iii) the sporadic simple group He of Held is the only faithful completion of

A(He).

Proof The assertion (i) is proved in section 12.3 in (Ivanov and Shpectorov2002). The assertions (ii) and (iii) have been established by coset enumerationon a computer (cf. proposition 12.3.6 in Ivanov and Shpectorov (2002) and Heiss(1991)).

Below we reproduce the presentations for the universal completions ofA(M24) and A(He) obtained by S. V. Shpectorov and the author in 1989. Thegenerators are 13 involutions ai, 1 ≤ i ≤ 13 such that the first 12 generate

Yb∼= CMb

(τ) ∼= 21+6+ : L3(2)

which is also the centralizer of a central involution in L5(2). These 12 generatorscorrespond to elementary 5 × 5 matrices over GF (2) as shown on the matrixbelow. It should be understood that the generator ai for 1 ≤ i ≤ 12 correspondsto the matrix with 0s everywhere except the diagonal and the position where ai

stands. This position is of course 1. It is easy to see that a7 generates the centre of

Geometries of M24 217

Yb, while a1, a2, a4, a7, a8, a9, a10 generate O2(Yb) ∼= 21+6+ and a3, a5, a6, a11, a12

generate an L3(2)-complement.1 0 0 0 0a1 1 a11 0 0a2 a3 1 a12 0a4 a5 a6 1 0a7 a8 a9 a10 1

Using the above matrix it is easy to calculate the commutators [ai, aj ] for

1 ≤ i, j ≤ 10 as given below (the missing commutators equal to the identity):

[a1, a3] = a2, [a1, a5] = a4, [a1, a8] = a7, [a2, a6] = a4,[a2, a9] = a7, [a2, a11] = a1, [a3, a6] = a5, [a3, a9] = a8,[a4, a10] = a7, [a4, a12] = a2, [a5, a10] = a8, [a5, a11] = a6,[a5, a12] = a3, [a6, a10] = a9, [a8, a11] = a9, [a9, a12] = a10.

These commutator relations together with the relations

(a3a11)3 = (a6a12)3 = (a11a12)4 = 1.

provide us with a presentation for Yb (this is basically a fragment of the Steinbergpresentation for L5(2)).

The additional generator a13 extends

〈ai | 1 ≤ i ≤ 11〉 and 〈ai | 1 ≤ i ≤ 12, i = 11〉(both isomorphic to 26 : (Sym4 × Sym2)) to Yt

∼= 26 : (Sym4 × Sym3) and toYs

∼= 26 : 3 × Sym6, respectively (where O2(Yt) = 〈a4, a5, a6, a7, a8, a9〉 andO2(Ys) = 〈a2, a3, a4, a5, a7, a8〉). The following relations are common for bothamalgams

[a2, a13] = a4a5, [a3, a13] = a4, [a7, a13] = a4, [a8, a13] = a5.

The extra relations for A(M24) are

[a1, a13] = a6, [a9, a13] = a6, (a10a13)3 = 1,

[a11, a13] = a6a4, (a12a13)5 = (a6a12a13)5 = (a10a13a12)5 = 1.

For A(He) they are

[a1, a13] = a6a7, [a6, a13] = a4, [a9, a13] = a4a6a7, [a11, a13] = a1a9a4a7,

(a2a7a10a13)3 = (a7a12a13)5 = (a6a12a13)5 = (a2a7a10a13a12a7)5 = 1.

Enumeration of the cosets of Ys gives

1, 771 = [M24 : Ys]

in the case of A(M24) and

29, 155 = [He : Ys]

in the case of A(He).



(i) each of the subgroups M∗22

∼= Aut (M22), M∗21

∼= PΓL3(4), Mb∼= 24 :

L4(2), Mt∼= 26 : (L3(2) × Sym3) and Ms

∼= 26 : 3 · Sym6 is maximal inM ∼= M24;

(ii) if CMb(τ) < F < M then F = Mb;

(iii) if NMt(R) < F < M then F = Mt.

11.3 Golay code and Todd modules

Let P24 denote the GF (2)-permutation module of M24 acting on P24 definedas in Section 3.2 and let C12 be the subspace in P24 generated by the octads(considered as elements of P24). Then C12 is 12-dimensional, known as the Golaycode module for M24. Besides the 759 octads and the empty set, C12 contains theset P24 itself and the 759 complements to the octads; the remaining 2576 elementsof C12 are the so-called dodecads. A dodecad is a 12-element subset of P24 whichcan be represented as the symmetric difference of two octads (intersecting in twoelements). The dodecads are transitively permuted by M24 and the stabilizer Md

of a dodecad is another Mathieu sporadic simple group M12 of order

|M12| = 26 · 33 · 5 · 11.

The complement of a dodecad is again a dodecad and the stabilizer M∗12 of a

complementary pair of dodecads is Aut (M12). The smallest Mathieu sporadicsimple group M11 is the stabilizer in Md of an element of P24, its order is

|M11| = 24 · 32 · 5 · 11.

The only composition series of P24 is

0 < Pc24 < C12 < Pe

24 < P24

and C11 = C12/Pc24 is known as the irreducible Golay code module. In that module

M24 has two orbits on the set of non-zero vectors indexed by the octads and pairsof complementary dodecads, respectively. The module

C12 = P24/C12

is called the Todd module. The non-zero vectors in the Todd module are indexedby the 1-, 2-, 3-element subsets of P24 and by the sextets. The submodule C11 =Pe

24/C12 of codimension 1 in the Todd module is called the irreducible Toddmodule. The non-zero vectors of C11 are indexed by 2-element subsets of P24 andby the sextets. It is clear that (C12, C12) and (C11, C11) are dual pairs.

For the proof of the following lemma see section 3.8 in (Ivanov 1999).

Lemma 11.3.1 Let Y = C11 and x = b, t or s. Then

1 < CY (Qx) < [Y, Qx] < Y

is the only composition series of Y as a module for Mx. Furthermore,

Shpectorov’s characterization of M22 219

(i) CY (Qb) ∼=∧2

Qb, [Y, Qb]/CY (Qb) ∼= Qb, Y/[Y, Qb] ∼= 2;(ii) CY (Qt) ∼= U∗

3 , [Y, Qt]/CY (Qt) ∼= Qt, Y/[Y, Qt] ∼= U2;(iii) CY (Qs) ∼= 2, [Y, Qs]/CY (Qs) ∼= Q∗

s, Y/[Y, Qs] is the natural symplecticmodule of Ms/O2,3(Ms) ∼= S4(2).

11.4 Shpectorov’s characterization of M22

In terms of the previous section let Ψ = (B, T,Σ) be a flag in H(M24), whereB ∈ B is an octad, T = B = B1, B2, B3 is a trio and Σ = S1, ..., S6 isa sextet and suppose that B1 = S1 ∪ S2, B2 = S3 ∪ S4, B3 = S5 ∪ S6. LetA(M24) = Mb, Mt, Ms be the amalgam of maximal parabolic subgroups inM = M24 associated with the flag Ψ. Let x, y ∈ S6 and let K be the setwisestabilizer of x, y in M ∼= M24. Then

K ∼= M∗22

∼= Aut (M22).

Let Ko = K ∩ Mb, Kt = K ∩ Mt, Kp = K ∩ Ms and put

A(Aut (M22)) = Ko, Kt, Kp.

Then Ko∼= 23 : L3(2) × 2, Kt

∼= (21+4+ × 2) : (Sym3 × 2) and Kp = 25 : S5, in

particular [Kt : Ko ∩ Kt] = 2.Let Γ(M22) be the graph whose vertices and edges are the cosets in K of Ko

and Kt, respectively; a vertex and an edge are incident if the corresponding cosetshave non-empty intersection (in terms of (2.3.2) Γ(M22) is Λ(Ko, Kt, ι, K),where ι is the identity mapping). Alternatively Γ(M22) is the subgraph in theoctad graph Γ(M24) induced by the set of images of B = B1 under K. This setof images consists of the 330 octads disjoint from x, y (these octads are calledoctets and therefore Γ(M22) is called the octet graph). Notice that two octets areadjacent in the octet graph Γ(M22) if and only if they are disjoint (for instanceB1, B2 is an edge). The octet graph is distance-transitive with the followingintersection diagram.

1 7 42 168 112

2 2 17 1 6 1 4 1 4 6

The action of K (and also of M22) is distance-transitive and locally projectiveof type (3, 2). The geometric cubic subgraphs in the octet graph are isomorphicto the Petersen graph and Kp is the stabilizer in K of one of these subgraphs. Thegeometry of vertices, edges, and Petersen subgraphs in Γ(M22) is the Petersentype geometry G(M22) with the diagram

P3 :1 P

2

2

and A(Aut (M22)) is the amalgam of maximal parabolic subgroups associatedwith the flag-transitive action of K on G(M22).


The octet graph possesses a 3-fold antipodal cover Γ(3·M22) known as Ivanov–Ivanov–Faradjev graph (Brouwer et al. 1989) whose intersection diagram is thefollowing:

1 7 42 168 336 336 84 14 2

2 2 1 2 2

7 1 6 1 4 1 4 2 4 4 1 4 1 6 1 7

The automorphism group K of Γ(3 · M22) is the non-split non-centralextension by K of a group of order 3, so that

K ∼= 3 · Aut M22.

The action of K on Γ(3 · M22) is also locally projective of type (3, 2), the geo-metric cubic subgraphs are still isomorphic to the Petersen graph; the geometryG(3 · M22) of vertices, edges, and Petersen subgraphs in G(3 · M22) is a coverof G(M22), in particular it also belongs to the diagram P3. Furthermore theamalgam of maximal parabolic subgroups of the flag-transitive action of K onG(3 · M22) is isomorphic to A(Aut (M22)).

The following beautiful combinatorial result was established almost 20 yearsago (cf. Shpectorov (1985) and proposition 3.5.5 in Ivanov (1999)). This resultlies in the foundation of the geometric approach to the sporadic simple groups.

Theorem 11.4.1 Let G be a geometry with the diagram P3 (which might ormight not be flag-transitive). Then the number of elements corresponding to theleftmost node on the diagram is at most 1898.

Theorem 11.4.2 Let Z = Zo, Zt, Zp be an amalgam of rank 3 which cor-responds to the diagram P3 and suppose that Zo

∼= Ko∼= 23 : L3(2) × 2.

Then

(i) Z ∼= A(Aut (M22));(ii) K ∼= Aut (M22) and K ∼= 3 · Aut (M22) are the only faithful completions

of A(Aut (M22)) (with the latter completion being the universal one).

Proof The assertion (i) was originally proved in Shpectorov (1985) and a proofis given in section 11.2 in Ivanov and Shpectorov (2002). Now (ii) is immediatefrom (11.4.1) since Γ(3 · M22) contains 990 vertices (which is more than half ofthe upper bound).

The classification of the flag-transitive geometries with diagram P3 obtainedin Shpectorov (1985) (see also Ivanov and Shpectorov (2002)) implies thefollowing.

Theorem 11.4.3 Let Θ be a graph of valency 7 and L be a group of auto-morphisms of Θ whose action on Θ is locally projective of type (3, 2). Supposefurther that a geometric cubic subgraph in Θ is isomorphic to the Petersen graph.

Shpectorov’s characterization of M22 221

Then one of the following holds:

(i) Θ is the octet graph Γ(M22) and L is either M22 or Aut M22;(ii) Θ is the Ivanov–Ivanov–Faradjev graph Γ(3 · M22) and L is either 3 · M22

or 3 · Aut (M22).

In terms introduced at the beginning of the section put P22 = P24 \ x, y,H = B3 \ x, y and let H be the set of images of H under K. Then (P22,H) isthe unique Steiner system S(3, 6, 22) whose blocks are called hexads. Let Kh bethe stabilizer in K of the hexad H. Then

Kh∼= 24 : Sym6

and Kh contains Kt with index 15.Let us say that two edges of the octet graph Γ(M22) are locally antipodal if

they are contained in a common geometric cubic subgraph Σ (isomorphic to thePetersen graph) and if they are antipodal in the line graph of Σ. Let ΦΓ(M22) bethe graph (called local antipodality graph) on the set of edges of Γ(M22) in whichtwo edges are adjacent if they are locally antipodal. The following result can beread from lemmas 3.4.4 and 7.1.7 in Ivanov (1999).

Lemma 11.4.4 Let e = Be1, B

e2 and f = Bf

1 , Bf2 be edges of the octet graph

Γ(M22). Then e and f are contained in the same connected component of thelocal antipodality graph ΦΓ(M22) if and only if

P22 \ (Be1 ∪ Be

2) and P22 \ (Bf1 ∪ Bf

2 )

is the same hexad of (P22,H).

In the local antipodality graph ΦΓ(3·M22) every connected component is iso-morphic to the triple cover (associated with 3 · Sym6) of the point graph of thegeneralized quadrangle of order (2,2).

Thus there is a bijection between the hexads in (P22,H) and the connectedcomponents of the local antipodality graph Φ = ΦΓ(M22). In these terms thecomponent Φc containing the edge B1, B2 corresponds to the hexad H =B3 \ x, y. Then Φc is the point graph of the unique generalized quadrangle oforder (2, 2) (the vertices are the 2-subsets of a 6-sets in which two of subsets areadjacent if they are disjoint). The triangles of Φc correspond to the 15 geometriccubic subgraphs in Γ(M22) whose edge-sets intersect the vertex-set of Φc. Thesesubgraphs naturally correspond the 2-element subsets of P22 contained in H.The group Kh (which is the stabilizer of H in K) induces the full automorphismgroup Sym6 of Φc; the kernel O2(Kh) stabilizes as a whole (but not vertexwisely)every edge of Γ(M22) contained in Φc.

The amalgam

B(Aut (M22)) = Kp, Ko, Khis the amalgam of maximal parabolic subgroups associated with theflag-transitive action of K on the Ronan–Smith geometry H(M22) (Ronan and


Smith 1980). The elements of H(M22) are the geometric cubic subgraphs inG(M22), the vertices of Γ(M22) and the connected components of the local anti-podality graph ΦΓ(M22). Alternatively elements are the pairs (2-element subsetsof P22), the octets and the hexads and the incidence relation is such that

(S6 \ x, y, B1, B3 \ x, y)

is a maximal flag (recall that S6 is the last tetrad in the sextet discussed inthe beginning of the section). The geometry H(M22) belongs to the followingdiagram

2 2

5

d t

H(Mat22) :

where the nodes from left to right correspond to pairs, octets and hexads,respectively.

The geometry H(M22) is convenient to view through the hexad graph. This isa graph Ξ on the set of 77 hexads in the Steiner system (P22,H) of type S(3, 6, 22)in which two hexads are adjacent if they intersect in a pair of elements of P22.The graph is strongly regular (which means distance-regular of diameter 2) withthe following intersection diagram:

1 1660

60 13+12+32

12 4515

The action on Ξ of the group K ∼= Aut (M22) is of rank 3, which means theaction is transitive both on the pairs of adjacent and on the pairs of non-adjacentvertices. The stabilizer of a vertex h is Kh

∼= 24 : Sym6. Thus Kh is transitiveon Ξ(h) consisting of the hexads intersecting h in pairs and on the set Ξ2(h)consisting of the hexads disjoint from h. We will need the following result whichis well known and easy to check.

Lemma 11.4.5 In the above terms O2(Kh) acts on Ξ(h) with orbits of length 4and its action on Ξ2(h) is transitive.

There are five hexads containing a given pair p. Clearly these five hexads forma complete subgraph in the hexad graph. This subgraph we denote by Ξ(p). Ifp ⊂ h then Ξ(p) \ h is an orbit of O2(Kh) on Ξ(h). The subgraphs Ξ(p)containing h are in the natural bijection with

(h2

). Since Kh/O2(Kh) induces the

symmetric group of h, the following lemma can be justified just by mere lookingat the above intersection diagram of Ξ. Alternatively it can be deduced from thefact that any two hexads are either disjoint or intersect in a pair.

Diagrams of H(M24) 223

Lemma 11.4.6 Let p1, p2 ∈(h2

). Then

(i) if |p1 ∩ p2| = 1 then every vertex from Ξ(p1) \ h is adjacent to everyvertex in Ξ(p2) \ h;

(ii) if |p1 ∩ p2| = 0 then every vertex from Ξ(p1) \ h is adjacent to a half ofthe vertices in Ξ(p2) \ h.

The following final result in this section is well known and easy to deduce.

Proposition 11.4.7 Each of the subgroups Kp∼= 25 : Sym5, Ko

∼= 23 : L3(2)×2and Kh

∼= 24 : Sym6 is maximal in K ∼= Aut (M22).

11.5 Diagrams of H(M24)

In this section we present the diagrams which describe the action of the Mathieugroup M ∼= M24 on its Ronan–Smith geometry H(M24). Let us explain theinformation given on the diagrams. The proof of the diagrams can be found insection 3.7 of (Ivanov 1999).

For x = b, t and s the diagram Dx(M24) describes the orbits of the maximalparabolic subgroup Mx on the elements of H(M24). An orbit is represented by acircle; the number inside the circle is the length of the orbit and the index at thisnumber (which is b, t, or s) indicates the type of elements in the orbit (octads,trios, or sextets, respectively). The length is given in the form 2lk which meansthat Qx = O2(Mx) acts with k orbits of length 2l each (when l = 0 the factor 1is suppressed).

Let O(1) and O(2) be orbits of Mx. Then the line joining the circles cor-responding to these orbits indicates that an element α ∈ O(1) is incident to anon-zero number m of elements in O(2). The number m is shown next to theline and closer to the circle of O(1). If the number m is shown as a sum of twonumbers, say m1 and m2 then the stabilizer of α in Mx acting on the set ofelements in O(2) incident to α has two orbits with lengths m1 and m2; other-wise the action is transitive. Notice that on each diagram every orbit is uniquelydetermined by its length and type, which enables us to refer to an orbit simplyby giving its length and type.

For instance consider the diagram Dt(M24). It shows that the subgroup Mt∼=

26 : (L3(2) × Sym3) acting on the set of octads has three orbits with lengths 3,84, and 672. The subgroup Qt = O2(Mt) acts trivially on the first of the orbits,has 21 orbits of length 4 on the second orbit and 21 orbits of length 32 on thethird orbit. Let α be an octad from the third orbit. Then among the 35 sextetsincident to α one is in the Mx-orbit of length 84 = 22 · 21, six are in the orbit oflength 336 = 23 · 42, and twenty-eight are in the orbit of length 1344 = 26 · 21.Finally the stabilizer of α in Mx acts on the set of sextets incident to α withfour orbits of lengths 1, 6, 12, and 16.


15t22 105t 24 210t

1

22 70b 24 28b1b

35s23 105s

24 56s

15

35

13

7

1

812

18

6 3 12

8

15

6

12

3

12

1

3

5

16

15

4

15

2

2 15b

2

1 14

1

20

10

28

1

7

6

Db(M24)

3b22 21b 25 21b

1

22 14t 26 42t1t

7s26 21s

3

7

13

7

1

812

1

2

3

1

6+8

12+16

2 21t

24 63t

22 21s 23 42s

14

281

6 1212

163

6

12

1 2 1 2 3

1

642

1661

6

31 6 12

8 13 6

128

31

12

Dt(M24)

Diagrams of H(M22) 225

15b23 45b 26 6b

1

25 45s1s

15t 26 45t

15

15

13

3

2 45s

18

3

6

24 15s

22 45t 24 45t

12

16

1 4 68

4+24 5

3015

3 121 6 8

1+6 8

1 6

8

312

3 12

31 2 4

13 6

11

62

Ds(M24)

11.6 Diagrams of H(M22)

In this section we present the diagrams describing the action of K ∼= Aut (M22)on the Ronan–Smith geometry H(M22). The notation is similar to that in theprevious section. The proofs of the diagrams can be found in section 3.9 of(Ivanov 1999).

2 15o 22 15o 24 15o

1

22 15h 24h1h

15p23 15p

24 6p

30

15

16

3

6

4

1

44 1

6

3 3 6

2

15

15

4

61

3

3

24

6

3

5

8

10

4

15

1

3

Dh(M22)


7h2 7h 23 7h

1

2 21o 24 7o1o

7p 22 21p 24 7p

7

7

1 3

3

12

12

62

13

!!!!!!!!!!!!!

7o 23 21o

22 7p

6

3

61

4

3

3

1

8

24

163

1612 8

3+12

21

4

1

2 4

16 2

1+4

1

2

4

3 3 2

61

2 4 44

3 4

14 4 6

1

Do(M22)

5h22 10h 25h

1

24 10p1p

10o 25 5o

5

10

1 3

6

44

3

15

2

10

2 15p

6

1

3

6

6

2

6

23 5p

23 15o 22 10o

24

8

16

1+6

16

12 5

15

4

4

4

1 4

46

3

3

1

11 2 2

44 3

1+64

1

Dp(M22)

Exercises

1. Deduce from (11.4.2) that the geometry H(M22) is simply connected.

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INDEX

actionlocally projective, 197, 205

amalgamlocally projective, 198representation of, 84

arc, 203automorphismcontragredient, 16

basishyperbolic, 3

centralizer, 201completionfaithful, 24generating, 24morphism, 24

cosetright, 201

coveringof graphs, 205

deck automorphisms, 30decompositionhyperbolic, 3

dent, 56diameter, 203, 204direct sum, 2

exterior square, 15

flag, 206formorthogonal, 1symplectic, 1induced, 8radical of, 8rank, 8

fundamental cycles, 121

geometry, 206rank, 206

Goldschmidt’s theorem, 43graphantipodal, 206coset, 51

distance-transitive, 204hexad, 149, 221Ivanov–Ivanov–Faradjev, 137, 219line, 206local antipodality, 220octad, 213octet, 136, 140, 218Petersen, 205

groupJ4, ixcohomologyfirst, 32

Conway Co1, 80extraspecial, 10of automorphismsflag-transitive, 208

pentad, 75symplectic, 4

heart, 34hexad, 220

intersection array, 204involution, 201

key commutator relation, 41, 46

Leech lattice, 80

maximal, 206modulederived, 200Golay code, 217irreducible, 217

hexacode, 214Todd, 217irreducible, 217

normalizer, 201

octad, 212orthogonalformminus type, 5plus type, 5

orthogonal group, 5

232

Index 233

partitioneven, 38odd, 38

path, 203in geometry, 207non-degenerate, 141

permutation module, 34perpendicular, 1planehyperbolic, 3

power set, 34productcentral, 11direct, 201semidirect, 201

quad, 213

relationantipodality, 206

sextet-line, 167Siegeltransformation, 6

spaceorthogonal, 1

symplectic, 1non-singular, 2singular, 2

standarddoubling, 127

subgraph, 203convex, 20, 203geodetically closed, 203geometric, viiiinduced, 203

subgroupfully normalized, 202normal, 201

suborbit diagram, 204subspacetotally isotropic, 3totally singular, 3, 6

transvection, 6treespanning, 121

valencies, 204vectornon-singular, 3singular, 3

Date post:	11-Sep-2021
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The Fourth Janko Group

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