Normal and non-normal Cayley graphs arisingfrom generalised quadrangles

Michael Giudici

joint work with John Bamberg

Centre for the Mathematics of Symmetry and Computation

Beijing Workshop on Group Actions on CombinatorialStructures, August 2011

Cayley graphs

• G a group

• S ⊂ G with 1 /∈ S and S = S−1

The Cayley graph Cay(G ,S) is the graph with

• vertex set G

• edges {g , sg} for all s ∈ S and g ∈ G .

Cay(G , S) is connected if and only if 〈S〉 = G .

Automorphisms

Let G be the right regular representation of G .

Then G 6 Aut(Cay(G , S)).

Call Cay(G ,S) a graphical regular representation of G (GRR) ifequality holds.

A graph Γ is a Cayley graph for G if and only if Aut(Γ) contains aregular subgroup isomorphic to G .

More automorphisms

Let Aut(G ,S) = {σ ∈ Aut(G ) | Sσ = S}.

Then Aut(G , S) is a group of automorphisms of Cay(G ,S).

For example, if G is abelian then

ι : G → Gg 7→ g−1

is a element of Aut(G ,S).

Normal Cayley graphs

G o Aut(G ,S) 6 Aut(Cay(G ,S))

If equality holds, we call Cay(G ,S) a normal Cayley graph.(Mingyao Xu 1998)

Otherwise called nonnormal.

Examples

• K4 is a normal Cayley graph for C 22 but a nonnormal Cayley

graph for C4.

• Kn for n ≥ 5 is a nonnormal Cayley graph for any group oforder n.

• Any GRR is a normal Cayley graph.

• Any Cayley graph of Cp other than Kp or pK1 is a normalCayley graph.(The full automorphism group is not 2-transitive so by atheorem of Wielandt has automorphism group contained inAGL(1, p).)

• The Hamming graph H(m, q) with q ≥ 4, is a nonnormalCayley graph for Cn

q .

Normal and nonnormal Cayley graphs

Mingyao Xu conjectured that almost all Cayley graphs are normal.

More precisely,

minG , |G |=n

# normal Cayley graphs of G

# Cayley graphs of G→ 1 as n→∞

A question

Yan-Quan Feng asked:

Is it possible for a graph to be both a normal and anonnormal Cayley graph for G?

Need to find Γ such that Aut(Γ) contains a normal regularsubgroup G and a nonnormal regular subgroup isomorphic to G .

Cay(G , S) is a CI-graph for G if all regular subgroups isomorphicto G are conjugate.

Li (2002) gave examples of Cayley graphs such that Aut(Γ) hastwo normal regular subgroups isomorphic to G .

Generalised quadrangles

A generalised quadrangle is a point-line incidence geometry Q suchthat

1 any two points lie on at most one line, and

2 given a line ` and a point P not incident with `, P is collinearwith a unique point of `.

If each line is incident with s + 1 points and each point is incidentwith t + 1 lines we say that Q has order (s, t).

If s, t ≥ 2 we say Q is thick.

An automorphism of Q is a permutation of the points that mapslines to lines.

Some graphs associated with a generalised quadrangle

Incidence graph: The bipartite graph with

• vertices the points and lines of Q• adjacency given by incidence.

Equivalent definition of a generalised quadrangle is a point-lineincidence geometry whose incidence graph has diameter 4 andgirth 8.

Has valency {t + 1, s + 1}

Some graphs associated with a generalised quadrangle II

Collinearity graph:

• Vertices are the points of Q• two vertices are adjacent if they are collinear.

Has valency s(t + 1).

The GQ axiom implies that any clique of size at least three iscontained in the set of points on a line.

Hence any automorphism of the point graph is an automorphism ofthe GQ.

The Classical GQ’s

Name Order Automorphism group

H(3, q2) (q2, q) PΓU(4, q)H(4, q2) (q2, q3) PΓU(5, q)W (3, q) (q, q) PΓSp(4, q)Q(4, q) (q, q) PΓO(5, q)Q−(5, q) (q, q2) PΓO−(6, q)

Take a sesquilinear or quadratic form on a vector space

• Points: totally isotropic 1-spaces

• Lines: totally isotropic 2-spaces

Groups acting regularly on GQ’s

Ghinelli (1992): A Frobenius group or a group with nontrivialcentre cannot act regularly on a GQ of order (s, s), s even.

De Winter, K. Thas (2006): Characterised the finite thick GQswith a regular abelian group of automorphisms.

Yoshiara (2007): No GQ of order (s2, s) admits a regular group.

p-groups

A p-group P is called special if Z (P) = P ′ = Φ(P).

A special p-group is called extraspecial if |Z (P)| = p.

Extraspecial p-groups have order p1+2n, and there are twoisomorphism classes for each order.

For p odd, one has exponent p and one has exponent p2.

Q8 and D8 are extraspecial of order 8.

Regular groups and classical GQs

Theorem

If Q is a finite classical GQ with a regular group G ofautomorphisms then

• Q = Q−(5, 2), G extraspecial of order 27 and exponent 3.

• Q = Q−(5, 2), G extraspecial of order 27 and exponent 9.

• Q = Q−(5, 8), G ∼= GU(1, 29).9 ∼= C513 o C9.

Use classification of all regular subgroups of primitive almostsimple groups by Liebeck, Praeger and Saxl (2010)

Alternative approach independently done by De Winter, K. Thasand Shult.

Ahrens-Szekeres-Hall quadrangles

Begin with W(3, q) defined by the alternating form

β(x , y) := x1y4 − y1x4 + x2y3 − y2x3

Let x = 〈(1, 0, 0, 0)〉.

Define a new GQ, Qx, with

• points: the points of W(3, q) not collinear with x , that is, all〈(a, b, c , 1)〉

• lines:

(a) lines of W (3, q) not containing x , and(b) the hyperbolic 2-spaces containing x .

Qx is a GQ of order (q − 1, q + 1).

This construction is known as Payne derivation.

Automorphisms

• PΓSp(4, q)x 6 Aut(Qx)

• Grundhofer, Joswig, Stroppel (1994): Equality holds forq ≥ 5.

• Note Sp(4, q)x consists of all matrices λ 0 0uT A 0z v λ−1

with

A ∈ GL(2, q) such that AJAT = J,

z ∈ GF(q) and u, v ∈ GF(q)2 such that u = λvJAT

where J =(

0 1−1 0

).

An obvious regular subgroup

Let

E =

1 0 0 0−c 1 0 0b 0 1 0a b c 1

∣∣∣ a, b, c ∈ GF(q)

C ΓSp(4, q)x

• |E | = q3,

• acts regularly on the points of Qx,

• elementary abelian for q even,

• special of exponent p for q odd (Heisenberg group).

So the collinearity graph of one of these GQs is a normal Cayleygraph for q ≥ 5.

But there are more . . .

q # conjugacy classes # isomorphism classes

3 2 2 (this is Q−(5, 2))4 58 305 2 17 2 18 14 89 5 5

11 2 113 2 116 231 -17 2 119 2 123 2 125 7 -

An element

For α ∈ GF(q), let

θα =

1 0 0 0−α 1 0 0−α2 α 1 0

0 0 α 1

∈ Sp(4, q)x.

Construction 1

Let {α1, . . . , αf } be a basis for GF(q) over GF(p).

P =

⟨1 0 0 00 1 0 0b 0 1 0a b 0 1

, θα1 , . . . , θαf

⟩

• P acts regularly on points and is not normal in Aut(Qx)

• nonabelian for q > 2

• not special for q even, special for q odd

• exponent 9 for p = 3

• P ∼= E for p ≥ 5

• So for p ≥ 5 the collinearity graph is both a normal andnonnormal Cayley graph for isomorphic groups.

Some other examples

• Halved folded 8-cube for C 62 (Royle 2008)

• a Cayley graph for C 23 oC 2

2 of valency 14 (Giudici-Smith 2010)

Construction 2

Let U ⊕W be a decomposition of GF(q), q = pf , f ≥ 2.

Let {α1, . . . , αk} be a basis for U over GF(p).

SU,W =

⟨1 0 0 00 1 0 0b 0 1 0a b 0 1

,

1 0 0 0−w 1 0 0

0 0 1 00 0 w 1

, θα1 , . . . , θαk

⟩

• SU,W acts regularly on points,

• for U a 1-space:• E 6∼= SU,W 6∼= P• SU,W is not special

Significance for GQs

The class of groups that can act regularly on the set of points of aGQ is richer/wilder than previously thought.

Such a group can be

• a nonabelian 2-group,

• a 2-group of nilpotency class 7,

• p-groups that are not Heisenberg groups,

• p-groups that are not special.