research-repository.uwa.edu.auresearch-repository.uwa.edu.au/files/3214321/Amarra...Abstract A graph...

Symmetric graphs of

diameter two

Maria Carmen Amarra

B.S., M.S. University of the Philippines – Diliman

This thesis is presented for

the degree of

Doctor of Philosophy

of

The University of Western Australia

School of Mathematics and Statistics

November 2012

Abstract

A graph Γ is G-symmetric if it admits an arc-transitive subgroup G of automorphisms,

and has diameter 2 if it is not a complete graph (that is, it has at least one pair of

nonadjacent vertices) and if any two nonadjacent vertices have a common neighbour. Using

normal quotient analysis, the study of G-symmetric diameter 2 graphs can be reduced to

the following cases:

(i) All nontrivial G-normal quotient graphs of Γ are complete graphs.

(ii) All nontrivial normal subgroups of G act transitively on the vertex set of Γ.

We consider in detail the pairs (Γ, G) that satisfy (i) where Γ may have diameter greater

than two, as well as those that satisfy (ii) where Γ has diameter 2 and G is maximal in

the symmetric group of the vertex set of Γ subject to being non-2-transitive.

For the first case, we show that if Γ has at least three nontrivial normal quotients,

then G corresponds to a finite transitive linear group H and Γ can be constructed from

the natural vector space of H. We classify all connected graphs arising from groups H

which are not subgroups of a one-dimensional affine group, and identify those which have

diameter greater than two. For the second case, the group G is given by C. E. Praeger’s

classification of quasiprimitive permutation groups, and we focus on the subcase where

G is of affine type. Such groups G correspond to irreducible subgroups of the general

linear group which, in turn, have been classified by M. Aschbacher. Moreover, a uniform

construction for Γ is known, so it only remains to determine which graphs have diameter

2. Using a case-by-case analysis, we are able to classify all diameter 2 graphs for some of

the Aschbacher classes; in the others we determine bounds on certain parameters in order

to have diameter 2, which reduce the number of unresolved cases.

iii

Dedicated to the memory of

Arturo Amarra, Mario Valencia,

and

Carmen Ponce-Villaroman

v

vi

This research was undertaken with the support of an Endeavour In-

ternational Postgraduate Research Scholarship (March 2008 to March

2012), a Samaha Top-Up Scholarship (March 2008 to September 2011),

a Scholarship for International Research Fees (April to June 2012), and a

University Postgraduate Award (International Students) from The Uni-

versity of Western Australia.

Acknowledgements

My deepest gratitude goes, naturally, to my supervisors, Professors Cheryl Praeger

and Michael Giudici. I can give them no praise or compliment that has not already been

given by others, except perhaps to say: I am a better mathematician (and writer) for

having worked with them. (Which is not to say, of course, that one has not much farther

to go.) I hope, throughout my professional life, to be able to do justice to the training I

received from them.

I thank The University of Western Australia for providing the financial support that

made this whole undertaking possible, and the School of Mathematics and Statistics —

especially the Centre for the Mathematics of Symmetry and Computation — for providing

a supportive intellectual environment. Special thanks to Dr. John Bamberg and Dr. Pablo

Spiga for pointing out connections with other areas; that their ideas were not developed

in this thesis is entirely my shortcoming.

And also to:

My family, without whom I would not be here — or anywhere; especially my mother

for saying Mangangarap ka na nga lang, taasan mo na (Aim high), and my father for his

incredible patience and understanding;

Gino and Manjo, for the friendship that has managed to survive the years and the

geographical distance;

Beth, for being there, and for all the other things that can never be repaid;

Chevs — the other Cheryl, for taking over the cleaning and the cooking, for practically

feeding her (useless) flatmate those last six months, and above all for the companionship

of the past four years;

And all my other friends in Perth, then and now, from whom I learned a lot —

including the meaning of bittersweet.

Maraming salamat.

vii

Contents

Abstract iii

Acknowledgements vii

Introduction xi

Chapter 1. Permutation groups 1

1.1. Basic concepts and notation 1

1.2. Blocks, primitivity and quasiprimitivity 5

1.3. The O’Nan-Scott Theorem for quasiprimitive groups 6

1.4. Rank 3 groups 12

Chapter 2. Algebraic graph theory 15

2.1. Basic concepts 15

2.2. Cayley graphs 17

2.3. Products of graphs 18

2.4. Normal quotients 24

Chapter 3. Linear algebra and geometries 31

3.1. Tensor product spaces 31

3.2. Linear, semilinear, and affine groups 38

3.3. Classical groups and their geometries 42

3.4. The exceptional group G2(q) and its geometry 51

3.5. The transitive finite linear groups 53

3.6. Aschbacher’s classification 54

Chapter 4. Quotient-complete symmetric graphs 59

4.1. Overview and main results 59

4.2. Examples and general structure 60

4.3. Connected quotient-complete symmetric graphs 68

4.4. Quotient-complete symmetric graphs with diameter 2 74

4.5. Quotient-complete symmetric graphs arising from H ≤ ΓL(1, q) 78

Chapter 5. Symmetric vertex-quasiprimitive graphs: affine case 81

5.1. Overview and main results 81

5.2. Class C8 82

5.3. Class C2 87

ix

x CONTENTS

5.4. Class C4 90

5.5. Class C5 93

5.6. Class C6 100

5.7. Class C7 102

Chapter 6. Other quasiprimitive types 107

6.1. Diagonal type subgroups 107

6.2. Quasiprimitive wreath products 111

6.3. Almost simple subgroups 114

Appendix A. Magma codes 115

A.1. Algorithms for Table 4.2 115

A.2. Algorithms for Example 4.5.4 118

A.3. Algorithms for Example 6.1.5 118

Bibliography 121

Introduction

A graph is an abstraction of a network, and consists of a set of vertices linked by edges.

Perhaps the best-known application of symmetric graphs is in the design of networks

for parallel computing, where graph properties such as symmetry, valency, and diameter

(among others) correlate with network properties such as efficiency, connectivity, and

resilience (see [2, 11, 26]). In this context, symmetric graphs with small diameters, with

their regular structure and high connectivity, are particularly desirable (see [12, 16]).

This thesis investigates the general structure of symmetric graphs with diameter 2 —

that is, those graphs whose automorphism group acts transitively on their arc set, and

in which any pair of vertices are connected or are connected to a common vertex. The

symmetric diameter 2 case is interesting since this includes all symmetric strongly regular

graphs, and in particular all of the rank 3 graphs (see Sections 1.4 and 6.3).

All symmetric graphs are vertex-transitive, and we set up our analysis by identifying

the basic vertex-transitive graphs. These are graphs whose structure provides insight on

the structure of any given vertex-transitive graph, and which can thus be considered as the

building blocks of vertex-transitive graphs. We achieve the above using normal quotient

reduction, which is described in Section 2.4, and obtain the following result:

Theorem 1. Let Γ be a connected graph with a vertex-transitive group G of automor-

phisms. Then there exists a proper normal subgroup N of G (possibly the trivial subgroup)

such that, relative to G/N , the corresponding normal quotient ΓN is either

(1) quotient-complete, or

(2) vertex-quasiprimitive and not complete.

The terms quotient-complete and vertex-quasiprimitive are defined in Section 2.4. We

note that the property of being quotient-complete or vertex-quasiprimitive (or, for that

matter, arc-transitive) is dependent on the subgroup of automorphisms being considered,

and gives extra useful information that we can exploit. Hence the characterisation of the

graphs that arise in Theorem 1 involves identifying the pairs (Γ, G), where Γ is a graph

and G ≤ Aut (Γ) with the desired property. Our broad aim is to classify all symmetric,

diameter 2 graphs that are either quotient-complete or vertex-quasiprimitive.

In the quotient-complete case we consider graphs which are symmetric with diameter

possibly greater than two. For symmetric quotient-complete graphs, a significant parameter

is the number k of nontrivial complete normal quotients. Infinite families of examples exist

xi

xii INTRODUCTION

for the cases k = 1 and k = 2, for which we give constructions in Section 4.2. A complete

classification, however, is difficult to achieve. For k ≥ 3 it turns out that the graph

structure is more restricted, and the corresponding automorphism groups arise from finite

transitive linear groups H acting on a vector space over a finite field. The main result for

this case is Theorem 2.

Theorem 2. Let Γ be a graph with a symmetric group G of automorphisms, such that

Γ is G-quotient-complete with k distinct nontrivial, complete G-normal quotients. If k ≥ 3

then Γ has order c2 for some prime power c, and all nontrivial complete normal quotients

have order c. Furthermore, either k = c and Γ ∼= c.Kc, or k ≤ √c+1. All pairs (Γ, G) are

known, up to possible additional examples associated with one-dimensional affine groups.

In addition to the above, we also determine which of the graphs in Theorem 2 have

diameter 2.

The vertex-quasiprimitive case further divides according to Praeger’s classification of fi-

nite quasiprimitive permutation groups (Theorem 1.3.8). Under this system the quasiprim-

itive groups are characterised by their socle, which is the direct product of all minimal

normal subgroups. We restrict our attention to the groups and graphs that satisfy Hy-

pothesis 3 below. (A permutation group is 2-transitive on a set if it is transitive on the

set of ordered pairs of distinct elements of that set.)

Hypothesis 3. For a nonempty set Ω, the group G ≤ Sym (Ω) is quasiprimitive on

Ω, and is maximal in Sym (Ω) such that G is not 2-transitive.

By Theorem 1.3.8, a group G satisfying Hypothesis 3 must be one of the following:

(i) a group of affine type;

(ii) a group of diagonal type;

(iii) a wreath product of symmetric groups in product action; or

(iv) an almost simple group.

We say that the quasiprimitive group G is maximal of affine type (respectively, of diagonal

type, product type, or almost simple type) in case (i) (respectively, (ii), (iii), or (iv)). The

socle of G is abelian if and only if G is of affine type.

Much of our work deals with the case where G is maximal of affine type. In this case

G is a subgroup of an affine group AGL(d, p) for some prime p, with the natural action of

AGL(d, p) on the underlying vector space V . The point stabiliser G0 of the zero vector is

an irreducible subgroup of GL(d, p) which is maximal with respect to being intransitive on

the set V # of nonzero vectors. The possibilities for G0 are given by Aschbacher’s Theorem

(see Theorem 3.6.1), which classifies the irreducible subgroups of the finite classical groups.

This result organises the irreducible subgroups into eight classes Ci, 2 ≤ i ≤ 9, and our

preliminary analysis shows that we only need to consider the subgroups of ΓL(n, q) and

INTRODUCTION xiii

ΓSp(n, q), where qn = pd, which are maximal in the classes Ci for 2 ≤ i ≤ 9, i 6= 3. Most

of the groups in classes C2 to C8 can be described as stabilising a particular geometric

configuration in V ; those in the class C9 do not have a uniform geometric description, and

so we do not consider this class in detail. Results in [21, 22] on the existence of regular

orbits imply order restrictions on G0; however, these are not strong enough to rule out

examples or to resolve this case.

Our main result for the affine case is Theorem 4.

Theorem 4. Let Γ be a connected graph with a symmetric group G of automorphisms,

where G is vertex-quasiprimitive and is maximal of affine type. Then Γ is isomorphic to

a Cayley graph Cay(V, S), where V = Fdp for some prime p and S is an orbit of the point

stabiliser G0 of the zero vector, with 〈S〉 = V and S = −S. The group G0 is a subgroup

of ΓL(n, q) or ΓSp(n, q), where qn = pd. If diam(Γ) = 2 then one of the following holds:

(1) (G0, S) are as in Tables 1 and 2;

(2) G0 satisfies the conditions in Table 3; or

(3) G0 belongs to the class C9.

Furthermore, all pairs (G0, S) in Tables 1 and 2 yield G-symmetric diameter 2 graphs

Cay(V, S).

In Tables 1 and 2, the sets Xs and Ys are as in (5.3.3) and (5.4.3), respectively, c(v) is

as in (5.5.4), Wβ is as in (5.3.7), S0 is as in (3.3.1), and S#, S, and S⊠ are as in (5.2.1).

Cayley graphs are defined in Section 2.2.

G0 ∩GL(n, q) S Conditions

1 GL(m, p) ≀ Sym (t), mt = d Xs qm > 2 and s ≥ t/2

2 GL(k, q)⊗GL(m, q), km = d Ys s ≥ 12min m,n

3 GL(n, q1/r) Zq−1, r > 2 and n > 2 vG0 c(v) = r − 1 or c(v) = r

4 GL(n, q1/r) Zq−1, r = 2 or n = 2 vG0 c(v) = 1

5 (Zq−1 (Z4 Q8)).Sp(2, 2), d = 2, q odd vG0 v ∈ V #

6 GL(m, q) ≀⊗ Sym (2), m2 = d Ys s ≥ m/2

7 GU(n, q), n ≥ 2 S0, S#

8 GO(n, q), n = 3 and q = 3 S0

9 GO(n, q), qn odd, n > 3 or q > 3 S0, S, or S⊠

10 GO+(n, q), n even, q odd, n > 2 or q > 2 S0 or S#

11 GO−(n, q), n even, q odd, n > 2 S0 or S#

Table 1. Symmetric diameter 2 graphs from maximal subgroups of ΓL(n, q)

If G is maximal of diagonal type then the corresponding graphs Γ can also be con-

structed as Cayley graphs Cay(T d−1, S), where T is a nonabelian simple group, d ≥ 2,

and S is a union of conjugacy classes of T d−1. We are not able to classify all diameter

2 graphs that arise, although we do know that diameter 2 graphs exist for d = 2 due to

the progress made on Thompson’s Conjecture and other related results (see Section 6.1).

xiv INTRODUCTION

G0 ∩GL(n, q) S Conditions

1 Sp(m, q)t.[q − 1].Sym (t), mt = d Xs qm > 2 and s ≥ t/2

2 GL(m, q).[2], 2m = d Wβ β ∈ Fq

3 (Zq−1 Q8).O−(2, 2), d = 2, q odd vG0 v ∈ V #

4 GO+(n, q), n = 2 and q = 2 S0

5 GO+(n, q), qn even, n > 2 or q > 2 S0 or S#

6 GO−(n, q), qn even, n > 2 S0 or S#

Table 2. Symmetric diameter 2 graphs from maximal subgroups of ΓSp(n, q)

G0 ∩GL(n, q) Conditions Restrictions

1 GSp(k, q)⊗GOǫ(m, q), m odd, q > 3 Proposition 5.4.4

2 GL(n, q1/r) Zq−1 c(v) 6= r − 1, r Proposition 5.5.3 (2), (3), (4)

3 (Zq−1 R).Sp(2t, r), d = rt R of type 1, t ≥ 2 Proposition 5.6.1 (1)

4 (Zq−1 R).Sp(2t, 2), d = rt R of type 2, t ≥ 2 Proposition 5.6.1 (2)

5 (Zq−1 R).O−(2t, 2), d = rt R of type 4, t ≥ 2 Proposition 5.6.1 (3)

6 GL(m, q) ≀⊗ Sym (t), mt = d t ≥ 3 Proposition 5.7.3

7 GSp(m, q) ≀⊗ Sym (t), mt = d, q odd t ≥ 3 Proposition 5.7.4

Table 3. Other conditions for diameter 2 (affine case)

For d ≥ 3 we obtain necessary conditions for diameter 2. Our main result for this case is

Theorem 5.


where G is vertex-quasiprimitive and is maximal of diagonal type. Then Γ ∼= Cay(T d−1, S),

where T is a nonabelian simple group and S is an orbit of Aut (T )×Sym(d). If diam(Γ) =

2 then one of the following holds:

(1) d = 2 and S ∪ S2 = T ;

(2) d = 3 and S does not contain (t, 1T ) for any t ∈ T ; or

(3) d ≥ 4, |T | is bounded above by a function of d, and S satisfies the restrictions

described in Proposition 6.1.6.

If G is maximal of product type then G ∼= Sym(k) ≀ Sym (m), k ≥ 5 and m ≥ 2, with

the product action (1.3.1), and the resulting graph is the distance-ν graph Hν(m,k) of

the Hamming graph H(m,k) for some ν ∈ 0, . . . ,m, which is defined in Section 6.2. We

thus have Theorem 6.


such that G is vertex-quasiprimitive and is maximal of product type. Then G = Sym(k) ≀Sym (m) with k ≥ 5, and Γ is a distance-ν graph Hν(m,k) of the Hamming graph H(m,k),

for some ν ∈ 0, . . . ,m. Furthermore, diam(Γ) = 2 if and only if ν ≥ 12m.

The rest of this thesis is organised as follows.

INTRODUCTION xv

Chapters 1, 2 and 3 present the basic concepts used throughout the thesis, as well

as related results: Chapter 1 on permutation groups and group actions, Chapter 2 on

algebraic graph theory, and Chapter 3 on linear algebra. We prove Theorem 1 in Section

2.4. We also present some classification theorems that form the framework of our analysis

of various cases — namely, the O’Nan-Scott Theorem for quasiprimitive groups (Theorem

1.3.8), Hering’s Theorem (Theorem 3.5.1), and Aschbacher’s Theorem (Theorem 3.6.1).

Chapter 4 deals with quotient-complete symmetric graphs. We give examples for the

cases where the parameter k has value 1 or 2, and prove Theorem 2 for the case where

k ≥ 3. The connected graphs that arise are presented in Tables 4.1.1 and 4.1.2; those

with diameter 2 are determined and are indicated in the table by the symbol “†”. We

do not treat completely the case corresponding to transitive subgroups of one-dimensional

affine groups, but instead consider only subcases corresponding to some infinite families

of subgroups.

Chapter 5 deals with vertex-quasiprimitive symmetric graphs with an automorphism

group that is maximal of affine type, and is devoted to the proof of Theorem 4.

Finally, in Chapter 6 we look briefly at the remaining vertex-quasiprimitive cases,

which are those where the automorphism group has nonabelian socle and is maximal with

respect to being non-2-transitive. We prove Theorems 5 and 6 and pose questions for

further research.

The publications arising from this thesis are [3] and [4].

CHAPTER 1

Permutation groups

In this chapter we introduce some terms, notation, and relevant results on permutation

groups. Most of the content is standard and can be found in [17]. The more specialised

material in Sections 1.2 and 1.3 can be found in [40].

1.1. Basic concepts and notation

Throughout this section assume that Ω is a finite nonempty set.

A permutation of Ω is a bijection from Ω to itself. The set of all permutations of Ω is

a group under composition, called the symmetric group on Ω and denoted by Sym (Ω). If

Ω = 1, . . . , n, we also write Sym (Ω) as Sym (n), the symmetric group on n letters. A

permutation group on Ω is a subgroup of Sym(Ω).

An action of a group G on Ω is a map G × Ω → Ω, (g, ω) 7→ ωg, with the following

properties: (i) ω1G = ω for all ω ∈ Ω; and (ii) ωgh = (ωg)h for all ω ∈ Ω and g, h ∈ G. We

say that “G acts on Ω” if G has an action on Ω.

For the rest of the section assume that the group G acts on Ω.

Every element g of G induces a permutation of the set Ω, given by the map g : ω 7→ ωg

for all ω ∈ Ω. The map ρ : G→ Sym (Ω), where ρ(g) = g for all g ∈ G, is a homomorphism

of groups and is called a permutation representation of G. We denote the image of ρ by

GΩ. If ker ρ = 1G then G ∼= GΩ (and so G is isomorphic to a subgroup of Sym(Ω))

and in this case we say that ρ is faithful. We shall frequently use the phrase “kernel of

the action” of G to refer to the kernel of the corresponding permutation representation,

which consists of all elements of G that fix every element of Ω under the action.

An orbit of an element ω ∈ Ω under the action of G is the set ωG := ωg | g ∈ G.Clearly, the set of G-orbits in Ω form a partition of Ω. If G has exactly one orbit in Ω

then its action is said to be transitive; otherwise, it is intransitive. Equivalently, G acts

transitively on Ω (or “G is transitive on Ω”) if for any two elements α, β ∈ Ω there is a

g ∈ G such that αg = β.

The stabiliser in G of a point ω ∈ Ω is the set Gω (alternatively, StabG(ω)) of all

elements of G which fix ω under the action. That is, Gω = g ∈ G | ωg = ω. The set Gω

is a subgroup of G. The action of G is said to be semiregular if Gω is the trivial subgroup

for all ω ∈ Ω; if the action is both semiregular and transitive then it is said to be regular.

The relationship between the orbit of a point and its stabiliser is captured in the

following fundamental result.

1

2 1. PERMUTATION GROUPS

Theorem 1.1.1 (Orbit-Stabiliser Theorem). [17, Theorem 1.4A (iii)] Let Ω be a finite

nonempty set, and G a group acting on Ω. Then for any ω ∈ Ω,

∣∣ωG∣∣ = |G : Gω|

The next theorem collects some basic results on transitive groups.

Theorem 1.1.2. [17, Corollary 1.4A, Exercise 1.4.1, Theorem 1.6A (iv)] Let Ω be a

finite nonempty set. If G is a group acting transitively on Ω, then the following hold.

(1) The point stabilisers in G form a single conjugacy class of subgroups of G. In

particular, Gωg = g−1Gωg for any ω ∈ Ω and g ∈ G.

(2) |G : Gω| = |Ω| for each ω ∈ Ω.

(3) The action of G is regular if and only if |G| = |Ω|.(4) Let H ≤ G and ω ∈ Ω. Then the following are equivalent: (i) G = GωH; (ii)

G = HGω; and (iii) H is transitive. In particular, the only transitive subgroup

of G containing Gω is G itself.

(5) If H ⊳G then the number of H-orbits in Ω divides |G : H|.

In this paragraph assume that G acts transitively on Ω. A suborbit of G is an orbit

of a point stabiliser Gω for any ω ∈ Ω. If g ∈ G and α, β ∈ Ω with αg = β, then by

statement 1 of Theorem 1.1.2 we have βGωg =(αGω

)g. So g induces a bijection from the

set of Gω-orbits to the set of Gωg -orbits, and since G is transitive the number of suborbits

is independent of the choice of ω. The number of its suborbits is called the rank of G,

and the lengths of the suborbits are its subdegrees. There is a one-to-one correspondence

between the set of suborbits of G and the set of G-orbits in the Cartesian set Ω×Ω under

the action

(α, β)g := (αg, βg) ∀ α, β ∈ Ω, g ∈ G. (1.1.1)

Indeed, it is not difficult to show that for any α ∈ Ω and G-orbit ∆ in Ω × Ω, the set

∆(α) := β | (α, β) ∈ ∆ is an orbit of Gα. The orbits of G on Ω×Ω are called its orbitals

on Ω. Clearly, the set ∆0 := (ω, ω) | ω ∈ Ω is a G-orbital; this is called the trivial or

diagonal orbital. For each orbital ∆ define ∆∗ := (β, α) | (α, β) ∈ ∆; if ∆ = ∆∗ then ∆

is said to be self-paired.

We now consider some generalisations of the concept of a point stabiliser. Suppose that

∆ is a nonempty proper subset of Ω, and for any g ∈ G denote by ∆g the set δg | δ ∈ ∆.The sets

G(∆) = g ∈ G | δg = δ ∀ δ ∈ ∆

and

G∆ = g ∈ G | δg ∈ ∆ ∀ δ ∈ ∆ = g ∈ G | ∆g = ∆

1.1. BASIC CONCEPTS AND NOTATION 3

are called the pointwise stabiliser and the setwise stabiliser, respectively, of ∆. Both G(∆)

and G∆ are subgroups of G, with G(∆) EG∆. It is easy to see that

G(∆) =⋂

δ∈∆Gδ,

and that the action of G on Ω induces an action of G∆ on ∆. The set ∆ is said to be

G-invariant if G∆ = G, that is, if ∆g = ∆ for all g ∈ G. Clearly, ∆ is G-invariant if and

only if it is a union of orbits of G. In this case the action of G restricted to ∆ is an action

of G on ∆ with kernel G(∆), so that with the notation above we have G/G(∆)∼= G∆.

Example 1.1.3. Consider the action of G on itself via conjugation, that is,

ag := g−1ag ∀ a, g ∈ G.

The point stabiliser of a ∈ G is the subgroup CG(a) of all group elements that commute

with a, and is called the centraliser of a in G. If H is a subgroup of G, the pointwise

stabiliser of H is the subgroup CG(H) where

CG(H) = x ∈ G | hx = xh ∀ h ∈ H,

and is likewise called the centraliser of H in G. (If H = G the group C(G) := CG(G) is

calld the center of G.) The setwise stabiliser of H is the subgroup NG(H) given by

NG(H) = x ∈ G | x−1hx ∈ H ∀ h ∈ H.

Observe that H E NG(H); indeed, NG(H) is the largest subgroup of G that contains H

as a normal subgroup. We call NG(H) the normaliser of H in G.

Some properties of the centraliser of a transitive group are given in Theorem 1.1.4.

Theorem 1.1.4. [17, Theorem 4.2A] Let G ≤ Sym (Ω) be transitive, ω ∈ Ω, and

C := CSym(Ω)(G). Then:

(1) C is semiregular, and C ∼= NG(Gω)/Gω.

(2) C is transitive if and only if G is regular.

(3) If C is transitive, then it is conjugate to G in Sym (Ω) and hence C is regular.

(4) C = 1 if and only if NG(Gω) = Gω.

(5) If G is abelian, then C = G.

Suppose that G acts transitively on Ω. A G-invariant partition of Ω is a partition

P such that P g ∈ P for all parts P ∈ P and g ∈ G. Thus G permutes the parts

of a G-invariant partition P, and if gP denotes the permutation of P induced by the

element g ∈ G, then the map G → Sym(P), given by g 7→ gP for all g ∈ G, is a

permutation representation of G on P with image GP . Clearly, the trivial partitions

Ω and ω | ω ∈ Ω are G-invariant, with GP trivial if P = Ω, and GP ∼= G if

P = ω | ω ∈ Ω.


It is sometimes necessary to compare two different actions of a group G. The actions

of G on nonempty sets Ω and ∆ are said to be equivalent if there is a bijection λ : Ω → ∆

such that

λ(ωg) = (λ(ω))g ∀ ω ∈ Ω, g ∈ G.

In this case the actions of G “differ only in the labelling of the points of the sets involved”

[17, Section 1.6]. If both actions are transitive, the lemma below gives a necessary and

sufficient condition for determining whether or not they are equivalent.

Lemma 1.1.5. [17, Lemma 1.6B] Let G be a group acting acting transitively on the

nonempty sets Ω and ∆, and let H be a point stabiliser in GΩ. Then the actions are

equivalent if and only if H is also a point stabiliser in G∆.

Example 1.1.6. [17, p. 22] Let G be a group, and let H be a subgroup of G. Consider

the action of G on the set ΓH of right cosets of H in G, given by

(Ha)g := Hag ∀ Ha ∈ ΓH , g ∈ G.

This action is transitive, and the stabiliser of the point Ha is the subgroup a−1Ha. In

particular, the subgroup H is the point stabiliser of H ∈ ΓH . We can also define an action

of G on the set of left cosets aH of H by

(aH)g := g−1aH ∀ a, g ∈ G,

and again H is the stabiliser of the point H. It follows from Lemma 1.1.5 that the action

of G on the set of right cosets of H and the action of G on the set of left cosets of H are

equivalent.

A consequence of Lemma 1.1.5 and Example 1.1.6 is that every transitive action of G

is equivalent to GΓH for some H ≤ G. Moreover, the transitive actions of G are given up

to equivalence by the actions GΓH , as H varies over the conjugacy classes of subgroups of

G.

The notion of equivalent actions of the same group can be generalised to involve actions

of two different groups. Suppose that G and H are groups acting on nonempty sets Ω

and ∆, respectively. Then G and H are permutation isomorphic if there is a bijection

λ : Ω → ∆ and a group isomorphism φ : G→ H such that

λ(ωg) = λ(ω)φ(g) ∀ ω ∈ Ω, g ∈ G.

In other words, the actions are “the same” except for the labelling of the points of the

sets and of the elements of the groups involved. The next result gives a criterion for two

groups, acting faithfully on the same set, to be permutation isomorphic.

Lemma 1.1.7. [17, Exercise 1.6.1] Let G and H be groups acting faithfully on Ω.

Then G and H are permutation isomorphic if and only if they are conjugate in Sym(Ω).

1.2. BLOCKS, PRIMITIVITY AND QUASIPRIMITIVITY 5

If G ≤ Sym (Ω) is regular, then by Example 1.1.6 its action on Ω is equivalent to its

action on itself by right multiplication and also to its action by left multiplication by the

inverse, which are given, respectively, by

ag := ag ∀ a, g ∈ G

and

ag := g−1a ∀ a, g ∈ G.

The representations of G into Sym (G) which correspond to each of the actions above are

called the right and left regular representations of G, respectively. The following results

on regular groups will be useful later on.

Theorem 1.1.8. Let G be a regular subgroup of Sym (Ω). Then X := NSym(Ω)(G) ∼=G ⋊ Aut (G), with the natural action of Aut (G) on G. The group X acts on G with the

right multiplication action of G on itself and the natural action of Aut (G) on G, and the

point stabiliser in X of 1G is Aut (G).

The group G⋊Aut (G) in Theorem 1.1.8 is called the holomorph of G and is denoted

by Hol (G).

1.2. Blocks, primitivity and quasiprimitivity

Throughout this section suppose that the group G acts transitively on the nonempty

set Ω.

A block of imprimitivity (or simply block) for G is a nonempty subset ∆ of Ω with the

property that for each g ∈ G, either ∆g = ∆ or ∆g ∩∆ = ∅. The entire set Ω, as well as

the single-element subsets of Ω, are clearly blocks for G, and are referred to as the trivial

blocks. Any other block is nontrivial. A transitive group G is said to be primitive if the

only blocks for G are the trivial blocks; otherwise, it is said to be imprimitive.

Let ∆ be a block for G. It is easy show that the setwise stabiliser G∆ is transitive

on ∆, and that for any g ∈ G, the set ∆g is also a block for G. We call D := ∆g | g ∈ Gthe system of blocks for G containing ∆. Clearly, a system of blocks for G is a G-invariant

partition of Ω, and each part of a G-invariant partition is a block for G. It thus follows

that G is primitive on Ω if and only if the only G-invariant partitions are the trivial ones.

The following theorem describes the relation between blocks for and subgroups of G,

and provides an alternative definition of primitivity. This result can be found in [17,

Theorem 1.5A]; we state here the version presented in [45].

Theorem 1.2.1. Let G be a group acting transitively on a set Ω, and let ω ∈ Ω. Then

there is a bijection between the collection of subgroups of G containing Gω, and the set of

blocks of G containing ω, defined by H 7→ ωH for any H ≤ G with Gω ≤ H.


In particular, G is primitive if and only if Gω is a maximal subgroup of G for any

ω ∈ Ω.

The partition consisting of the orbits of a normal subgroup of G is G-invariant, and

is called a G-normal partition. These will be of interest later on (see Section 2.4). The

following theorem from [17], which we restate here with the assumption that G is finite,

gives the basic properties of G-normal partitions.

Theorem 1.2.2. [17, Theorem 1.6A] Let G be a finite group acting transitively on a

set Ω, and let N ⊳G. Then the following hold:

(1) The orbits of N form a system of blocks for G.

(2) If ∆ and Φ are two N -orbits then N∆ and NΦ are permutation isomorphic.

(3) If any point in Ω is fixed by all elements of N , then N lies in the kernel of the

action of G.

(4) The group N has at most |G : N | orbits, and the number of orbits of N divides

|G : N |.(5) If G acts primitively on Ω then either N is transitive or N lies in the kernel of

the action.

The action of G is said to be quasiprimitive if and only if the only G-normal parti-

tions are the trivial ones. Clearly, all primitive groups are quasiprimitive, but there are

quasiprimitive groups which are imprimitive. An example of such a group is given below.

Example 1.2.3. Let G be a nonabelian simple group and let H be a nontrivial and

non-maximal subgroup of G. Consider the action of G by right multiplication on the

set ΓH of right cosets of H in G. Since the only normal subgroups of G are the trivial

subgroup and G itself, all G-normal partitions are trivial. Hence GΓH is quasiprimitive.

Since H is a point stabiliser in GΓH (see Example 1.1.6) and H is not maximal in G, it

follows from Theorem 1.2.1 that GΓH is imprimitive.

1.3. The O’Nan-Scott Theorem for quasiprimitive groups

The structure of a finite quasiprimitive permutation group can be studied by con-

sidering the subgroup generated by its minimal normal subgroups. A nontrivial normal

subgroup of a group G is minimal if it does not properly contain any nontrivial normal

subgroup of G. The subgroup of G generated by all of its minimal normal subgroups is

called the socle of G and is denoted by soc(G). Theorem 1.3.1 lists some basic results on

the structure of the socle of a finite group.

Theorem 1.3.1. [17, Theorem 4.3A] Let G be a nontrivial finite group.

1.3. THE O’NAN-SCOTT THEOREM FOR QUASIPRIMITIVE GROUPS 7

(1) If N is a minimal normal subgroup of G and M is any normal subgroup of G,

then either N ≤M or 〈M,N〉 =M ×N .

(2) There exist minimal normal subgroups N1, . . . , Nk of G such that soc(G) = N1 ×. . . ×Nk.

(3) If N is a minimal normal subgroup of G then there exist simple groups T1, . . . , Tℓ

which are conjugate under G such that N = T1 × . . . × Tℓ.

(4) If the subgroups Ni in (2) are nonabelian, then these are the only minimal normal

subgroups of G. Similarly, if the subgroups Tj in (3) are nonabelian, then these

are the only minimal normal subgroups of N .

The next result is an immediate consequence of Theorem 1.3.1.

Corollary 1.3.2. [17, Corollary 4.3A] If N is a minimal normal subgroup of a finite

group, then either N is elementary abelian or Z(N) = 1.

In the case where the group G is quasiprimitive, the structure of the socle is more

constrained, as can be seen in the next two results. The first one, discovered by Burnside,

concerns 2-transitive groups and is considered to be a precursor of the O’Nan-Scott Theo-

rem. A group G ≤ Sym(Ω) is said to be 2-transitive if the G-action on the set of ordered

pairs of distinct elements of Ω, with the action defined in (1.1.1), is transitive. It is known

that all 2-transitive groups are primitive.

Theorem 1.3.3. [17, Theorem 4.1] A finite 2-transitive group has a unique minimal

normal subgroup N . Moreover, N is either a regular elementary abelian p-group for some

prime p, or a nonregular nonabelian simple group.

The next result, which can be found in [17, Theorem 4.3B], is originally stated for

primitive groups, but also applies to quasiprimitive groups.

Theorem 1.3.4. Let G be a group acting quasiprimitively on Ω and let N be a minimal

normal subgroup of G. Then exactly one of the following holds:

(1) N is regular and elementary abelian of order pd for some prime p and integer d,

and soc(G) = N = CG(N);

(2) N is regular and nonabelian, CG(N) is a minimal normal subgroup of G which

is permutation isomorphic to N , and soc(G) = N × CG(N);

(3) N is nonabelian (possibly regular or nonregular), CG(N) = 1, and soc(G) = N .

Each case in Theorem 1.3.4 determines certain possible quasiprimitive actions, which

are described in Theorem 1.3.8. Before we state this, we first describe in some detail

certain constructions which arise in some of the cases in Theorem 1.3.8.


1.3.1. Product actions. Let H andK be groups withK acting on a finite nonempty

set Θ with m ≥ 2 elements, say Θ = 1, . . . ,m. The action of K on Θ induces an action

of K on Hm, as follows : for any (h1, . . . , hm) ∈ Hm and x ∈ K,

(h1, . . . , hm)x := (h1′ , . . . , hm′) , where i′ := ix−1 ∀ i. (1.3.1)

The wreath product H ≀K (with respect to the K-action on Θ) is the semidirect product

Hm⋊K, with the K-action (1.3.1) on Hm. The group Hm is called the base group of the

wreath product, and K is called the top group.

In the wreath product H ≀K, assume that the group H acts on a finite nonempty set

Λ. Set Ω := Λm. The product action of H ≀K on Ω is defined by

(λ1, . . . , λm)(h,x) :=(λ1′

h1′ , . . . , λm′hm′

), where i′ := ix

−1 ∀ i, (1.3.2)

for all (λ1, . . . , λm) ∈ Ω, h := (h1, . . . , hm) ∈ Hm, and x ∈ K. Hence we may think of the

product action as a “composition” of two actions: the natural componentwise action of

the base group Hm on Ω, and the action of K by permutation of the components of Ω.

The product action of a wreath product is primitive under the conditions given in

Theorem 1.3.5.

Theorem 1.3.5. [17, Lemma 2.7A] Suppose that H and K are nontrivial groups

acting on the finite sets Λ and Θ, respectively, with |Θ| = m ≥ 2. Then the wreath product

H ≀ K = Hm⋊K (with respect to the K-action on Θ) is primitive on the product action on

Ω := Λm if and only if the K-action on Θ is transitive and the H-action on Λ is primitive

but not regular.

1.3.2. Diagonal type subgroups. Let T be a group in its right regular action, and

let k ≥ 2. Let D := (c, . . . , c) | c ∈ T ≤ T k and let Ω be the set of cosets of D

in T k. Then Ω can be identified with T k−1. The product action of T ≀ Sym(k) on T k,

which is imprimitive by Theorem 1.3.5, induces the following faithful action on Ω: for any

(t1, . . . , tk−1) ∈ Ω, a := (a1, . . . , ak) ∈ T k, and π ∈ Sym(k),

(t1, . . . , tk−1)a.π :=

(ak′

−1tk′−1t1′a1′ , . . . , ak′

−1tk′−1t(k−1)′a(k−1)′

)(1.3.3)

where i′ := iπ−1

for all i and tk := 1T . The group A := (τ, . . . , τ) | τ ∈ Aut (T ) ≤Aut (T )k acts on Ω by

(t1, . . . , tk−1)(τ,...,τ) := (t1

τ , . . . , tk−1τ ) (1.3.4)

for all (t1, . . . , tk−1) ∈ T k and τ ∈ Aut (T ), and this action commutes with that of Sym (k).

Note that the subgroup D induces the subgroup (τ, . . . , τ) | τ ∈ Inn(T ) ≤ A.

If T is a nonabelian simple group and H := T k, then by [17, Lemma 4.5B], the group

W := NSym(Ω)(H) can be written as

W =(a1, . . . , ak).π | π ∈ Sym(k) , ai ∈ Aut (T ) , aia

−1j ∈ Inn(T ) ∀i, j

. (1.3.5)


Hence W ∼= (T ≀ Sym(k)).Out (T ), and the stabiliser in W of ω := (1T , . . . , 1T ) ∈ Ω is

Wω = A× Sym (k) ∼= Aut (T )× Sym(k) .

A group G is said to be of diagonal type if

T k = H ≤ G ≤W = NSym(Ω)(H)

for some nonabelian simple group T . In this case Inn(T ) . Gω . Aut (T )× Sym (k), and

G acts by conjugation on the set T := T1, . . . , Tk of simple direct factors of H.

A group of diagonal type is primitive under the conditions given in Theorem 1.3.6.

Theorem 1.3.6. [17, Theorem 4.5A] Using the notation above, suppose that G ≤Sym(Ω) is of diagonal type. Then G is primitive if and only if k = 2, or k ≥ 3 and the

conjugation action of G on T is primitive. In particular, W is primitive for all k ≥ 2.

1.3.3. Twisted wreath product. Let T and P be groups, and Q ≤ P such that

there exists a homomorphism ϕ : Q → Aut (T ). Denote by Fun(P, T ) the set of all

functions f : P → T , which is a group under pointwise multiplication, and define

H :=f ∈ Fun(P, T ) f(xy) = f(x)ϕ(y) ∀ x ∈ P, y ∈ Q

.

Then H ≤ Fun(P, T ) and H ∼= T k, where k = |P : Q|. The group P acts on H by

f z(x) := f(zx) (1.3.6)

for all f ∈ H and x, z ∈ P , and it can be shown that this action preserves the group

operation on H so P . Aut (H). The twisted wreath product T twrϕ P is defined to be

the semidirect product H ⋊ P with the action in (1.3.6).

The group T twrϕ P acts on H as follows: H acts on itself by right multiplication, and

P acts on H by (1.3.6). The stabiliser in T twrϕ P of 1H (= f where f(x) = 1T for all

x ∈ P ) is precisely P .

Theorem 1.3.7 gives a special case of the twisted wreath product that is primitive.

Theorem 1.3.7. [17, Lemma 4.7A] Using the notation above, let G = T twrϕP ≤Sym(H) where T is a nonabelian simple group, P ≤ Sym(k) is primitive with point

stabiliser Q, and Imϕ ≥ Inn(T ) but Imϕ is not a homomorphic image of P . Then G is a

primitive group with regular socle H and point stabiliser isomorphic to P .

1.3.4. The O’Nan-Scott quasiprimitive types. The O’Nan-Scott Theorem for

quasiprimitive groups, which was established by C.E. Praeger in [39], identifies eight

quasiprimitive actions and asserts that these are the only possible ones. This is a gen-

eralisation of the O’Nan-Scott Theorem for primitive groups [17, Theorem 4.1A], and in

most cases the quasiprimitive types can be described in a similar way as the corresponding

primitive types.


Theorem 1.3.8 (O’Nan-Scott Theorem for quasiprimitive groups). [39, Theorem 1]

Each finite quasiprimitive group is permutation isomorphic to a group of exactly one of

the quasiprimitive types HA, HS, HC, AS, TW, SD, CD and PA.

The descriptions of the different O’Nan-Scott quasiprimitive types, which we give

below, can be found in [39, Section 2] and [40, Section 12]. We present them according

to the cases in Theorem 1.3.4 to which they correspond.

Assume throughout thatG ≤ Sym(Ω) is a quasiprimitive group with soc(G) = H ∼= T d

for some simple group T and integer d ≥ 1, and that N is a minimal normal subgroup of

G. Then N is transitive and by Theorem 1.1.2 (4) we have G = N.Gω for any ω ∈ Ω.

As it happens, all quasiprimitive groups belonging to Cases 1 and 2 (that is, of types

HA, HS, HC) are primitive.

Case 1: H = N abelian and regular. Then H is elementary abelian, say H = Zdp for

some prime p, and we can identify H with a vector space V (d, p) of dimension d over Fp.

It follows from Theorem 1.1.8 that G ≤ Hol (H) = AGL(V ) and the point stabiliser of the

zero vector is contained in GL(V ). This case corresponds to one quasiprimitive type.

HA (holomorph of an abelian group): The group soc(G) = H is regular and ele-

mentary abelian, say H = Zdp for some prime p; Ω = V (d, p); and H ≤ G ≤ AGL(V )

with G = H.G0, where H is identified with the group of translations on V and the

point stabiliser G0 of the zero vector is an irreducible subgroup of GL(V ). The group

G acts on Ω via the natural action of AGL(V ) on V .

Case 2: H = N × CG(N), N nonabelian and regular. Again by Theorem 1.1.8 we

have G ≤ Hol (N). We have two types, corresponding to each of the subcases N = T and

N = T k for some k ≥ 2.

HS (holomorph of a simple group): The group N = T is regular and nonabelian,

and soc(G) = H ∼= T 2; Ω = T ; and T 2 ≤ G ≤ T 2.Out (T ) ≤ W , where W is as in

(1.3.5) with k = 2. The group G acts on Ω with the action of W defined in (1.3.3)

and (1.3.4). Equivalently, T.Inn(T ) ≤ G ≤ Hol (T ) with the following action: for any

t ∈ Ω, a ∈ T and τ ∈ Aut (T ),

ta.τ := tτaτ

If ω = 1T ∈ Ω, then G = T.Gω where Inn(T ) ≤ Gω ≤ Aut (T ).

HC (holomorph of a compound group): The group N = T k, for some k ≥ 2, is

regular and nonabelian, and soc(G) = H ∼= T 2k; Ω = T k; and T 2k ≤ G ≤ U ≀ Sym (k),

where U := T 2.Out (T ) ≤ Sym(T ) is a quasiprimitive group of type HS. The group

G acts on Ω with the product action of U ≀ Sym(k) defined in (1.3.2). Equiva-

lently, N.Inn(N) ≤ G ≤ Hol (N) with the following action: for any (t1, . . . , tk) ∈ Ω,


a := (a1, . . . , ak) ∈ T k and σ ∈ Aut (N) = Aut (T ) ≀ Sym(k),

(t1, . . . , tk)a.σ := (t1a1, . . . , tkak)

σ .

If ω = 1N = (1T , . . . , 1T ) ∈ Ω then G = N.Gω, where Inn(N) ≤ Gω ≤ Aut (N) and

Gω acts transitively by conjugation on the simple direct factors of N .

Case 3: H = N nonabelian, CG(N) = 1. In this case H may be regular or nonregular.

Hence the map ψ : G → Aut (H), where ψ(g) : h 7→ g−1hg for any h ∈ H and g ∈ G,

is an endomorphism, and G . Aut (H) = Aut (T ) ≀ Sym(d). The case where H is simple

corresponds to one quasiprimitive type.

AS (almost simple): The group soc(G) = T is nonabelian simple, and may be

regular or nonregular; T ≤ G ≤ Aut (T ) and G = TGω, for some ω ∈ Ω.

For the remaining cases H = T d for some d ≥ 2. If H is regular then we get another

quasiprimitive type.

TW (twisted wreath): The group soc(G) = H ∼= T d is regular and nonabelian,

with d ≥ 2; Ω = H; and G = T twrϕ P , where H, P and ϕ are as defined in Section

1.3.3. Furthermore,

coreP(ϕ−1(Inn(T ))

)=⋂

x∈Pϕ−1(Inn(T ))x = 1.

The group G acts on Ω as follows: H acts on itself by right multiplication and P acts

on H by the action defined in (1.3.6). If ω := 1H , then Gω = P .

The case where H is nonregular corresponds to three quasiprimitive types.

SD (simple diagonal): The group soc(G) = H = T d is nonregular and nonabelian,

with d ≥ 2; Ω = T d−1; and T d ≤ G ≤W , where W is as defined in (1.3.5) with k = d.

The group G acts on Ω with the action of W defined in (1.3.3) and (1.3.4), with G

acting transitively (not necessarily primitively) by conjugation on the d simple direct

factors of H. If ω := 1H then Hω = I ≤ Gω ≤ A×Sym(d), where I := (t, . . . , t) | t ∈Inn(T ) and A := (a, . . . , a) | a ∈ Aut (T ).

CD (compound diagonal): The group soc(G) = H = T d is nonregular and

nonabelian, with d ≥ 4; Ω = Λm for some divisor m of d, m ≥ 2; and T d ≤G ≤ U ≀ Sym(m), where U ≤ Sym(Λ) is a quasiprimitive group of type SD with

soc(U) = T d/m. The group G acts on Ω with the product action defined in (1.3.2),

and acts transitively by conjugation on the d simple direct factors of H.


PA (product action): The group soc(G) = H = T d is nonregular and nonabelian,

with d ≥ 2; Ω = Λd; and T d ≤ G ≤ U ≀Sym (k), where U ≤ Sym (Λ) is a quasiprimitive

group of type AS with soc(U) = T and U nonregular. The group G acts on Λd with

the product action defined in (1.3.2), and acts transitively by conjugation on the d

simple direct factors of H. If ω = (λ, . . . , λ) ∈ Λd then Hω ≤ Gω ≤ Uλ ≀ Sym (k).

Also P = Λd is a fixed G-invariant partition of Ω, and for a fixed λ ∈ Λ, there is a

(possibly trivial) G-invariant patition P ′ of Ω such that Hδ = (Tλ)d for some δ ∈ P ′,

and for α ∈ δ the point stabiliser Hα is a subdirect product of (Tλ)d (i.e., Hα projects

surjectively onto each of the direct factors of (Tλ)d).

1.4. Rank 3 groups

We discuss here briefly the quasiprimitive rank 3 groups, which give rise to symmetric,

vertex-quasiprimitive graphs with diameter 2 (see Section 6.3). Recall from Section 1.1

that the rank of a permutation group G ≤ Sym (Ω) is the number of G-orbitals on Ω;

hence G has rank 3 if it has exactly two orbits on the set of ordered pairs of distinct

elements in Ω.

The following result by P. Cameron gives a relationship between the rank of a primitive

permutation group G with nonabelian socle, and the number of simple direct factors in

soc(G).

Theorem 1.4.1. [10, Proposition 5.1] Let G be a primitive group with soc(G) = T d

for some nonabelian simple group T . If G has rank r, then d ≤ r − 1.

Together with Theorem 1.3.1, this implies that if G is primitive with rank 3 then one

of the following holds:

(i) soc(G) is regular and elementary abelian (i.e., G is of type HA);

(ii) soc(G) = T where T is nonabelian simple (i.e., G is of type AS);

(iii) soc(G) = T 2 where T is nonabelian simple.

The primitive rank 3 groups satisfying (i) are classified by M. Liebeck in [34]. Those

satisfying (ii) are classified in [7] for the case where T is an alternating group, [28] for

T a classical group, and [35] for T an exceptional group of Lie type or a sporadic group.

The groups satisfying (iii) are subgroups of a wreath product U ≀ Sym(2) where U is 2-

transitive with soc(U) = T , and the classification of all 2-transitive groups (see [10]) gives

all primitive rank 3 permutation groups in this case.

The quasiprimitive rank 3 groups which are imprimitive are determined by A. Devillers,

et.al. in [15]. They show that if G is a group which is quasiprimitive and imprimitive

with rank 3, then the following conditions are satisfied.

(i) The group G has a unique system D of blocks.

(ii) The action of G on its block system is faithful (and thus GD ∼= G).

1.4. RANK 3 GROUPS 13

(iii) The group G is a subgroup of the wreath product H ≀ X, where X ∼= GD and

H = G∆∆ for some ∆ ∈ D, and both H and X are 2-transitive.

(iv) The group G is almost simple.

So the classification of imprimitive quasiprimitive rank 3 groups follows from the classifi-

cation of 2-transitive groups.

Thus we have:

Theorem 1.4.2. [15, Corollary 1.4] All quasiprimitive rank 3 permutation groups are

known. They are either primitive, or imprimitive and almost simple.

CHAPTER 2

Algebraic graph theory

In this chapter we introduce some definitions, notation and relevant results on permu-

tation groups. The material in the first section is standard, and can be found in [9]. In

the last section we discuss briefly the progress made on Thompson’s Conjecture, which

may yield examples of symmetric diameter 2 graphs for some of the vertex-quasiprimitive

cases. The content of this section can also be found in [29] (for Section 2.3), [40, Section

4] and [41, Section 7] (for Section 2.4).

2.1. Basic concepts

A graph Γ consists of a nonempty set V (Γ) of nodes or vertices, and a set E(Γ) of

edges that connect some (possibly none or all) pairs of distinct vertices. The graph Γ is

finite if V (Γ) is a finite set. Two vertices α and β which are connected by an edge are

said to be adjacent, and α is called a neighbour of β and vice versa. An edge is usually

denoted as an unordered pair of adjacent vertices, that is, the edge connecting adjacent

vertices α and β is α, β. An arc is a directed edge and is represented by an ordered pair

of adjacent vertices; thus each edge α, β determines two arcs, namely, (α, β) and (β, α).

We will sometimes write α ∼Γ β to indicate that α, β ∈ E(Γ).

A graph ∆ is said to be a subgraph of Γ if V (∆) ⊆ V (Γ) and E(∆) ⊆ E(Γ). If

S ⊆ V (Γ), the induced subgraph of Γ on S is the graph [S] with V ([S]) = S and E([S]) =

α, β ∈ E(Γ) | α, β ∈ S, i.e., the set of all edges of Γ which join vertices in S.

A path of length n in Γ is a sequence [γ0, γ1, . . . , γn] of n + 1 vertices of Γ such that

γi, γi+1 ∈ E(Γ) for all i ∈ 0, . . . , n and γi−1 6= γi+1 for all i ∈ 1, . . . , n−1. Hence anedge determines a path of length 1. A graph is connected if any two distinct vertices are

connected by a path; otherwise, it is disconnected. The distance between two vertices α

and β is the length of the shortest path that joins α and β. The diameter of a connected

graph is the smallest positive integer d such that any two vertices are connected by a

path of length at most d. In particular, a graph has diameter one if every pair of distinct

vertices are adjacent; such a graph is also called complete. A graph has diameter 2 if it is

not complete, that is, if it has at least one pair of distinct vertices which are not adjacent,

and if any two nonadjacent vertices α and β have at least one common neighbour.

An automorphism of Γ is a bijection x of the set V (Γ) which sends adjacent pairs of

vertices to adjacent pairs, and nonadjacent pairs to nonadjacent pairs, that is, αx, βx ∈E(Γ) if and only if α, β ∈ E(Γ). The automorphisms of Γ form a group, which we

denote by Aut (Γ). Any subgroup of Aut (Γ) is called an automorphism group of Γ. The

15

16 2. ALGEBRAIC GRAPH THEORY

graph Γ is vertex-transitive if for any α, β ∈ V (Γ) there is an element x ∈ Aut (Γ) with

αx = β. It is arc-transitive or symmetric if, for any pair of arcs (α, β) and (γ, δ), we can

find y ∈ Aut (Γ) with αy = γ and βy = δ. If the above statements hold when Aut (Γ) is

replaced by some subgroup G ≤ Aut (Γ), then Γ is, respectively, G-vertex-transitive and

G-arc-transitive (or G-symmetric).

It follows from the definitions above that, in a vertex-transitive graph, every vertex

has the same number of neighbours, say v. In this case we say that the graph is regular

with valency v. Also, a connected symmetric graph is necessarily vertex-transitive; the

converse, however, is generally not true.

The following theorem is a special case of [9, Theorem 17.5], and gives a necessary

and sufficient condition for G to be symmetric.

Theorem 2.1.1. [9, Theorem 17.5] A graph Γ is G-symmetric for some G ≤ Aut (Γ)

if and only if G is transitive on V (Γ) and, for any α ∈ V (Γ), Gα is transitive on the

neighbours of α.

An isomorphism of graphs Γ and ∆ is a bijection from V (Γ) onto V (∆) which maps

edges of Γ to edges of ∆, and non-edges of Γ (i.e., non-adjacent pairs of vertices) to

non-edges of ∆. If such a map exists then Γ and ∆ are isomorphic graphs, written as

Γ ∼= ∆.

A digraph, or directed graph, is a generalisation of a graph, in which two adjacent

vertices are connected by arcs instead of edges, and arcs of the form (γ, γ) (which are

called loops) are allowed. A digraph is a graph if it has no loops, and if (α, β) is an arc

whenever (β, α) is an arc.

Let G ≤ Sym(Ω). To each nontrivial G-orbital ∆ (see Section 1.1) we can associate

a digraph which has vertex set Ω and arc set ∆, and is called the orbital digraph for G

associated with ∆. We also denote this digraph by ∆. Note that ∆ is a graph if and only

if the orbital ∆ is self-paired. Clearly, ∆ admits G as a subgroup of automorphisms. The

next theorem is another characterisation of arc-transitive graphs that follows easily from

the preceding discussion.

Theorem 2.1.2. [40, Theorem 2.1] A graph Γ is G-symmetric for some G ≤ Aut (Γ)

if and only if Γ is an orbital graph for G, namely, for the nontrivial self-paired orbital

(α, β) | α, β ∈ E(Γ).

A rank 3 graph is a graph Γ with an automorphism group G that acts as a rank 3

group on its vertex set. In this case the nontrivial G-orbitals on V (Γ) are precisely the

arc set of Γ and the set of nonadjacent vertices of Γ. So Γ is an orbital graph for G

and is therefore G-symmetric. A connected rank 3 graph has diameter 2. Indeed, if α

and β are nonadjacent vertices, then distΓ(α, β) = distΓ(αg, βg) for any g ∈ G. Since a

2.2. CAYLEY GRAPHS 17

connected rank 3 graph clearly contains vertices with distance 2, it follows that all pairs

of nonadjacent vertices have distance 2, and thus diam(Γ) = 2. Recall from Section 1.4

that all quasiprimitive rank 3 groups are known; those which contain an involution give

rise to vertex-quasiprimitive symmetric diameter 2 graphs.

2.2. Cayley graphs

Notation. We denote by G# the set of all nonidentity elements of a group G.

Definition 2.2.1. Let G be a group and let S be a nonempty subset of G. The Cayley

digraph of G relative to S is the digraph with vertex set G and arc set (g, sg) | g ∈ G,

s ∈ S.

Let Γ be a Cayley digraph as defined above. It is easy to see from the definition above

that Γ contains loops if and only if S contains the identity element 1G. If s ∈ S then

(1G, s) is an arc of Γ, and (s, 1G) is also an arc if and only if it can be written as (g, s′g)

for some g ∈ G and s′ ∈ S. Hence we must have 1G = s′s, and s′ = s−1. This implies that

s−1 ∈ S whenever s ∈ S, that is, S−1 = s−1 | s ∈ S = S. Conversely, if S−1 = S then

for any g ∈ G and s ∈ S, (g, sg) and (g, s−1g) are arcs and thus so is (sg, g). This shows

that a Cayley digraph is a graph if and only if the set S consists of nonidentity elements

and S is closed under taking inverses. We have the following definition.

Definition 2.2.2. Let G be a group and let S be a nonempty subset of G# such that

S−1 = S. The Cayley graph of G relative to S, denoted by Cay(G,S), is the graph with

vertex set G and edges g, sg, for all g ∈ G and s ∈ S. Also, a subset S of G# such that

S−1 = S is called a Cayley subset of G.

All Cayley graphs are vertex-transitive graphs, as shown below. For a finite group G

and a subset S of G, define Aut (G,S) to be the subgroup of elements of Aut (G) which

fix S setwise.

Theorem 2.2.3. [9, Proposition 16.2] Let G be a group and let S be a Cayley subset

of G. Then:

(1) The graph Cay(G,S) is vertex-transitive. (In particular, G is a subgroup of

Aut (Cay(G,S)), acting regularly on itself by right multiplication.)

(2) The group Aut (G,S) is a subgroup of the stabiliser in Aut (Cay(G,S)) of the

vertex 1G.

The following theorem characterises graphs which can be constructed as Cayley graphs,

in terms of their automorphism groups.


Theorem 2.2.4. [9, Lemma 16.3] A graph has an automorphism group G acting reg-

ularly on V (Γ) if and only if Γ ∼= Cay(G,S) for some Cayley subset S of G.

The following is a direct application of the results above.

Lemma 2.2.5. Let Γ be a Cayley graph Cay(G,S) for some finite group G and Cayley

subset S of G. Then:

(1) Γ is connected if and only if S generates G. In particular, Γ has diameter 2 if

and only if S 6= G \ 1G and S2 ∪ S = G.

(2) If Aut (G,S) is transitive on S, then Γ is symmetric.

Proof. To prove (1), recall from the previous section that Γ is connected if and only if

any two distinct vertices are connected by a path; equivalently, there is a path between the

vertex 1G and any other vertex g of Γ. Such a path has form [1G, s1, s2s1, . . . , snsn−1 · · · s1],for some integer n ≥ 1 and elements s1, . . . , sn ∈ S whose product is g. So Γ is connected if

and only if each element of G is expressible as a product of elements of S, i.e., S generates

G. Clearly, the set of all neighbours of the vertex 1G is S, and diam(Γ) ≥ 2 if and only if

S ⊂ G \ 1G. Also, S2 = s1s2 | s1, s2 ∈ S contains all vertices at distance 2 from 1G,

and it follows that diam(Γ) = 2 if and only if S 6= G \ 1G and S2 ∪ S = G.

Statement (2) follows immediately from Theorems 2.1.1 and 2.2.4.

Remark. Observe that the condition S−1 = S implies that |S2| ≤ |S|2 − |S| + 1.

Hence if G = S ∪ S2 then |G| ≤ |S|2 + 1. This fact will be frequently used.

2.3. Products of graphs

In this section we describe three different ways of combining two graphs to form a

third. These will be used later to construct some infinite families of symmetric diameter

2 graphs (see Examples 2.4.1 and 2.4.2).

Definition 2.3.1. Let Γ and ∆ be graphs.

(1) The lexicographic product Γ[∆] of Γ and ∆ is the graph with vertex set V (Γ) ×V (∆), and with (γ, δ) ∼Γ[∆] (γ

′, δ′) if and only if either γ ∼Γ γ′, or γ = γ′ and

δ ∼∆ δ′.

(2) The direct product Γ×∆ of Γ and ∆ is the graph with vertex set V (Γ)× V (∆),

and with (γ, δ) ∼Γ×∆ (γ′, δ′) if and only if γ ∼Γ γ′ and δ ∼∆ δ′.

(3) The Cartesian product Γ∆ of Γ and ∆ is the graph with vertex set V (Γ)×V (∆),

and with (γ, δ) ∼Γ∆ (γ′, δ′) if and only if either γ = γ′ and δ ∼∆ δ′, or δ = δ′

and γ ∼Γ γ′.

Each of these products is illustrated in Figure 1.

2.3. PRODUCTS OF GRAPHS 19

γ

γ′

Γ

δ

δ′

δ′′

∆

(γ, δ) (γ, δ′) (γ, δ′′)

(γ′, δ) (γ′, δ′) (γ′, δ′′)

(γ, δ) (γ, δ′) (γ, δ′′)

(γ′, δ) (γ′, δ′) (γ′, δ′′)

(γ, δ) (γ, δ′) (γ, δ′′)

(γ′, δ) (γ′, δ′) (γ′, δ′′)

Γ[∆]

Γ×∆Γ∆

Figure 1. Lexicographic, direct, and Cartesian product of graphs

It is easy to see that for any g ∈ Aut (Γ) and h ∈ Aut (∆), the map

V (Γ)× V (∆) → V (Γ)× V (∆)

(γ, δ) 7→ (γ, δ)(g,h) :=(γg, δh

)(2.3.1)

is an automorphism of Γ[∆], Γ×∆, and Γ∆. Hence Aut (Γ)×Aut (∆) ≤ Aut (Σ), where

Σ is one of Γ[∆], Γ×∆, and Γ∆, acting on V (Σ) via the map (2.3.1).

Observe also that the lexicographic product Γ[∆] can be constructed by replacing

each vertex γ in Γ by a copy ∆γ of ∆, and each edge α, β of Γ by all possible edges

(α, δ), (β, δ′) where δ ∈ ∆α and δ′ ∈ ∆β. For any g ∈ Aut (Γ) and h ∈ J , where

J :=∏

α∈V (Γ)Aut (∆α), define the map

V (Γ)× V (∆) → V (Γ)× V (∆)

(γ, δ) 7→ (γ, δ)(g,h) :=(γg, δh(γ

g)), (2.3.2)

where h(γg) is the component of h that belongs to Aut (∆γg). Then (2.3.2) defines an

action of J ⋊Aut (Γ) ∼= Aut (∆) ≀Aut (Γ) on V (Γ)×V (∆) which is also an automorphism

of Γ[∆]. Hence Aut (∆) ≀ Aut (Γ) ≤ Aut (Γ[∆]).

We now proceed to determine necessary and sufficient conditions in order for each of

the graph products in Definition 2.3.1 to have diameter 2.

Lemma 2.3.2. Let Γ and ∆ be graphs, each with at least two vertices for (1) and (3)

and at least three vertices for (2). Then:

(1) diam(Γ[∆]) = 2 if and only if Γ is connected with diameter at most two and at

least one of Γ and ∆ is not a complete graph.


(2) diam(Γ×∆) = 2 if and only if Γ and ∆ are both connected with diameter at most

two, and every edge in Γ and in ∆ lies in a triangle of Γ and of ∆, respectively.

(3) diam(Γ∆) = 2 if and only if Γ and ∆ are both complete graphs.

Proof. To prove (1), suppose that diam(Γ) ≤ 2 and either Γ or ∆ is not complete.

We consider each case separately.

Case 1.1: Assume that Γ is not complete. Then diam(Γ) = 2 and there are distinct

γ, γ′ ∈ V (Γ) which are not adjacent. It follows from Definition 2.3.1 that, for any δ ∈V (∆), we have (γ, δ) ≁Γ[∆] (γ

′, δ′), and thus diam(Γ[∆]) ≥ 2. Let (α, β) and (α′, β′) be

any pair of distinct nonadjacent vertices of Γ[∆]. Then again by Definition 2.3.1 we have

α ≁Γ α′, and since Γ has diameter 2 there is an α′′ ∈ V (Γ) such that α ∼Γ α

′′ ∼Γ α′. It

follows that (α, β) ∼Γ[∆] (α′′, β) ∼Γ[∆] (α

′, β′), and so diam(Γ[∆]) = 2.

Case 1.2: Assume that Γ is complete. Then diam(∆) = 2, and thus there are distinct

and nonadjacent δ, δ′ ∈ V (∆). For any γ ∈ V (∆) the vertices (γ, δ) and (γ, δ′) are distinct

and nonadjacent in Γ[∆], so that diam(Γ[∆]) ≥ 2. Let (α, β), (α′, β′) be any pair of distinct

nonadjacent vertices of V (Γ[∆]). Since Γ is complete, either α ∼Γ α′ or α = α′; if α ∼Γ α

′

then (α, β) ∼Γ[∆] (α′, β′), a contradiction. So α = α′, which implies that β and β′ are

distinct and nonadjacent in ∆. Any α′′ ∈ V (Γ) with α′′ 6= α is adjacent to α, so that

(α, β) ∼Γ[∆] (α′′, β) ∼Γ[∆] (α, β

′). Therefore diam(Γ[∆]) = 2.

Conversely, suppose that diam(Γ[∆]) = 2. Let γ, γ′ be distinct vertices of Γ, and let

δ ∈ V (∆). Then (γ, δ) and (γ′, δ) are distinct vertices of Γ[∆], and hence they are either

adjacent or are connected by a path of length 2. If (γ, δ) ∼Γ[∆] (γ′, δ) then γ ∼Γ γ

′ by

Definition 2.3.1. If (γ, δ) ≁Γ[∆] (γ′, δ) then γ ≁Γ γ

′, and there exists (γ′′, δ′′) ∈ V (Γ[∆])

such that (γ, δ) ∼Γ[∆] (γ′′, δ′′) ∼Γ[∆] (γ

′, δ). Hence γ′′ 6= γ, γ′, and γ ∼Γ γ′′ ∼Γ γ

′. Thus

diam(Γ) ≤ 2. If Γ and ∆ are both complete then Γ[∆] is complete; since diam(Γ[∆]) = 2,

it follows that either Γ or ∆ is not complete. This ends the proof of (1).

To prove (2), suppose that Γ and ∆ have diameter at most two, and that every edge

in Γ and in ∆ lies in a triangle. Observe that Γ×∆ contains distinct nonadjacent vertices

(for instance, the vertices (γ, δ) and (γ, δ′), where δ 6= δ′), so diam(Γ×∆) ≥ 2. Let (α, β)

and (α′, β′) be distinct nonadjacent vertices of Γ×∆. It follows from Definition 2.3.1 that

one of the following holds: α = α′ and β 6= β′; β = β′ and α 6= α′; α ≁Γ α′; or β ≁∆ β′.

Since Γ and ∆ both have diameter at most two, it follows that for each of these cases there

exist α′′ ∈ V (Γ) and β′′ ∈ V (∆) such that α ∼Γ α′′ ∼Γ α′ and β ∼Γ β′′ ∼Γ β′. Hence

(α, β) ∼Γ×∆ (α′′, β′′) ∼Γ×∆ (α′, β′), and therefore diam(Γ×∆) = 2.

Now suppose that diam(Γ×∆) = 2. Clearly the complete graph has diameter less than

two and has the property that every edge lies in a triangle, so assume that either Γ or ∆ is

not complete. Without loss of generality suppose that Γ is not complete. Let γ, γ′ ∈ V (Γ)

be distinct, and let δ ∈ V (∆). Then (γ, δ), (γ′, δ) are distinct nonadjacent vertices of

Γ ×∆, and there exists (γ′′, δ′′) ∈ Γ ×∆ such that (γ, δ) ∼Γ×∆ (γ′′, δ′′) ∼Γ×∆ (γ′, δ). It


follows from Definition 2.3.1 that γ ∼Γ γ′′ ∼Γ γ

′. If γ ≁Γ γ′ then [γ, γ′′, γ′] is a path of

length 2 in Γ, and since γ, γ′ are arbitrary we conclude that diam(Γ) = 2. If γ ∼Γ γ′ then

[γ, γ′′, γ′, γ] is a triangle in Γ×∆, and thus every edge of Γ lies in a triangle. If ∆ is not

a complete graph then using similar arguments we can show that diam(∆) = 2 and every

edge of ∆ lies in a triangle. This proves (2).

Finally we prove (3). Suppose that Γ and ∆ are complete graphs. The graph Γ∆

contains distinct nonadjacent vertices (such as (γ, δ) and (γ′, δ′), where γ 6= γ′ and δ 6= δ′),

so diam(Γ∆) ≥ 2. Now, by Definition 2.3.1, two vertices (γ, δ), (γ′, δ′) of Γ∆ are

distinct and nonadjacent if and only if one of the following holds: γ ≁Γ γ′, δ ≁∆ δ′, or

γ 6= γ′ and δ 6= δ′. Since both Γ and ∆ are complete the last case must hold, and in

particular γ ∼Γ γ′ and δ ∼∆ δ′. Hence (γ, δ) ∼Γ∆ (γ, δ′) ∼Γ∆ (γ′, δ′), and therefore

diam(Γ∆) = 2.

Conversely, suppose that diam(Γ∆) = 2. Let γ, γ′ and δ, δ′ be pairs of distinct

vertices of Γ and of ∆, respectively. Then (γ, δ) and (γ′, δ′) are distinct and nonadjacent in

Γ∆, and there exists (γ′′, δ′′) ∈ V (Γ∆) such that (γ, δ) ∼Γ∆ (γ′′, δ′′) ∼Γ∆ (γ′, δ′).

Then either (γ′′, δ′′) = (γ, δ′) or (γ′′, δ′′) = (γ′, δ). Hence γ ∼Γ γ′ and δ ∼∆ δ′. Therefore

Γ and ∆ are complete graphs, which proves (3).

The next result gives sufficient (but not necessary) conditions in order for the products

in Definition 2.3.1 to be symmetric. A graph is said to be empty if it contains no edges.

Lemma 2.3.3. Let Γ and ∆ be graphs.

(1) If Γ and ∆ are both symmetric and at least one of them is an empty graph, then

the lexicographic product Γ[∆] is symmetric. In particular, Γ[∆] is G-symmetric

for the group G := Aut (∆) ≀ Aut (Γ).(2) If Γ and ∆ are both symmetric then the direct product Γ ×∆ is symmetric. In

particular, Γ×∆ is H-symmetric where H := Aut (Γ)×Aut (∆).

(3) If Γ and ∆ are both symmetric and Γ ∼= ∆, then the Cartesian product Γ∆ is

symmetric. In particular, Γ∆ is K-symmetric for the group K := (Aut (Γ)×Aut (∆))⋊ Z2.

Proof. To avoid confusion we shall denote (γ, δ) ∈ V (Γ)× V (∆) by γδ.

We first show (1). Recall from the discussion before Lemma 2.3.2 that the group

J :=∏

x∈V (Γ) Aut (∆x) is a subgroup of Aut (Γ[∆]), with the action on V (Γ[∆]) given

in (2.3.2). If both Γ and ∆ are empty graphs then Γ[∆] is empty, and is therefore G-

symmetric, so from now on assume that one of Γ and ∆ is not empty. We consider each

case separately.

Case 1.1: Suppose that Γ is not an empty graph. Then ∆ is empty. If (αβ, α′β′) and

(γδ, γ′δ′) are distinct arcs of Γ[∆], then α ∼Γ α and γ ∼Γ γ′, possibly α,α′ = γ, γ′.

(Indeed, by Definition 2.3.1 either α ∼Γ α′, or α = α′ and β ∼∆ β′; since ∆ is empty the


first case must hold, and similarly for γ and γ′.) Since Γ is symmetric there exists g ∈Aut (Γ) such that (α,α′)g = (γ, γ′). Let σ, σ′ ∈ Aut (∆) such that βσ = δ and (β′)σ

′= δ′,

and take h ∈ J such that h(αg) = σ and h((α′)g) = σ′. Then (g, h) ∈ G ≤ Aut (Γ[∆]),

with

(αβ, α′β′)(g,h) =(αgβh(α

g), α′gβ′h(α′g))= (γδ, γ′δ′).

Therefore Γ[∆] is G-symmetric.

Case 1.2: Suppose that ∆ is not empty, so that Γ is empty. Again let (αβ, α′β′) and

(γδ, γ′δ′) be distinct arcs of Γ[∆]. Then α = α′, γ = γ′, β ∼∆ β′, and δ ∼∆ δ′. Since

∆ is symmetric there exists ρ ∈ Aut (∆) such that (β, β′)ρ = (δ, δ′). Take g ∈ Aut (Γ)

such that αg = γ, and h ∈ J such that h(αg) = ρ. Then as in Case 1.1 we have

(g, h) ∈ G ≤ Aut (Γ[∆]) with

(αβ, α′β′)(g,h) =(αgβh(α

g), αgβ′h(α′g))= (γδ, γδ′),

and again Γ[∆] is G-symmetric. This completes the proof of (1).

To prove (2) let (αβ, α′β′) and (γδ, γ′δ′) be distinct arcs of Γ × ∆. Then α ∼Γ α′,

γ ∼Γ γ′, β ∼∆ β′, and δ ∼∆ δ′, and so there exist g ∈ Aut (Γ) and h ∈ Aut (∆) with

(α,α′)g = (γ, γ′) and (β, β′)h = (δ, δ′). Hence (g, h) ∈ H ≤ Aut (Γ×∆), and

(αβ, α′β′)(g,h) =(αgβh, α′gβ′h

)= (γδ, γ′δ′).

Therefore Γ×∆ is H-symmetric.

We now prove (3). Let (αβ, α′β′) and (γδ, γ′δ′) be distinct arcs of Γ∆, and let

τ : Γ → ∆ be a graph isomorphism. We have two cases.

Case 3.1: Suppose that the two arcs have no common vertex. Then one of the following

holds: (i) α = α′ and γ = γ′, or (ii) β = β′ and δ = δ′. If (i) holds then β ∼Γ β′ and

δ ∼∆ δ′, and there exist g ∈ Aut (Γ) and h ∈ Aut (∆) with αg = γ and (β, β′)h = (δ, δ′).

Then (g, h) ∈ Aut (Γ∆) and (αβ, α′β′)(g,h) = (γδ, γ′δ′). If (ii) holds then the same result

is obtained using similar arguments.

Case 3.2: Suppose that the two arcs have a common vertex. Then either: (i) α = α′

and δ = δ′, or (ii) β = β′ and γ = γ′. If (i) holds then β ∼∆ β′ and γ ∼Γ γ′. Take

g ∈ Aut (Γ) and h ∈ Aut (∆) such that αg = δτ−1

and (β, β′)h = (γ, γ′)τ . Define the map

ζ : V (Γ∆) → V (Γ∆) by (xy)ζ := yτ−1

xτ for all x ∈ V (Γ) and y ∈ V (∆). It is easy

to show that ζ ∈ Aut (Γ∆). We have

(αβ, αβ′)(g,h)ζ =(αgβh, αgβ′h

)ζ=(δτ

−1

γτ , δτ−1

γ′τ)ζ

= (γδ, γ′δ).

If (ii) holds then using similar arguments we can find g′ ∈ Aut (Γ) and h′ ∈ Aut (∆)

with (αβ, α′β)(g′,h′)ζ = (γδ, γ′δ). Together with Case 3.1 this implies that Γ∆ is K-

symmetric, which proves (3).

We denote the complete graph on n vertices by Kn, and the empty graph with n

vertices by Kn.


A graph is said to be bipartite if its vertex set can be partitioned into two parts P

and Q such that no two vertices in the same part are adjacent. It is said to be complete

bipartite if, in addition to the above, every vertex in P is adjacent to every vertex in Q,

and vice versa. We denote the complete bipartite graph by Km,n, where m = |P | andn = |Q|.

In the case where m = n, the complete bipartite graph Kn,n is isomorphic to the

lexicographic productK2

[Kn

], which is symmetric with diameter 2 by Lemmas 2.3.2 and

2.3.3. As we show in Lemma 2.3.4, a bipartite graph which is symmetric with diameter 2

is necessarily isomorphic to Kn,n for some n ≥ 2.

Lemma 2.3.4. Let Γ be a symmetric graph with diameter 2. If Γ is bipartite, then Γ

is a complete bipartite graph Kn,n for some n ≥ 2.

Proof. Let P and Q be the two elements of the bipartition, and let γ ∈ P and

γ′ ∈ Q. Then either γ and γ′ are adjacent, or they are connected by a path of length 2,

say γ, δ, γ′. The latter is impossible, since δ must belong to either P or Q, but by the

definition of a bipartite graph no two vertices in P or in Q are joined by an edge. So every

vertex in P is adjacent to every vertex in Q. Since Γ is symmetric it must be regular, and

thus |P | = |Q| = n for some n. Therefore Γ = Kn,n.

Remark. If both Γ and ∆ are nonempty graphs, it is unclear whether the lexicographic

product Γ[∆] is M -symmetric for any subgroup M of Aut (Γ[∆]). The converse, however,

is clearly not true: for instance, if Γ and ∆ are both complete graphs, then Γ[∆] is also a

complete graph and is clearly symmetric. For the Cartesian product of Γ and ∆, if Γ∆

is symmetric it is not necessary that Γ ∼= ∆, unless Γ and ∆ are both complete, as shown

in Lemma 2.3.5 below. For example, if C4 denotes the 4-cycle then the graph C4 K2 is

the cube, which is symmetric.

Lemma 2.3.5. The graph Km Kn is symmetric if and only if m = n.

Proof. If m = n then Km Kn is symmetric by Lemma 2.3.3. Suppose that Σ :=

Km Kn is symmetric. As in the proof of Lemma 2.3.3 we denote (γ, δ) ∈ V (Km)×V (Kn)

by γδ. Fix γδ ∈ V (Σ), and let Γ and ∆ be the subgraphs of Σ induced on γ′δ | γ′ ∈V (Km) and γδ′ | δ′ ∈ V (Kn), respectively. Then Γ ∼= Km and ∆ ∼= Kn, and moreover

any complete subgraph of Σ which contains γδ must be a subgraph of either Γ or ∆.

(Indeed, if Λ is a complete subgraph of Σ which contains γδ, then V (Λ) ⊆ V (Γ) ∪ V (∆).

Clearly any vertex in V (Γ) \ γδ is not adjacent to any vertex in V (∆) \ γδ, so either

V (Λ) ⊆ V (Γ) or V (Λ) ⊆ V (∆). Since Σ is symmetric, for any γ′δ ∈ V (Γ) \ γδ and

γδ′ ∈ V (∆) there exists σ ∈ Aut (Σ)γδ such that (γ′δ)σ = γδ′. Then σ(Γ) is a complete

subgraph of ∆, so n ≥ m. By a similar argument we have m ≥ n. Therefore m = n,

which completes the proof.


2.4. Normal quotients

Notation. If Γ is a graph and G is a group acting on V (Γ), we denote by GΓ the

group induced by the action of G on V (Γ).

Let Γ be a graph, G a vertex-transitive subgroup of Aut (Γ), and P a G-invariant

partition of V (Γ) (see Section 1.1). The quotient graph of Γ relative to P is the graph ΓP

whose vertices are the parts of P, and whose edges are the pairs P,P ′ such that there

is at least one edge of Γ which joins a vertex in P and a vertex in P ′. Hence the graph

ΓP can be obtained from Γ by collapsing into one vertex all the vertices of Γ belonging to

the same part P , with the corresponding effect on the edges joining vertices in different

parts (all edges between vertices in the same part are ignored; see Figure 2). The quotient

graph ΓP is nontrivial if it has more than one vertex, and is proper if |V (ΓP )| < |V (Γ)|.Thus ΓP is nontrivial and proper if and only if P is a nontrivial partition. In particular, if

G acts primitively on V (Γ), then Γ has no proper nontrivial quotient graph corresponding

to a G-invariant partition.

P1 P2

P3

V (Γ)

P1 P2

P3

V (ΓP)

Figure 2. Quotient graph of Γ with respect to the partition P

It is known that quotient graphs “inherit” some of the properties of the original

graph. For instance, ΓP is connected if the original graph Γ is connected, and in this

case diam(ΓP ) ≤ diam(Γ). We have GP ≤ Aut (ΓP), and if Γ is G-vertex-transitive or

G-symmetric, then ΓP is GP -vertex-transitive and GP -symmetric, respectively.

If Γ is connected and G-symmetric then all edges of Γ join vertices which lie in different

parts of P; that is, there is no edge joining vertices which belong in the same part. In

other words, the induced subgraph (see Section 2.1) on any P ∈ P is an empty graph.

Recall from Theorem 1.2.2 in Section 1.1 that the set of orbits of a normal subgroup

N of G forms a G-invariant partition; the quotient graph relative to such a partition is

called a G-normal quotient of Γ (or simply normal quotient, if no confusion arises), and is

denoted by ΓN . It is easy to see that ΓN is nontrivial exactly when N is intransitive on

V (Γ), and is proper if and only if N does not lie in the kernel of the action of G on V (Γ).

One property of normal quotients which is not always guaranteed for ordinary quotient

graphs is the following: If Γ is a connected G-symmetric graph and N ⊳G, then Γ is an

ℓ-multicover of ΓN for some positive integer ℓ, that is, for any two adjacent N -orbits P

and P ′, each vertex of Γ belonging in P has exactly ℓ neighbours in P ′, and vice versa.

2.4. NORMAL QUOTIENTS 25

The existence of G-normal quotients depends on the choice of the automorphism group

G, as illustrated in Examples 2.4.2 and 2.4.3 below.

Example 2.4.1. Let Σ := Γ[∆], where Γ and ∆ are nontrivial graphs, and let G :=

Aut (∆)≀Aut (Γ) = Aut (∆)|V (Γ)|⋊Aut (Γ), acting as in (2.3.2). Take N := Aut (∆)|V (Γ)|⊳

G. Then the N -orbits are the sets (γ, δ) | δ ∈ V (∆) = V (∆γ) for each γ ∈ V (Γ) (see

discussion before Lemma 2.3.2). It follows that ΣN∼= Γ. The graph Σ is an ℓ-multicover

of ΣN , where ℓ = |V (∆)|.Consider the special case where Γ = Km and ∆ = Kn with m,n ≥ 2. In this case

G = Sym(n) ≀ Sym (m) and N = Sym (n)m. By Lemma 2.3.2 the graph Σ has diameter

2, and by Lemma 2.3.3 the graph Σ is G-symmetric. Furthermore Σ is an n-multicover of

ΣN .

Example 2.4.2. Let Σ := Γ×∆, where Γ and ∆ are nontrivial graphs with at least

three vertices each, and let G := Aut (Γ)×Aut (∆). Let M := Aut (Γ) and N := Aut (∆).

Then the M -orbits are the sets (γ, δ) | γ ∈ V (Γ) for each δ ∈ V (∆), while the N -orbits

are (γ′, δ′) | δ′ ∈ V (∆) for each γ′ ∈ V (Γ). It follows that ΣM∼= ∆ and ΣN

∼= Γ. The

graph Σ is an ℓ1-multicover of ΣM where ℓ1 = val (Σ), and an ℓ2-multicover of ΣN where

ℓ2 = val (∆).

If Γ = Km and ∆ = Kn, wherem,n ≥ 3, then G = Sym(m)×Sym (n),M = Sym(m),

and N = Sym(n). Then Σ is G-symmetric with diameter 2 by Lemmas 2.3.2 and 2.3.3.

The graphs ΣM and ΣN are G-normal quotients isomorphic to Kn and Km, respectively;

moreover, Σ is an (m − 1)-multicover of ΣM and an (n − 1)-cover of ΣN . If m = n then

Aut (Σ) = Sym (n) ≀ Z2, which has rank 3 action on V (Σ). The graph Σ does not have a

proper nontrivial H-normal quotient for H = Aut (Σ).

Example 2.4.3. Let Σ := Γ∆ and G := Aut (Γ) ×Aut (∆). Taking M := Aut (Γ)

and N := Aut (∆), we again get the G-normal quotients ΣM∼= ∆ and ΣN

∼= Γ. Observe

that the induced subgraph on each M -orbit is isomorphic to Γ and the induced subgraph

on each N -orbit is isomorphic to ∆. Note that Σ is not G-symmetric for our choice of G.

If we take Γ = Kn = ∆ for some n ≥ 2, then it follows from Lemmas 2.3.2 and 2.3.3 that

Σ has diameter 2 and is H-symmetric for H = Aut (Γ) ≀ Z2, and as in Example 2.4.2, the

group H has rank 3 action on V (Σ). However, Σ has no nontrivial H-normal quotient.

Lemma 2.4.4 and Proposition 2.4.5 give some properties of G-symmetric graphs Γ with

a proper nontrivial G-normal quotient.


Lemma 2.4.4. Let Γ be a G-symmetric graph with diameter 2, and suppose that there

is an N⊳G which acts intransitively and nontrivially on V (Γ). Then Γ is an ℓ-multicover

of ΓN for some ℓ ≥ 2.

Proof. Let α ∈ V (Γ) and α′ ∈ αN \ α. Then α ≁Γ α′ by the discussion above. By

our assumption we can find a third vertex β such that α ∼Γ β ∼Γ α′. So βN 6= αN , and

β is adjacent to at least two different vertices in αN . Therefore ℓ ≥ 2.

Proposition 2.4.5. Let Γ be a G-symmetric graph with diameter 2, and suppose that

there is an N ⊳ G which acts intransitively and nontrivially on V (Γ). Then one of the

following holds:

(1) Γ ∼= ΓN

[Kn

], where diam(ΓN ) ≤ 2 and n := |γN | ≥ 2 for any γ ∈ V (Γ).

(2) N has at least three orbits in V (Γ) and every edge of ΓN lies in a triangle.

Proof. If N has exactly two orbits in V (Γ), then ΓN∼= K2 and Γ is bipartite with

the two N -orbits forming the bipartition (since the N -orbits contain no edge of Γ). It

follows from Lemma 2.3.4 that Γ ∼= Kn,n∼= K2

[Kn

]where n is the length of an N -orbit,

with n ≥ 2 since the action of N is nontrivial.

Assume from now on that N has at least three orbits, and set n := |γN | for any

γ ∈ V (Γ). Suppose first that Γ is an n-multicover of ΓN . Then α ∼Γ β if and only if

αN ∼ΓNβN . For each N -orbit A, label the elements of A by α1, α2, . . . , αn. It is easy to

check that Γ ∼= ΓN

[Kn

]via the identification

V (Γ) −→ V (ΓN

[Kn

]) = V (ΓN )× V (Kn)

αi 7−→ (A, i)

Now suppose that Γ is an ℓ-multicover of ΓN for some ℓ < n. Then for any A,B ∈E(ΓN ), there is at least one pair α, β of vertices of Γ with α ∈ A, β ∈ B, and α ≁Γ β.

Since diam(Γ) = 2, there is a third vertex γ such that α ∼Γ γ ∼Γ β. Then γ /∈ A ∪B. If

C := γN it follows that A ∼ΓNC and B ∼ΓN

C. Hence A, B and C form a triangle in

ΓN . Therefore every edge of ΓN lies in a triangle.

A special case of Proposition 2.4.5 is considered in Proposition 2.4.6.

Proposition 2.4.6. Let Γ be a G-symmetric graph with diameter 2, and suppose that

there exists an N⊳G which acts nontrivially on V (Γ), such that the following are satisfied:

(1) the group N has exactly three orbits in V (Γ); and

(2) the graph Γ is an (n − 1)-multicover of ΓN , where n :=∣∣γN

∣∣ ≥ 3, γ ∈ V (Γ).

Then Γ ∼= K3 ×Kn.

Proof. Let A, B and C be the distinct N -orbits in V (Γ). It follows from Proposition

2.4.5 (2) that ΓN∼= K3. Fix α1 ∈ A, and take β1 ∈ B and γ1 ∈ C such that β1 and γ1 are


the unique vertices in B and in C which are not adjacent to α1. We claim that β1 ≁Γ γ1.

Suppose not, and let γ′ ∈ C \ γ, so that α1 ∼Γ γ′ and β1 ∼Γ γ

′ (see Figure 3). Then

A C B

α1 γ1

γ′

β1

Figure 3. Diagram for the proof of Proposition 2.4.5

there exists g ∈ G such that (α1, γ′)g = (β1, γ1), which implies that Cg = C and g swaps A

and B setwise. Hence β1g ∈ A, and β1

g is the unique vertex in A which is not adjacent to

α1g = β. Therefore β1

g = α1. This in turn implies that α1 = β1g ∼Γ (γ′)g = γ1, contrary

to the choice of γ1. Therefore β1 and γ1 must be nonadjacent.

Set P1 := α1, β1, γ1. Choose α2 ∈ A \ α1, and β2, γ2 such that β2 and γ2 are the

unique vertices in B \ β1 and C \ γ1, respectively, which are not adjacent to α2, and

set P2 := α2, β2, γ2. (Note that β2 6= β1 and γ2 6= γ1 since α2 6= α1.) Then arguing as in

the above we again conclude that β2 and γ2 are nonadjacent. Continuing the process we

eventually obtain a partition P of V (Γ) consisting of sets Pi = αi, βi, γi, i = 1, . . . , n,

where αi ∈ A, βi ∈ B and γi ∈ C and αi, βi, γi are pairwise nonadjacent. Clearly P is

G-invariant, and ΓP is a complete graph on n vertices. It is also easy to see that the map

V (Γ) −→ V (ΓN × ΓP) = A,B,C × P

δ 7−→(δN , Pi

), δ ∈ Pi

is a bijection. If δ, δ′ ∈ V (Γ) with δ ∈ Pi and δ′ ∈ Pj , then

δ ∼Γ δ′ ⇔ δN 6= (δ′)N and Pi 6= Pj

⇔ δN ∼ΓN(δ′)N and Pi ∼ΓP

Pj

Therefore Γ ∼= ΓN × ΓP ∼= K3 ×Kn.

It is still unclear what happens in the case where N has more than three orbits in

V (Γ). Under some additional conditions we have the following.

Proposition 2.4.7. Let Γ be a G-symmetric graph with diameter 2, where G :=M×Nand M and N act intransitively and nontrivially on V (Γ). Suppose further that Γ is an

(m − 1)-multicover of ΓM , where m :=∣∣γM

∣∣, and an (n − 1)-multicover of ΓN , where

n :=∣∣γN

∣∣. Then Γ ∼= Kn ×Km.

Proof. We first show that αM ∩αN = α for any α ∈ V (Γ). Suppose that αM ⊆ αN

for some α ∈ V (Γ). Then for any g ∈ M there exists h ∈ N such that αg = αh, and

equivalently g ∈ Gαh. So M ≤ GαN , and hence MN ≤ GαN . Since MN = G then


GαN =MN = G and thus N is transitive, a contradiction. Therefore αM * αN , and by

a similar argument αN * αM . Take β ∈ αM\αN and suppose that there is a γ ∈ αM ∩αN

with γ 6= α. Since Γ is an (n− 1)-cover of ΓN , α is the unique element of αN which is not

adjacent to β. So β must be adjacent to γ, which contradicts the fact that αM contains

no edges of Γ. Therefore αM ∩ αN = α.It follows from the preceding paragraph that the map

V (Γ) −→ V (ΓM × ΓN ) = V (ΓM )× V (ΓN )

α 7−→(αM , αN

)

is a bijection, so that |V (ΓM )| = n and val (ΓM ) = n − 1, and thus ΓM∼= Kn. Likewise

ΓN∼= Km. For any α, β ∈ V (Γ), we have

α ∼Γ β ⇔ αM 6= βM and αN 6= βN

⇔ αM ∼ΓMβM and αN ∼ΓN

βN

⇔(αM , βM

)∼ΓM×ΓN

(αN , βN

)

Therefore Γ ∼= ΓM × ΓN = Kn ×Km.

2.4.1. Normal quotient reduction. The family of symmetric diameter 2 graphs

being rather large, it is convenient for us to focus our study on those graphs which are, in

a sense, the building blocks of all the graphs in the family. We identify these building blocks

through normal quotient reduction, which has become an important tool in analysing the

structure of certain families of graphs (see, for instance, [19, 36, 37, 39]). The general

idea behind this process is as follows. We begin with a family F of finite graphs which is

closed under taking normal quotients - that is, for any graph Γ ∈ F , the normal quotients

of ΓN are also members of F , as are the normal quotients of these normal quotients, and

so on. Now, the class F contains graphs which are considered to be “degenerate” — in

many cases the trivial graph K1 is degenerate, as is each empty graph, and depending on

the context there may be other graphs as well. If Γ is nondegenerate then, since Γ is finite,

the process of taking normal quotients repeatedly will eventually yield a graph which is

nondegenerate but all of whose proper normal quotients are degenerate. These graphs,

which we call basic, are the building blocks in which we are interested.

Let F denote the family of symmetric diameter 2 graphs. Recall that it is possible

for the diameter of a normal quotient graph to be less than the diameter of the original

graph (see the beginning of this section). In particular, it is possible for a diameter 2

graph to have a complete graph as a normal quotient. Hence, to ensure that our family is

closed under normal quotient reduction, we include the complete graphs in F as degenerate

graphs.

The basic graphs in F are those nondegenerate graphs Γ, which admit an arc-transitive

G ≤ Aut (Γ), which satisfy either of the following:


(1) all G-normal quotients of Γ are complete graphs; or

(2) all nontrivial G-normal quotients of Γ are trivial graphs.

In the first case the graph is said to be G-quotient-complete; in the second it is G-

vertex-quasiprimitive (from the fact that this case occurs if and only if all nontrivial

normal subgroups of G are vertex-transitive — in other words, the action of G on V (Γ) is

quasiprimitive). It turns out that these two cases also arise as basic graphs in the family

of vertex-transitive graphs, of which F is a subfamily, and this fact is stated in Theorem

1. Hence we define quotient-complete graphs formally (accommodating the possibility of

disconnected graphs), as follows:

Definition 2.4.8. Let Γ be a graph and let G ≤ Aut (Γ). We say that Γ is G-quotient-

complete if it has at least one nontrivial complete G-normal quotient, and if each of its

other proper G-normal quotients is either a complete graph or an empty graph.

Clearly, the complete graphKn on n vertices isG-quotient-complete withG = Sym (n);

on the other hand its complement, the empty graph Kn on n vertices, is not quotient-

complete for any n. Note also that there is no restriction on the orders of complete graphs

that occur as normal quotient graphs, and there is no upper bound on the number of

complete normal quotients of a graph. For instance, for arbitrary positive integers m and

n, Example 2.4.2 gives a connected graph Γ with complete normal quotients of orders m

and n. Also, for a prime power c, Example 4.2.1 (1) gives a subgroup of Aut (c.Kc) =

Sym(c) ≀Sym (c) that admits c+1 normal quotients, with c of them isomorphic to Kc and

the remaining one isomorphic to Kc.

The proof of Theorem 1 makes use of the following observation. Suppose that N ≤ G

is intransitive on V (Γ), and let M E G with M ≥ N . Then any M -orbit in V (Γ) is

a union of N -orbits, so ΓM is itself a GΓN -normal quotient of ΓN . Thus the minimal

normal quotients with at least two vertices, which are obtained after a sequence of these

operations, can also be obtained by choosing the normal subgroup N to be maximal in G

such that N is intransitive on V (Γ) (possibly N = 1). For such an N , the quotient ΓN is

called a minimal G-normal quotient.

Note that in the above, it is possible to have ΓM = ΓN with M 6= N . For example,

if Γ is the cycle C4 and G = D8, then for N = Z(G) ∼= Z2, we have ΓN = C2. The

kernel of the action of G on V (ΓN ) is M ∼= Z2 × Z2, giving ΓM = ΓN . In this case ΓN

is a minimal G-normal quotient with N properly contained in the maximal intransitive

normal subgroup M .

If G acts quasiprimitively on V (Γ), then all nontrivial normal subgroups of G are

transitive and Γ is its own unique minimal G-normal quotient.


Lemma 2.4.9. Let Γ be a graph, G ≤ Aut (Γ), and N EG such that N is intransitive

on V (Γ). If N is maximal subject to being intransitive on V (Γ), then GΓN ∼= G/N and

acts quasiprimitively on V (Γ).

Proof. Suppose that ΓN is a minimal G-normal quotient of Γ, with N is maximal in

G with respect to being vertex-intransitive on Γ. Then N is the kernel of the action of G

on V (ΓN ), and hence GΓN ∼= G/N . Also, by the discussion above, ΓN has no nontrivial

GΓN -normal quotients, and so is GΓN -vertex-quasiprimitive.

The main result in this section is Theorem 1, which we now prove.

Proof of Theorem 1. If Γ is G-vertex-quasiprimitive then we have case (2) if Γ is

not complete and case (1) if it is, with N = 1 in each case. If Γ is G-quotient-complete then

we have case (1), again with N = 1. So assume that Γ is neither G-vertex-quasiprimitive

nor G-quotient complete. Let N be a maximal vertex-intransitive normal subgroup of G.

Then by Lemma 2.4.9, G/N ≤ Aut (ΓN ) and is quasiprimitive on V (ΓN ). If ΓN is not

complete then we have case (2), so suppose that ΓN is complete for all maximal vertex-

intransitive N E G. Since Γ is not G-quotient complete, it follows from Definition 2.4.8

that there exists a nontrivial normal subgroup M of G such that M is vertex-intransitive

on Γ and ΓM is not complete; we choose M such that it is maximal in G having these

properties. Since Γ is connected so is ΓM , and it follows from the maximality of M

that G/M ≤ Aut (ΓM ) and ΓM is not G/M -vertex-quasiprimitive. We claim that ΓM is

G/M -quotient-complete.

Let L′ be a nontrivial normal subgroup of G/M such that L′ is intransitive on V (ΓM ).

Then (ΓM )L′ is a proper nontrivial G/M -normal quotient of ΓM . Now L′ = L/M for

some L such that M < LEG and L is intransitive on V (Γ), and (ΓM )L′ = ΓL. Since M

is maximal in G such that ΓM is not complete and nontrivial, ΓL must be complete. It

follows that all proper (G/M)-normal quotients of ΓM are complete graphs, and thus ΓM

is G/M -quotient-complete as claimed. Therefore case (1) holds, as required.

CHAPTER 3

Linear algebra and geometries

It is shown in Chapter 4 and 5 that some families of basic symmetric diameter 2

graphs (in particular, the quotient-complete graphs and the graphs with affine vertex-

quasiprimtive automorphism group; see Theorem 1) can be organised according to the

classification of finite transitive linear groups and of subgroups of finite classical groups,

which are given, respectively, by Hering’s Theorem and Aschbacher’s Theorem. This

chapter is devoted to the exposition of these two theorems, together with some related

technical results. Section 3.1 discusses tensor product spaces, which are structures pre-

served by some of the subgroups in Aschbacher’s classification. Sections 3.2 to 3.4 give

background on linear, semilinear, affine, and classical groups, as well as the exceptional

group G2(q), which are among the types of subgroups described in both theorems. Finally,

we present Hering’s Theorem (Theorem 3.5.1) in Section 3.5 and Aschbacher’s Theorem

(Theorem 3.6.1) in Section 3.6.

Notation. Throughout this chapter F denotes a finite field; if we need to be specific

about the order then we write Fq for a field of order q. If V is a vector space over F

and S ⊆ V , then 〈S〉F is the F-span of S. The dimension of V over F is written dim (V )

or dimF(V ), as necessary. The zero vector is denoted by 0V , or simply 0 if there is no

confusion. The set of nonzero vectors is denoted by V #, and similarly the set of nonzero

field elements is written as F# or F#q , as appropriate.

If a1, . . . , an ∈ F then diag(a1, . . . , an) is the n×n matrix [gij ]n×n with diagonal entries

gii = ai and 0 everywhere else. The n × n identity matrix is denoted by In. The general

linear group of V is denoted by GL(V ).

3.1. Tensor product spaces

Let U and W be vector spaces over F, with bases u1, . . . , um and w1, . . . , wn,respectively. The tensor product of U and W is the vector space U ⊗W of dimension mn

over F, with basis B := ui ⊗ wj | 1 ≤ i ≤ m, 1 ≤ j ≤ n. For u ∈ U and w ∈ W , with

u =∑m

i=1 αiui and w =∑n

j=1 βjwj, we write

u⊗ w =m∑

i=1

n∑

j=1

αiβj(ui ⊗ wj), (3.1.1)

and such elements of U ⊗W are called simple. By [27, Corollary IV.5.3], the definition of

U ⊗W is independent of the choice of bases for U and W , and the map U ×W → U ⊗W ,

31

32 3. LINEAR ALGEBRA AND GEOMETRIES

(u,w) 7→ u⊗ w, is bilinear. That is, for any u, u′ ∈ U , w,w′ ∈W and α ∈ F we have

(u+ u′)⊗ w = u⊗ w + u′ ⊗ w ;

u⊗ (w + w′) = u⊗ w + u⊗ w′ ; and

αu⊗ w = u⊗ αw = α(u⊗ w).

We shall refer to the basis B as a tensor product basis of U ⊗W .

It follows from (3.1.1) that the group GL(U)×GL(W ) acts on U ⊗W , where, for any

g ∈ GL(U), h ∈ GL(W ), and v =∑

i,j αi,jui ⊗ wj ,

v(g,h) =∑

i,j

αi,jugi ⊗ wh

j .

In particular, for u ∈ U and w ∈W ,

(u⊗ w)(g,h) := ug ⊗wh.

This action is not faithful, and its kernel is the subspace K := (λIm, λ−1In) | λ ∈ F#.If g = [gij ]m×m and h = [hij ]n×n are the matrices of g and h with respect to the ordered

bases (u1, . . . , um) and (w1, . . . , wn), respectively, then with respect to the ordered basis

(u1 ⊗ w1, . . . , u1 ⊗ wn, u2 ⊗ w1, . . . , um ⊗ wn) of U ⊗W , (g, h) in its action on U ⊗W

corresponds to the matrix in GL(U ⊗W ) = GL(mn,F) with block structure

g11h . . . g1mh...

...

gm1h . . . gmmh

(3.1.2)

The matrix in (3.1.2) is usually denoted by g ⊗ h; extending this notation, we denote

the image of GL(U) × GL(W ) in its action on U ⊗W by GL(U) ⊗ GL(W ). Note that

GL(U)⊗GL(W ) ∼= (GL(U)×GL(W ))/K, and thatK is isomorphic to a diagonal subgroup

of Z(GL(U)) × Z(GL(W )). Note that K is contained in the centre of GL(U) × GL(W )

and that (λIm, In)K = (Im, λIn)K for all λ ∈ F#. The group GL(U) ⊗ GL(W ) is called

a central product of GL(U) and GL(W ), and is written as GL(U) GL(W ).

Recall from above that a simple vector in U ⊗W is a nonzero vector which can be

written as u ⊗ w for some u ∈ U and w ∈ W . The weight wt (v) of a nonzero vector

v ∈ U ⊗W is the least number k such that v is the sum of k simple vectors.

Lemma 3.1.1. [20, Lemma 4.3] Let V = U ⊗W , where U and W are vector spaces

over a finite field F with (possibly not equal) finite dimensions, and let v ∈ V # with v :=∑k

i=1 (ui ⊗ wi). Then wt (v) = k if and only if u1, . . . , uk and w1, . . . , wk are linearly

independent over F. Furthermore, if v has another representation v =∑k

i=1 (u′i ⊗ w′

i),

then 〈u1, . . . , uk〉F = 〈u′1, . . . , u′k〉F and 〈w1, . . . , wk〉F = 〈w′1, . . . , w

′k〉F .

3.1. TENSOR PRODUCT SPACES 33

The operation of taking tensor products is associative [27, Theorem IV.5.8], that is,

for vector spaces U , V and W over F,

(U ⊗ V )⊗W ∼= U ⊗ (V ⊗W ).

Hence we can extend the definition of a tensor product of two F-spaces to that involving

t spaces for arbitrary t ≥ 2, as follows. For each i ∈ 1, . . . , t, let Ui be a vector space

over F with dimension mi, and let Bi := bi,1, . . . , bi,mi be a basis for Ui. The tensor

product of U1, . . . , Ut is the vector space ⊗ti=1Ui over F with dimension

∏ti=1mi and basis

B := ⊗ti=1bi,ji | 1 ≤ ji ≤ mi. In particular, for ui =

∑miji=1 αi,jibi,ji ∈ Ui we write

⊗ti=1ui =

m∑

j1=1

· · ·m∑

jt=1

α1,j1 · · ·αt,jt

(⊗t

i=1bi,ji).

It follows that the map U1 × · · · × Ut → ⊗ti=1Ui, (u1, . . . , ut) 7→ ⊗t

i=1ui, is t-linear. That

is, for any i ∈ 1, . . . , t, ui, u′i ∈ Ui, and α ∈ F,

x⊗ (ui + u′i)⊗ y = x⊗ ui ⊗ y + x⊗ u′i ⊗ y and

x⊗ αui ⊗ y = α(u1 ⊗ · · · ⊗ ut),

where x := ⊗i−1j=1uj and y := ⊗t

j=i+1uj . The simple vectors in ⊗ti=1Ui are those nonzero

vectors which can be written as ⊗ti=1ui for some ui ∈ Ui, and the (tensor) weight wt (v)

of a nonzero vector v is the smallest number k such that v is the sum of k simple vectors.

The basis B is again called a tensor product basis.

The group GL(U1)× · · · ×GL(Ut) acts on ⊗ti=1Ui with

(⊗t

i=1ui)(g1,...,gt) := ⊗t

i=1uigi

for any ui ∈ Ui and gi ∈ GL(Ui), extended linearly to the whole space. We denote

the image of (g1, . . . , gt) under the corresponding representation by ⊗ti=1gi, and the im-

age of GL(U1) × · · · × GL(Ut) by ⊗ti=1GL(Ui). The kernel of this representation is

K =(λ1Im1

, . . . , λtImt) λ1 · · ·λt = 1, λi ∈ F#, which is a subgroup of the centre, so

⊗ti=1GL(Ui) is again the central product GL(U1) · · · GL(Ut).

The next result is a basic property of ⊗ti=1Ui, and is proved in [45].

Lemma 3.1.2. [45, Lemma 3.1.4] Let t ≥ 2 and V = ⊗ti=1Ui, where Ui is a finite-

dimensional vector space over F for all i. Let u,w ∈ V # with u := ⊗ti=1ui and w :=

⊗ti=1wi. Then u + w is simple if and only if ui is a scalar multiple of wi for all but at

most one i.

The following are easy observations about the relationship between the tensor weights

of the same vector in two different tensor product decompositions of V . To distinguish

between these weights, we use the notation wtU⊗W (v) to denote the weight of v in U⊗W .


Lemma 3.1.3. Let t ≥ 3 and V = ⊗ti=1Ui, where Ui is a finite-dimensional vector

space over F for all i.

(1) If X := ⊗si=1Ui and Y := ⊗t

i=s+1Ui for some s ∈ 1, . . . , t−1, then wt⊗ti=1Ui

(v) ≥wtX⊗Y (v) for any v ∈ V #.

(2) If v =∑k

i=1 (vi,1 ⊗ · · · ⊗ vi,t−1 ⊗ w), then wt⊗tj=1Uj

(v) ≤ wt⊗t−1j=1

Uj(u), where u :=

∑ki=1 (vi,1 ⊗ · · · ⊗ vi,t−1).

Proof. Statement (1) follows immediately from the fact that a simple vector in

⊗ti=1Ui is also simple in X ⊗ Y .

To prove statement (2), let X := ⊗t−1j=1Uj. Then in X ⊗ Ut, v can be written as

v = u ⊗ w, and u in turn can be written as a sum of ℓ simple vectors in ⊗t−1j=1Uj, where

ℓ := wt⊗t−1j=1Uj

(u). Since the tensor product of a simple vector in X and a vector in Ut is

a simple vector in ⊗tj=1Uj , it follows that wt⊗t

j=1Uj(v) ≤ ℓ.

For the rest of the section, assume that mi = m for all i. In this case the symmetric

group Sym (t) acts on ⊗ti=1Ui by permuting the tensor factors. Thus we get an action

of GL(Ui) ≀ Sym(t) on ⊗ti=1Ui that is similar to the product action defined in (1.3.2); we

denote the image of this wreath product in this action by GL(Ui) ≀⊗ Sym(t).

Lemma 3.1.4. Let t ≥ 2 and V = ⊗ti=1Ui, where Ui = Fm for all i. Then for any

v ∈ V #, g ∈ ⊗ti=1GL(Ui) and π ∈ Sym(t), we have wt (vg.π) = wt (v).

Proof. Let u := ⊗ti=1ui be a simple vector in V . Then for any g := ⊗t

i=1gi ∈⊗t

i=1GL(Ui) and π ∈ Sym(t), ug.π = ⊗ti=1ui′

gi′ where i′ := iπ−1

for each i. So ug.π is also

simple. It follows that wt (vg.π) ≤ wt (v) for any v ∈ V . Similarly, since v, g and π are

arbitrary, wt (v) = wt((vg.π)(g.π)

−1)≤ wt (vg.π). Therefore wt (vg.π) = wt (v).

Lemma 3.1.5. Let t ≥ 2 and V = ⊗ti=1Ui, where Ui = Fm for all i, and let v =

∑ki=1

(⊗t

j=1vi,j

)∈ V #. If wt (v) = k, then the set

vi,1 ⊗ · · · ⊗ vi,ℓ−1 ⊗ vi,ℓ+1 ⊗ · · · ⊗ vi,t | 1 ≤ i ≤ k

is an F-linearly independent subset of ⊗j 6=ℓUj for each ℓ ∈ 1, . . . , t.

Proof. If t = 2 then this is true by Lemmas 3.1.1 and 3.1.2, so assume from now on

that t ≥ 3. It follows from Lemma 3.1.4 that we only need to prove the result for ℓ = 1. Let

A := ⊗tj=2vi,j | 1 ≤ i ≤ k. Suppose that A is F-dependent. Then a := dim (〈A〉F) < k.

Reorder the elements of A so that ⊗tj=2vi,j | 1 ≤ i ≤ a is a basis for 〈A〉F , and for each

i ∈ 1, . . . , k write

⊗tj=2vi,j =

a∑

r=1

αi,r

(⊗t

j=2ur,j),


where αi,r ∈ F for all i and r. Applying Lemma 3.1.2 we get

v =k∑

i=1

(vi,1 ⊗

a∑

r=1

αi,r

(⊗t

j=2vr,j))

=a∑

r=1

((k∑

i=1

αi,rvi,1

)⊗ vr,2 ⊗ · · · ⊗ vr,t

),

so that wt (v) = a < k, a contradiction. Hence A is F-independent, and it follows from

Lemma 3.1.4 that so is ⊗j 6=ℓvi,j | 1 ≤ i ≤ k for all ℓ ∈ 2, . . . , t.

If t = 2 then by Lemma 3.1.1 the converse of Lemma 3.1.5 holds, but it does not hold

in general when t ≥ 3. A special case for which the converse is true is the following.

Lemma 3.1.6. Let t ≥ 2 and V = ⊗ti=1Ui, where Ui = Fm for all i. Let v :=

∑ki=1 (⊗t

j=1vi,j), where k ≤ m and vi,j | 1 ≤ i ≤ k is linearly independent in Uj for all

j. Then wt (v) = k.

Proof. Clearly wt (v) ≤ k. Consider the decomposition V = U1 ⊗W , where W :=

⊗tj=2Uj , and for each i ∈ 1, . . . , k set wi := ⊗t

j=2vi,j. Now the set wi | 1 ≤ i ≤ k is

linearly independent in W , so by Lemma 3.1.1 applied to U1⊗W , the vector v has weight

k in U1 ⊗W . Therefore wt (v) ≥ k by Lemma 3.1.3 (1), and thus wt (v) = k.

Lemma 3.1.5 implies that wt (v) ≤ mt−1 for all v ∈ V #, and by Lemma 3.1.1, this

bound is achieved if t = 2. It is not achieved when t ≥ 3, as implied by the next result,

which gives an improvement on this bound when t ≥ 3.

Lemma 3.1.7. Let t ≥ 3 and V = ⊗ti=1Ui, where Ui = Fm and m ≥ 2. Then for any

v ∈ V #,

wt (v) ≤ mt−3(m2 −

⌊m2

⌋).

Proof. Assume first that t = 3. For each i ∈ 1, . . . , t let

Bi := bi,ji | 1 ≤ ji ≤ m (3.1.3)

be a basis for Ui. Then b2,j2 ⊗ b3,j3 | 1 ≤ ji ≤ m for i = 2, 3 is a basis for U2 ⊗ U3, and

applying Lemma 3.1.2 we can write

v =m∑

j3=1

m∑

j2=1

(uj2,j3 ⊗ b2,j2 ⊗ b3,j3)

,

with uj2,j3 ∈ U1 for all j2 and j3. For each j3 set wj3 :=∑m

j2=1 (uj2,j3 ⊗ b2,j2 ⊗ b3,j3) and

Aj3 := uj2,j3 | 1 ≤ j2 ≤ m (that is, Aj3 is the set of U1-projections of the terms in wj3).

Then wt (v) ≤ ∑mj3=1 wt (wj3). Claim: For all j3 ∈ 1, . . . ,m, wt (wj3) ≤ dim (〈Aj3〉F).

Indeed, if d(j3) := dim (〈Aj3〉F), then by Lemma 3.1.3 (2),

wt (wj3) ≤ wtU1⊗U2(uj2,j3 ⊗ b2,j2) ≤ min d(j3),m = d(j3).


In particular, wt (wj3) ≤ m if Aj3 is linearly independent, and wt (wj3) ≤ m− 1 if Aj3 is

linearly dependent.

In finding an upper bound for wt (v) we consider two cases, according to the number

ℓ of sets Aj3 , where 1 ≤ j3 ≤ m, such that Aj3 is linearly independent.

Case 1: Suppose that ℓ = 0 or ℓ = 1. It follows from the claim that

wt (v) ≤ ℓm+ (m− ℓ)(m− 1)

= m(m− 1) + ℓ

≤ m2 − (m− 1)

≤ m2 − ⌊m/2⌋

Case 2: Suppose that ℓ ≥ 2. Without loss of generality suppose that Aj3 is linearly

independent for 1 ≤ j3 ≤ ℓ, and linearly dependent otherwise. Form the pairs A1, A2,A3, A4, . . . up to Aℓ−1, Aℓ if ℓ is even, or Aℓ−2, Aℓ if ℓ is odd. For any pair Ar, Astake a ∈ U1 such that a /∈ 〈Ar \ u1,r〉F ∪ 〈As \ u1,s〉F . (Such an a exists because two

proper subspaces do not cover U1.) Then a, u2,r, . . . , um,r and a, u2,s, . . . , um,s are

bases for U1. Writing u1,r = α1a+∑m

j2=2 αj2uj2,r and u1,s = β1a+∑m

j2=2 βj2uj2,s, where

αj2 , βj2 ∈ F for all j2, we get

wr + ws =

m∑

j2=1

(uj2,r ⊗ b2,j2 ⊗ b3,r) +

m∑

j2=1

(uj2,s ⊗ b2,j2 ⊗ b3,s)

=

α1a+

m∑

j2=2

αj2uj2,r

⊗ b2,1 ⊗ b3,r +

m∑

j2=2

(uj2,r ⊗ b2,j2 ⊗ b3,r)

+

β1a+

m∑

j2=2

βj2uj2,s

⊗ b2,1 ⊗ b3,s +

m∑

j2=2

(uj2,s ⊗ b2,j2 ⊗ b3,s)

= a⊗ b2,1 ⊗ (α1b3,r + β1b3,s) +

m∑

j2=2

(uj2,r ⊗ (αj2b2,1 + b2,j2)⊗ b3,r)

+m∑

j2=2

(uj2,s ⊗ (βj2b2,1 + b2,j2)⊗ b3,s) .

So wt (wr + ws) ≤ 2m− 1. This together with the claim gives us

wt (v) ≤ ⌊ℓ/2⌋(2m − 1) + (ℓ− 2⌊ℓ/2⌋)m + (m− ℓ)(m− 1)

= m2 −m+ ℓ− ⌊ℓ/2⌋

≤ m2 − ⌊ℓ/2⌋.

This completes the proof for the case where t = 3.

Now suppose that t > 3. Let Bi be as in (3.1.3) for i ∈ 1, . . . , t. Then B :=

⊗ti=4bi,ji | 1 ≤ ji ≤ m is a basis for ⊗t

i=4Ui, and b2,j2 ⊗ b3,j3 ⊗ b | b ∈ B; 1 ≤ j2, j3 ≤ m


is a basis for ⊗ti=2Ui. Hence we can write

v =∑

b∈B

∑

1≤j2,j3≤m

(uj2,j3,b ⊗ b2,j2 ⊗ b3,j3 ⊗ b

),

for some uj2,j3,b ∈ U1. For each b set wb :=∑

1≤j2,j3≤m (uj2,j3,b ⊗ b2,j2 ⊗ b3,j3 ⊗ b), and

consider the decomposition V = U1 ⊗ U2 ⊗ U3 ⊗ Y , where Y := ⊗ti=4Ui. If u :=

∑1≤j2,j3≤m (uj2,j3,b ⊗ b2,j2 ⊗ b3,j3), then wt

(wb

)≤ wt⊗3

i=1Ui(u) by Lemma 3.1.3 (2). It

was shown above that the weight of any vector in ⊗3i=1Ui is at most m2 − ⌊m/2⌋. Hence

wt (v) ≤∑

b∈B

wt(wb

)≤ mt−3

(m2 − ⌊m/2⌋

).

The next result shows that the upper bound in Lemma 3.1.7 is met when (m, t) = (2, 3).

Moreover, computer calculations using Magma [1] have shown that it is also met when

(m, t) = (2, 4) and F = F2.

Lemma 3.1.8. Let V = ⊗3i=1Ui, where Ui = F2 for all i. For each i let bi,1, bi,2 be

a basis of Ui, and define

v := b1,1 ⊗ b2,1 ⊗ b3,1 + b1,2 ⊗ b2,1 ⊗ b3,2 + b1,1 ⊗ b2,2 ⊗ b3,2.

Then wt (v) = 3.

Proof. Let W := U2 ⊗ U3. Then V = U1 ⊗W , and in this decomposition we can

write

v = b1,1 ⊗ (b2,1 ⊗ b3,1 + b2,2 ⊗ b3,2) + b1,2 ⊗ (b2,1 ⊗ b3,2).

Both b1,1, b1,2 and b2,1 ⊗ b3,1 + b2,2 ⊗ b3,2, b2,1 ⊗ b3,2 are linearly independent by the

choice of the vectors bi,j, so wtU1⊗W (v) = 2 by Lemma 3.1.1. Hence wt⊗3j=1Uj

(v) ≥ 2 by

Lemma 3.1.3 (1).

Suppose that wt⊗3j=1Uj

(v) = 2. Then v =∑2

i=1 (⊗3j=1vi,j) for some simple vectors

⊗3j=1v1,j ,⊗3

j=1v2,j ∈ V . By Lemma 3.1.1 and the above, 〈v1,1, v1,2〉F = 〈b1,1, b1,2〉F and

〈v1,2⊗v1,3, v2,2⊗v2,3〉F = 〈b2,1⊗b3,1+b2,2⊗b3,2, b2,1⊗b3,2〉F . Then for some α, β, γ, δ ∈ F

we have v1,1 = αb1,1 + βb1,2 and

v1,2 ⊗ v1,3 = γ (b2,1 ⊗ b3,1 + b2,2 ⊗ b3,2) + δ (b2,1 ⊗ b3,2)

= b2,1 ⊗ (γb3,1 + δb3,2) + b2,2 ⊗ γb3,2.

By Lemma 3.1.2 the set γb3,1 + δb3,2, γb3,2 is linearly dependent, so γ = 0 and δ 6= 0.

Let u := ⊗3j=1v2,j = v −⊗3

j=1v1,j . Then

u = v − δ (αb1,1 + βb1,2)⊗ b2,1 ⊗ b3,2

= b1,1 ⊗ (b2,1 ⊗ b3,1 + b2,2 ⊗ b3,2 − αδb2,1 ⊗ b3,2) + (1− βδ)b1,2 ⊗ b2,1 ⊗ b3,2.

We have two cases.


Case 1: Suppose that βδ = 1. Then 1− βδ = 0 and

u = b1,1 ⊗ b2,1 ⊗ (b3,1 − αδb3,2) + b1,1 ⊗ b2,2 ⊗ b3,2.

Note that b2,1, b2,2 and b3,1 − αδb3,2, b3,2 are both linearly independent, and thus

wt⊗3j=1Uj

(u) > 1 by Lemma 3.1.2, a contradiction.

Case 2: Suppose that βδ 6= 1. In this case b1,1, (1 − βδ)b1,2 and b2,1 ⊗ b3,1 + b2,2 ⊗b3,2−αδb2,1⊗b3,2, b2,1⊗b3,2 are both linearly independent, so wtU1⊗W (u) = 2 by Lemma

3.1.1. It follows from Lemma 3.1.3 (1) that wt⊗3j=1Uj

(u) ≥ 2, a contradiction.

From cases 1 and 2 we conclude that wt⊗3j=1Uj

(v) 6= 2. Therefore wt⊗3j=1Uj

(v) = 3 by

Lemma 3.1.7.

3.2. Linear, semilinear, and affine groups

In this section assume that V = Fnq with q = pℓ for some prime p. Then the general

linear group GL(V ) of V may be identified with the group GL(n, q) of all invertible n×n

matrices over Fq, and the centre of GL(V ) consists of all nonzero scalar transformations (or

all scalar matrices λIn, λ ∈ F#q ). The centre of GL(V ), which we denote by Zq−1, is thus

isomorphic to F#q , which is a cyclic group of order q − 1. The special linear group SL(V )

(or SL(n, q)) of V is the subgroup of GL(V ) consisting of all elements with determinant

1, and its centre is Zq−1∩SL(n, q) = λIn | λ ∈ F#q , λn = 1. The group SL(V ) is normal

in GL(V ).

The general semilinear group of V is the group ΓL(V ) (or ΓL(n, q)) of all invertible

Fq-semilinear transformations of V , that is, all maps g : V → V that satisfy the following:

(i) there exists σ(g) ∈ Aut (Fq), dependent only on g, such that

(λu+ v)g = λσ(g)ug + vg ∀ λ ∈ Fq and u, v ∈ V ;

and

(ii) v ∈ V | vg = 0V = 0V .The map

σ : ΓL(n, q) → Aut (Fq) , (3.2.1)

with σ(g) as in (i) for any g, is an epimorphism with kernel GL(n, q), and ΓL(n, q) =

GL(n, q) ⋊ Aut (Fq). The group Aut (Fq) is a cyclic group of order ℓ generated by the

Frobenius automorphism τ , which is the map τ : α 7→ αp for all α ∈ Fq, and the action of

ΓL(V ) on V is determined by the actions on V of GL(V ) and of τ . For any fixed basis

B := v1, . . . , vn of V , we can define an action of τ on V by setting(

n∑

i=1

λivi

)τ

:=n∑

i=1

λτi vi (3.2.2)

for any λ1, . . . , λn ∈ Fq. Note that this action of τ depends on the choice of the basis B,and all actions of τ obtained as B varies over all bases of V are equivalent (i.e., conjugate

by elements of GL(V )).

3.2. LINEAR, SEMILINEAR, AND AFFINE GROUPS 39

The orders of SL(n, q), GL(n, q) and ΓL(n, q) are summarised in the next lemma.

Lemma 3.2.1. [43, Section 3.3] Let ℓ, n ≥ 1 and q = pℓ for some prime p. Then:

(1) |GL(n, q)| =∏n−1i=0 (qn − qi) = qn(n−1)/2

∏ni=1 (q

i − 1)

(2) |SL(n, q)| = |GL(n, q)|/(q − 1)

(3) |ΓL(n, q)| = ℓ|GL(n, q)|

Let X ∈ GL,SL,ΓL. Then X(V ) induces a permutation action on the set of 1-

dimensional subspaces of V , and the kernel of this action is X(V )∩Zq−1. The permutation

group induced by X(V ) on the set of 1-spaces of V is called the projective semilinear group

(respectively, projective general linear group and projective special linear group) of V if

X = ΓL (respectively, X = GL and X = SL), and is denoted by PX(V ) (or PX(n, q)).

Hence

PX(n, q) ∼= X(n, q)/(X(n, q) ∩ Zq−1).

Let TV denote the translation group of V . The affine semilinear group (or affine

general linear group, affine special linear group) of V is the group AX(V ) (or AX(n, q))

generated by TV and X(V ), where X = ΓL (respectively, X = GL and X = SL). So AX(V )

consists of all maps tw,g : v 7→ vg + w (v ∈ V ), where w ∈ V and g ∈ X(V ). The group

TV is a regular normal subgroup of AX(V ), and we have

AX(V ) ∼= TV ⋊X(V ).

Since Aut (V ) = GL(V ), we have from Theorem 1.1.8 that

AGL(V ) ∼= NSym(V )(TV ),

and GL(V ) is the stabiliser in AGL(V ) of 0V . Thus AGL(V ) = Hol (V ), as defined in

Section 1.2, where V is identified with TV . This fact will be used later.

For any subfield Fq0 of Fq and X ∈ GL,ΓL, it is easy to see that the group X(n, q)

contains a subgroup isomorphic to X(n, q0). Likewise, if k > 1 is a divisor of n, then

X(n, q) contains a subgroup isomorphic to X(n/k, qk). Indeed, V can be viewed as a

vector space of dimension m := n/k over the extension field Fqk of Fq. If Fqk = Fq[ω] and

B := (u1, . . . , um) is an ordered Fqk -basis of V , then

B0 := ωiuj | 0 ≤ i ≤ k − 1, 1 ≤ j ≤ m

is an Fq-basis of V . For any g ∈ GL(m, qk) and v =∑

i,j λijωiuj ∈ V (where λij ∈ Fq

for all i and j) we have vg =∑

i,j λijωi(uj)

g, so g induces an Fq-linear map on Fnq . So

GL(n/k, qk) . GL(n, q). If τ ′ is the Frobenius automorphism of Fqk , acting on V with

respect to the basis B, then τ ′ = hτ , where h ∈ GL(n, q) and (ωiuj)h = (ωi)puj for each

i and j, and τ acts as in (3.2.2) with B = B0. Hence ΓL(n/k, qk) . ΓL(n, q). It follows

immediately that AX(n, q0) . AX(n, q) and AX(n/k, qk) . AX(n, q) for X ∈ GL,ΓL.Observe also that the Fqk -automorphism (τ ′)ℓ is Fq-linear on V . Hence GL(n, q) contains


a subgroup isomorphic to GL(n/k, qk) ⋊ Gal(Fqk/Fq

), where Gal

(Fqk/Fq

)= 〈(τ ′)ℓ〉 is

the Galois group of Fqk over Fq, and is cyclic of order k. In particular, if q = p, then

ΓL(n/k, pk) . GL(n, p).

The next two results concern certain families of subgroups of AGL(V ). In particular,

we consider the case where dim (V ) is even and look at subgroups of the form TV ⋊G0 for

some G0 ≤ GL(V ). In this case V has a direct sum decomposition U ⊕U for some vector

space U ; a diagonal subspace of V (with respect to this decomposition) is a subspace of

the form (u, uϕ) | u ∈ U for some ϕ ∈ GL(U). For any subgroup L of TV , we denote by

VL the subspace of V identified with L.

Lemma 3.2.2. Let U be a finite vector space, V := U ⊕ U , G0 ≤ GL(V ), and G =

TV ⋊ G0. Suppose that TU⊕0U and T0U⊕U are minimal normal subgroups of G, and

let L < G such that L 6= TU⊕0U, T0U⊕U . Then L is a minimal normal subgroup of G

if and only if L < TV and VL is a G0-invariant diagonal subspace of V . In particular, if

(u, u) | u ∈ U is G0-invariant, then the following hold:

(1) the group G0 = (h, h) | h ∈ H for some H ≤ GL(U) which is irreducible on

U#; and

(2) the minimal normal subgroups of G are precisely the groups TW , where W is the

subspace U ⊕ 0U, 0U ⊕ U , or (u, uϕ) | u ∈ U for some ϕ ∈ CGL(U)(H).

Proof. We first show that L is a minimal normal subgroup of G, distinct from

TU⊕0U and T0U⊕U , if and only if L < TV and VL is a G0-invariant diagonal subspace

of V . Since G0 acts faithfully on V , each minimal normal subgroup L of G is contained

in TV . Also, a subgroup of TV is normal in G if and only if the corresponding subspace is

G0-invariant. Thus it remains only to show that L is minimal normal in G if and only if

VL is a diagonal subspace.

Suppose that L is minimal normal in G distinct from TU⊕0U and T0U⊕U . Denote by

π1 and π2 the projection maps from VL to U⊕0U and to 0U⊕U , respectively, and let

X := π1(VL). Since VL is G0-invariant, so is X, and hence TX EG. Now TX ≤ TU⊕0U,

and thus, since TU⊕0U is minimal normal in G, either TX = 1 or TX = TU⊕0U.

Equivalently, either X = 0V or X = U ⊕ 0U. If X = 0V then VL ≤ 0U ⊕ U , so

L ≤ T0U⊕U . Since T0U⊕U is minimal normal equality holds, contrary to the choice of

L. So π1(VL) = X = U ⊕ 0U, and similarly π2(VL) = 0U ⊕ U ; therefore π1VLand

π2VLare surjective. Observe that L ∩ TU⊕0U is trivial since L and TU⊕0U are distinct

minimal normal subgroups of G, so that VL ∩ 0U = 0V and the kernel of π1VLis

trivial. Hence π1VLis one-to-one, and similarly so is π2VL

. Therefore π1VLand π2VL

are

bijections, which shows that VL is a diagonal subspace of V .

Conversely, suppose that VL is a G0-invariant diagonal subspace of V . Then L is a

nontrivial normal subgroup of G. Let L′ be a minimal normal subgroup of G which is

contained in L. Then by the preceding paragraph VL′ is a G0-invariant diagonal subspace

3.2. LINEAR, SEMILINEAR, AND AFFINE GROUPS 41

of V , and VL ≥ VL′ . It follows that VL = VL′ and thus L = L′. Therefore L is minimal

normal in G.

Now suppose thatW0 := (u, u) | u ∈ U is G0-invariant. Then for any u ∈ U and any

(g1, g2) ∈ G0 with g1, g2 ∈ GL(U), we have (u, u)(g1,g2) = (ug1 , ug2) ∈W0. So ug1 = ug2 for

all u ∈ U , and hence g1 = g2. Thus G0 = (h, h) | h ∈ H for some H ≤ GL(U), where

H is irreducible because of the minimality of TU⊕0U and T0U⊕U . This proves (1).

Finally we show (2). Let L be a minimal normal subgroup of G. We have already

proved that L is TU⊕0U, T0U⊕U , or TY (ρ) for some G0-invariant diagonal subspace

Y (ρ) := (u, uρ) | u ∈ U of V , and to complete the proof we need to show that the G0-

invariant diagonal subspaces are precisely the Y (ρ) with ρ ∈ CGL(U)(H). Suppose that

Y (ρ) is G0-invariant. Then for all u ∈ U and h ∈ H we have (u, uρ)(h,h) =(uh, uρh

)∈

Y (ρ). It follows from the definition of Y (ρ) that uρh = uhρ for all u ∈ U and h ∈ H. Thus

ρh = hρ for all h ∈ H; that is, ρ ∈ CGL(U)(H). Conversely, if ρ ∈ CGL(U)(H), then for

any u ∈ U and h ∈ H we have (u, uρ)(h,h) =(uh, uρh

)=(uh, uhρ

)∈ Y (ρ). So Y (ρ) is

G0-invariant, which completes the proof of (2).

The next result gives more information about the action of G0 in Lemma 3.2.2 in the

case where H is transitive on U#.

Lemma 3.2.3. Let U be a finite vector space, V = U ⊕ U , H ≤ GL(U) which is

transitive on U#, G0 = (h, h) | h ∈ H ≤ GL(V ), and S a G0-orbit in V #. Then

〈S〉 = V if and only if S is not (U ⊕ 0U)#, (0U ⊕ U)#, or W# for a G0-invariant

diagonal subspace W of V .

Proof. Throughout this proof 〈S〉 denotes the subspace of V generated by S ⊆ V .

Claim 1: 〈S〉 = V if and only if 〈S〉 6= S ∪ 0V . Suppose that 〈S〉 = V . Since

(U ⊕ 0U)# is a proper G0-invariant subset of V#, G0 is not transitive on V # and thus

S ∪ 0U 6= V = 〈S〉. Conversely, suppose that 〈S〉 6= S ∪ 0V . Let v = (u1, u2) ∈ S.

If u2 = 0U then we must have u1 6= 0U , so that v ∈ (U ⊕ 0U)#. Since H is transitive

on U# we have S = vG0 = (U ⊕ 0U)#, so 〈S〉 = U ⊕ 0U = S ∪ 0V , contrary to

our choice of S. Similarly if u1 = 0U then 〈S〉 = U ⊕ 0U, also a contradiction. Thus

u1, u2 6= 0U , and moreover (U ⊕ 0U)# , (0U ⊕ U)# ⊆ V \ S. Let π1 and π2 be the

projection maps from S to U ⊕ 0U and to 0U ⊕ U , respectively. Arguing as before

we get that πi(S) ⊆ U# for each i; moreover, since each πi(S) is H-invariant and H is

transitive on U#, we have πi(S) = U# for each i. It follows that πi(〈S〉) = 〈πi(S)〉 = U .

For each u ∈ U# let B(u) := v′ ∈ S | π1(v′) = u. Then B(u) is nonempty for each u

and is a block of imprimitivity for the action of G0 on S. Hence

|S| = (|U | − 1)|B(u)| ≥ |U | − 1.

Clearly we have |S| + 1 ≤ |〈S〉|, so if 〈S〉 were a diagonal subspace then |〈S〉| = |U | ≤|S| + 1 and thus |S| + 1 = |〈S〉|. Consequently 〈S〉 = S ∪ 0V , a contradiction, and


therefore 〈S〉 is not a diagonal subspace. Since π1(〈S〉) = U by the above, there exist

vectors u,w1, w2 ∈ U with w1 6= w2 and (u,w1), (u,w2) ∈ 〈S〉. So (0U , w1 − w2) ∈ 〈S〉,and (0U , w1 − w2)

G0 = (0U ⊕ U)# ⊆ 〈S〉. Similarly (U ⊕ 0U)# ⊆ 〈S〉, and hence

V = U ⊕ U ⊆ 〈S〉. Therefore 〈S〉 = V , which proves Claim 1.

Claim 2: 〈S〉 = S ∪ 0V if and only if S is (U ⊕ 0U)#,(0U ⊕ U#

), or is a

G0-invariant diagonal subspace. It was shown above that if 〈S〉 has one of these forms

then 〈S〉 = S ∪ 0V . Suppose now that 〈S〉 = S ∪ 0V . Then 〈S〉 is G0-invariant,

with G0 transitive on 〈S〉# and thus irreducible on 〈S〉. It follows that T〈S〉 is a minimal

normal subgroup of G, and by Lemma 3.2.2, 〈S〉 is U ⊕0U, 0U⊕U , or a G0-invariant

diagonal subspace. Therefore Claim 2 holds, which completes the proof.

3.3. Classical groups and their geometries

Throughout this section V = Fnq where q is a prime power.

Let f : V × V → Fq. Then f is a left-linear form on V if

f(αu+ βv,w) = αf(u,w) + βf(v,w) ∀ u, v, w ∈ V, ∀ α, β ∈ Fq.

That is, for each v ∈ V the map V → Fq, u 7→ f(u, v), is linear. Similarly, f is a right-

linear form if for each u ∈ V the map V → Fq, v 7→ f(u, v) is linear. The map f is said to

be

bilinear if it is both left-linear and right-linear;

symmetric if f(u, v) = f(v, u) ∀ u, v ∈ V ;

skew-symmetric if f(u, v) = −f(v, u) ∀ u, v ∈ V ;

alternating if f(v, v) = 0 ∀ v ∈ V ;

conjugate-symmetric sesquilinear if there exists σ ∈ Aut (Fq), σ 6= 1, such that

f(u, v) = f(v, u)σ ∀ u, v ∈ V .

Note that the definition of conjugate-symmetric implies that σ2 = 1; in this case q must be

a square and σ : α 7→ α√q for all α ∈ Fq. The left radical of f is the set u ∈ V | f(u, v) =

0 ∀ v ∈ V , and its right radical is v ∈ V | f(u, v) = 0 ∀ u ∈ V . The map f is

nondegenerate if its left and right radicals are both zero. It is easy to see that if f is

symmetric, skew-symmetric, alternating, or conjugate-symmetric, then its left and right

radicals coincide, and the radical of f is denoted by rad(f).

A quadratic form on V is a map Q : V → Fq such that

Q(αu+ v) = α2Q(u) +Q(v) + αf(u, v) ∀ u, v ∈ V, ∀ α ∈ Fq,

where f is a symmetric bilinear form on V . The map f in this case is called the associated

bilinear form of Q. It follows from the definition that 2Q(v) = f(v, v) for all v ∈ V .

Hence if char (Fq) 6= 2 then Q(v) = 12f(v, v), that is, f uniquely determines the quadratic

form Q. On the other hand, if char (Fq) = 2 then f is alternating, and there is more

than one quadratic form Q having f as its associated bilinear form. The quadratic form

Q is nondegenerate if f is nondegenerate (i.e., rad(f) = 0V ), and is nonsingular if

3.3. CLASSICAL GROUPS AND THEIR GEOMETRIES 43

the set rad(Q) := v ∈ rad(f) | Q(v) = 0 is zero. In odd characteristic, the terms

“nondegenerate” and “nonsingular” are equivalent; in characteristic 2, all nondegenerate

forms are nonsingular, but not conversely.

We write (V, φ) to signify that there is a map φ : V × V → Fq or φ : V → Fq defined

on V . We are interested in the following cases:

(1) (V, f), where f is a nondegenerate alternating bilinear form;

(2) (V, f), where f is nondegenerate conjugate-symmetric sesquilinear form;

(3) (V,Q), whereQ is a nondegenerate quadratic form with associated bilinear form f .

The space V and its underlying geometry is called symplectic in case (1), unitary in case

(2), and orthogonal in case (3).

If f is a left-linear form on V and U is any subspace of V , define

U⊥ := v ∈ V | f(u, v) = 0 ∀ u ∈ U.

Then U⊥ is a subspace of V . If f is nondegenerate, then by [31, Lemma 2.1.5], the spaces

U and U⊥ have the following properties:

(i) dim (U) + dim(U⊥) = dim (V );

(ii) (U⊥)⊥ = U ;

(iii) U is totally isotropic if and only if U ≤ U⊥;

(iv) U is nondegenerate (that is, the restriction f |U is nondegenerate) if and only if

V = U ⊕ U⊥.

The space V is an orthogonal direct sum of its subspaces U and W , written V = U ⊥W ,

if V = U ⊕W and f(u,w) = 0 for all u ∈ U , w ∈ W . A nonzero vector v is said to be

isotropic if f(v, v) = 0, and anisotropic otherwise. If f is symplectic or unitary, then an

isotropic vector is also called singular. If f is symmetric bilinear with quadratic form Q,

then a singular vector is a nonzero vector v such that Q(v) = 0. Hence, in general, all

isotropic vectors are singular and vice versa, unless V is orthogonal and q is even. If V

is orthogonal and q is even then all vectors are isotropic but not all are singular. If f is

symplectic or unitary then a subspace U is totally isotropic or totally singular if f |U ≡ 0;

if f is symmetric bilinear with quadratic form Q, then U is totally isotropic if f |U ≡ 0

and totally singular if Q|U ≡ 0. On the other extreme, a subspace U is anisotropic if all

nonzero vectors in U are anisotropic. A hyperbolic pair in V is a pair x, y of singular

vectors such that f(x, y) = 1. A hyperbolic plane is a two-dimensional subspace spanned

by a hyperbolic pair.

Suppose that f1 and f2 are two left-linear forms on V which are of the same type (i.e.,

symplectic, unitary, or symmetric bilinear associated with a nonsingular quadratic form).

Then f1 and f2 are similar if there exists g ∈ GL(V ) such that, for some λ(g) ∈ F#q which

depends only on g,

f1(ug, vg) = λ(g)f2(u, v) ∀ u, v ∈ V.


Likewise, two quadratic forms Q1 and Q2 on V are similar if there is a g ∈ GL(V ) such

that for some λ(g) ∈ F#q ,

Q1(vg) = λ(g)Q2(v) ∀ v ∈ V.

In both cases the element g is called a similarity from (V, φ1) to (V, φ2), where φi = fi or

φi = Qi for i = 1, 2. If λ(g) = 1 then g is called an isometry. Observe that a similarity from

(V,Q1) to (V,Q2) is a similarity from (V, f1) to (V, f2), where fi is the associated bilinear

form of Qi; the converse is true only if char (Fq) is odd. If φ1 = φ2 = φ then a similarity

(respectively, isometry) of (V, φ) is called a φ-similarity (respectively, φ-isometry).

Theorem 3.3.1. [31, Propositions 2.3.2, 2.4.1, 2.5.3] Let V = Fnq , and let f be a left-

linear form on V which is symplectic, unitary, or a symmetric bilinear form associated

with a nondegenerate quadratic form Q. Then

V = 〈x1, y1〉 ⊥ . . . ⊥ 〈xm, ym〉 ⊥ U

where xi, yi is a hyperbolic pair for each i and U is an anisotropic subspace. Moreover:

(1) If f is symplectic then U = 0. Hence n is even and, up to equivalence, there is a

unique symplectic geometry in dimension n over Fq.

(2) If f is unitary then U = 0 if n is even and dim (U) = 1 if n is odd. Hence up to

equivalence, there is a unique unitary geometry in dimension n over Fq.

(3) If f is symmetric bilinear with quadratic form Q and n is odd, then dim (U) = 1

and there are two isometry classes of quadratic forms in dimension n over Fq, one

a non-square multiple of the other. Hence all orthogonal geometries in dimension

n over Fq are equivalent.

(4) If f is symmetric bilinear with quadratic form Q and n is even, then U = 0 or

dim (U) = 2. For each n there are exactly two isometry classes of orthogonal

geometries over Fq, which are distinguished by dim (U).

In Theorem 3.3.1 (4), the quadratic form Q and the corresponding geometry is said

to be of plus type if U = 0, and of minus type if dim (U) = 2. The order of the isom-

etry group of (V, f) or (V,Q) is determined by counting the number of possible bases

x1, . . . , xn/2, y1, . . . , yn/2 satisfying the conditions of Theorem 3.3.1; these orders are

given in Table 3.3.2.

The group of isometries of (V, φ) is called the symplectic group if φ is symplectic,

the unitary group if φ is unitary, and the orthogonal group if φ is nonsingular quadratic.

Following the convention of [31], we denote the isometry group by I(V ), I(V, φ), I(n, q)

or I(n, q, φ), as convenient, where I is given in Table 3.3.1. In particular, note that our

notation for the unitary group is nonstandard - in the literature, including [31], U(n, q) (or,

in many cases, GU(n, q)) denotes the unitary group on a vector space of dimension n over

Fq2 , whereas we use U(n, q) to refer to the unitary group on Fnq . Recall that if q is even and


φ I

trivial GL

symplectic Sp

unitary U

nonsingular quadratic, odd dimension O

nonsingular quadratic, even dimension, plus type O+

nonsingular quadratic, even dimension, minus type O−

Table 3.3.1. Isometry groups I(V, φ)

φ = Q is nonsingular quadratic, then the associated bilinear form f of Q is alternating,

and any Q-isometry is an f -isometry. Observe that the restriction Q : rad(f) → Fq is

semilinear with (trivial) kernel rad(Q), so dim (rad(f)) = dim(ImQrad(f)

)≤ 1. If Q

is nondegenerate then f is symplectic, so n is even and O±(n, q) < Sp(n, q). If Q is

degenerate (but nonsingular) then dim (rad(f)) = 1, and f induces a symplectic form f ′

on V/rad(f). So n− 1 is even, that is, n is odd. It can be shown that the isometry groups

of (V,Q) and (V/rad(f), f ′) are isomorphic, so O(n, q) ∼= Sp(n− 1, q). For this reason we

only consider the case where Q is nondegenerate, so that n is even whenever q is even.

Group Order

Sp(n, q) qn2/4∏n/2

i=1 (q2i − 1)

U(n, q) qn(n−1)/4∏n

i=1 (qi/2 − (−1)i)

O(n, q) 2q(n−1)2/4∏(n−1)/2

i=1 (q2i − 1)

O+(n, q) 2qn(n−2)/4(qn/2 − 1)∏n/2−1

i=1 (q2i − 1)

O−(n, q) 2qn(n−2)/4(qn/2 + 1)∏n/2−1

i=1 (q2i − 1)

Table 3.3.2. Orders of the isometry groups

The next theorem is a well-known and fundamental result on classical geometries, and

which can be found in various sources such as [31, Proposition 2.1.6] and [43, Theorem

3.3].

Theorem 3.3.2 (Witt’s Lemma). Let V be a vector space and φ a form on V , where

φ is a symplectic, unitary, or nondegenerate quadratic form. Then any isometry between

subspaces of V extends to an isometry of V .

The following is a consequence of Witt’s Lemma, and can also be found in [44].

Theorem 3.3.3. [44, Propositions 3.11, 5.12, 6.8 and 7.10] Let V = Fnq and φ a

symplectic, unitary, or nondegenerate quadratic form on V . Then the orbits in V # of the

isometry group of (V, φ) are the sets Sλ for each λ ∈ Imφ, where

Sλ := v ∈ V # | φ(v) = λ (3.3.1)


and

φ(v) =

f(v, v) if φ = f is symplectic or unitary;

Q(v) if φ = Q is quadratic.(3.3.2)

Proof. Let I be as given in Table 3.3.1, and let v ∈ Sλ. Then clearly vI(V ) ⊆ Sλ.

For any other w ∈ Sλ, define the map g : 〈v〉 → 〈w〉 by αv → αw for any α ∈ Fq. Then

g is an isometry from 〈v〉 to 〈w〉, and by Theorem 3.3.2, g extends to an isometry of V .

So w ∈ vI(V ), and thus Sλ ⊆ vI(V ). Therefore Sλ = vI(V ), and the sets Sλ are all the

I(V )-orbits in V #.

Recall that if φ = f is symplectic then the map φ in (3.3.2) is identically zero, so that

V # = S0 and Sp(V ) is transitive on V # by Theorem 3.3.3. For φ unitary or orthogonal,

the image of φ is given by the next result. Statement (2) of Lemma 3.3.4 follows from the

arguments in the proof of [31, Lemma 2.5.2 (ii)], which we present here. We denote by

Fq the set of all squares in F#

q , and by F⊠q the set F#

q \ Fq .

Lemma 3.3.4. Let V = Fnq , φ a unitary or nondegenerate quadratic form on V , and

φ as in (3.3.2).

(1) If φ is unitary then Imφ = Fq0, the subfield of index 2 in Fq.

(2) If φ is quadratic then Imφ = Fθq ∪ 0 for some θ ∈ ,⊠ if n = 1, and

Imφ = Fq if n ≥ 2.

Proof. Suppose first that φ = f is unitary. Then f(v, v)√q = f(v, v) for any v ∈ V ,

so Imφ ≤ Fq0. Observe that Theorem 3.3.1 implies that V contains a nonsingular vector,

say u. So f(αu, αu) = α√q+1f(u, u) = η(α)f(u, u) for any α ∈ Fq, where η : Fq → Fq0 is

the norm map. Since η is surjective so is φ, which proves (1).

Now suppose that φ = Q is quadratic. If n = 1 then V is anisotropic, and for some

nonsingular vector u we have Imφ = α2Q(u) | α ∈ Fq = Q(u)Fq ∪ 0. If n ≥ 2 then

Theorem 3.3.1 implies that the space V contains nonzero vectors x and y such that either

x, y is a hyperbolic pair, or 〈x, y〉 is a nondegenerate anisotropic subspace. If x, y is

a hyperbolic pair then Q(x+ αy) = α for all α ∈ Fq, so Imφ = Fq, as required. Suppose

that 〈x, y〉 is anisotropic. Then in particular Q(x) 6= 0, and since 〈x, y〉 is nondegeneratewe can choose y such that f(x, y) = Q(x), where f is the associated bilinear form of Q.

Then for all λ ∈ Fq we have Q(λx − y) = Q(x)(λ2 − λ + ζ), where ζ := Q(y)Q(x)−1.

Now Q(λx − y) 6= 0 since 〈x, y〉 is anisotropic, so the polynomial P (X) := X2 − X + ζ

is irreducible over Fq. Let ω be a root of P in the extension field Fq2 of Fq. Then ωq is

the other root of P , so P (X) = (X − ω)(X − ωq), which gives ω + ωq = 1 and ωq+1 = ζ.

Hence for any λ, µ ∈ Fq, we have Q(λx+ µy) = Q(x)(λ2 + µ2ζ + λµ) = Q(x) η′(λ+ µω),

where η′ : Fq2 → Fq is the norm map. Since η′ is surjective so is φ. This completes the

proof of (2).


We are also interested in determining Imφ|〈v〉⊥ for any v ∈ V #. This is given in Corol-

lary 3.3.5, which follows immediately from Lemma 3.3.4 and the following observations.

Suppose that f is a symplectic, unitary, or nondegenerate symmetric bilinear form on

V with quadratic form Q. If v ∈ V # is nonsingular, then 〈v〉⊥ is nondegenerate, and by

the remarks above, V = 〈v〉 ⊥ 〈v〉⊥. If v is singular then 〈v〉 is totaly singular, so that

〈v〉 ≤ 〈v〉⊥. By the remarks in [31, pp. 17-18], the form f induces a nondegenerate form

fU , of the same type as f , on the space U := 〈v〉⊥/〈v〉, defined by fU(x+ 〈v〉, y + 〈v〉) :=f(x, y) for all x, y ∈ 〈v〉⊥. For f symmetric bilinear, the quadratic form Q likewise induces

a nondegenerate quadratic form QU on U , where Q(x+ 〈v〉) := Q(x) for all x ∈ 〈v〉⊥, andfU is the associated bilinear form of QU . It follows from Theorems 3.3.2 and 3.3.1 that all

maximal totally isotropic subspaces of V have the same dimension, which, in all cases, is

at most n/2, so in particular v⊥ contains a nonsingular vector whenever n ≥ 3.

Corollary 3.3.5. Let V = Fnq , φ a unitary or nondegenerate quadratic form on V , φ

as in (3.3.2), and v ∈ V #.

(1) Suppose that φ is unitary, and let Fq0 denote the subfield of Fq of index 2. Then

Imφ|〈v〉⊥ = Fq0 if v is nonsingular and n ≥ 2, or if v is singular and n ≥ 3.

(2) Suppose that φ is quadratic.

(i) If q is even, then Imφ|〈v〉⊥ = Fq if v is nonsingular and n ≥ 2, or if v is

singular and n ≥ 3.

(ii) If q is odd, then Imφ|〈v〉⊥ = Fq if v is nonsingular and n ≥ 3, or if v is

singular and n ≥ 4.

(iii) If v is singular and n = 3, then Imφ|〈v〉⊥ = Fθq ∪ 0 for some θ ∈ ,⊠.

Proof. This follows immediately from Lemma 3.3.4 applied to 〈v〉⊥, and the remarks

above.

Recall that for each dimension n, there are exactly two isometry classes of orthogonal

geometries over Fq. If n is even these are distinguished by the dimension of a maximal

totally isotropic subspace (which is n/2 when Q is of plus type, and (n − 2)/2 when Q

is of minus type). If n is odd then there is a unique quadratic form up to similarity, and

the isometry classes are distinguished by the value of Q(u) (mod Fq ), where 〈u〉 = U ,

the 1-dimensional anisotropic subspace in part (3) of Theorem 3.3.1. Another way of

distinguishing the two types of orthogonal geometries in odd characteristic for a given n is

via the discriminant, which is defined as follows. Let f be the associated bilinear of Q, and

let B := v1, . . . , vn be a fixed ordered basis of V . If fB is the matrix of f with respect

to B, the discriminant D(Q) of Q is defined to be the coset of Fq in F#

q which contains

det fB. The discriminant of a quadratic form is well defined, and for any n and q, where

q is odd, two quadratic forms Q1 and Q2 are isometric if and only if D(Q1) = D(Q2) [31,


Proposition 2.5.10]. Moreover, we have the following from [31, Propositions 2.5.11 (ii)

and 2.5.12].

Theorem 3.3.6. Let V = Fnq with q odd, and let Q be a nondegenerate quadratic form

on V .

(1) If V = V1 ⊥ . . . ⊥ Vt, where each Vi is a nondegenerate subspace of V , then

D(Q) =

t∏

i=1

D(Q|Vi),

with the multiplication in the quotient group F#q /F

q .

(2) There is a basis B for V such that

fB =

In if D(Q) = F

q ;

diag(µ, 1, . . . , 1) if D(Q) = F⊠q ,

where µ ∈ F#q and 〈µ〉 = F#

q .

3.3.1. Other classical groups. A semisimilarity of (V, φ) is an element g ∈ ΓL(V )

such that, for some γ(g) ∈ F#q and σ(g) ∈ Aut (Fq) which both depend only on g,

f(ug, vg) = γ(g)f(u, v)σ(g) ∀ u, v ∈ V (3.3.3)

if φ = f is symplectic or unitary, and

Q(vg) = γ(g)Q(v)σ(g) ∀ v ∈ V

if φ = Q is quadratic. The set of semisimilarities of (V, φ) forms a subgroup of ΓL(V )

which is denoted by ΓI(V ) or ΓI(n, q), with I as given in Table 3.3.1. It can be shown

that, with a suitable choice of basis, σ(g) = σ(g) for all g ∈ ΓI(V ), where σ(g) is as given

in (3.2.1). Hence the map σ : ΓI(n, q) → Aut (Fq) is an epimorphism whose kernel is the

similarity group GI(n, q) of V , which we call the general symplectic, general unitary or

general orthogonal group, according to the type of φ. The map γ : GI(n, q) → F#q is an

epimorphism whose kernel is the isometry group I(n, q).

If φ is unitary or orthogonal, the subgroup I(V )∩SL(V ) of elements with determinant

1 is denoted by SI(V ). If φ is symplectic then Sp(n, q) ≤ SL(n, q) for all even n, with

equality if n = 2.

The next theorem further describes the relationship among the groups I(V ), GI(V ),

and ΓI(V ).

Theorem 3.3.7. [31, Propositions 2.3.4, 2.4.3, 2.6.2, 2.7.1] Let n ≥ 2 and let q be a

prime power, subject to the appropriate restrictions such that V = Fnq is equipped with a

symplectic, unitary or nonsingular quadratic form φ. Let τ be the Frobenius automorphism


I GI(n, q) ΓI(n, q)

Sp

I(n, q)× Zq−1 if q is even

I(n, q)⋊ 〈g〉 if q is odd,GI(n, q)⋊Aut (Fq)

g =

µIn/2 0

0 In/2

U I(n, q) Zq−1 = (I(n, q).Zq−1)/〈µ√q−1In/2〉 GI(n, q)⋊Aut (Fq)

O I(n, q) Zq−1 = (I(n, q).Zq−1) /〈−In/2〉 GI(n, q)⋊Aut (Fq)

O+


I(n, q)⋊ 〈g〉 if q is odd,GI(n, q)⋊Aut (Fq)

g =

µIn/2 0

0 In/2

O−


I(n, q)⋊ 〈h〉 if q is odd,

GI(n, q)⋊ Zq−1 if D(φ) = F

q

GI(n, q).〈mτ〉 if D(φ) = F⊠q ,

h =

diag(h2, . . . , h2︸︷︷︸n/2

) if D(φ) = Fq

diag(h1, h2, . . . , h2︸︷︷︸n/2−1

) if D(φ) = F⊠q

m = diag(µ(q−1)/2, 1, . . . , 1)

Table 3.3.3. Similarity and semisimilarity groups

of Fq, µ ∈ F#q such that F#

q = 〈µ〉, and h1 :=(0 µ

1 0

). If q is odd, let α, β ∈ F#

q such that

α2 +β2 = µ and h2 :=

(α β

β −α

). Then GI(n, q) and ΓI(n, q) are as given in Table 3.3.3.

The special orthogonal group SO(V ) or SO±(V ) contains a subgroup of index 2, which

is usually denoted by Ω(V ) or Ω±(V ), as appropriate. For a more detailed description of

this subgroup see [31, Section 2.5] or [43, Sections 3.7 and 3.8].

Let X ∈ Ω,Ω±,SI, I,GI,ΓI. Then X(V ) modulo scalars yields the projective group

PX(V ). It is known that the simple classical groups are precisely the following:

PSL(n, q), n ≥ 2, (n, q) 6= (2, 2), (2, 3);

PSp(n, q), n even and n ≥ 4, (n, q) 6= (4, 2);

PSU(n, q), n ≥ 3, (n, q) 6= (3, 4);

PΩ(n, q), n ≥ 7, nq odd;

PΩ±(n, q), n even and n ≥ 8.

3.3.2. Tensor products. Suppose that V = U ⊗W , where U = Fkq and W = Fm

q ,

possibly k 6= m. Recall from Section 3.1 that GL(U) ⊗ GL(W ) ≤ GL(V ); if U and W

are equipped with forms φU and φW , respectively, then it can be shown analogously that,


I(U, φU ) I(W,φW ) I(U ⊗W,φU ⊗ φW )

Sp Oǫ

Sp if the characteristic is odd;

O+ else

Sp Sp O+

Oǫ1 Oǫ2

O+ if ǫi = + for some i, or ǫi = − for both i;

O if dim (U) and dim (W ) are odd;

O− else

U U U

Table 3.3.4. Tensor products of classical groups

under certain conditions, the inclusion I(U, φU )⊗ I(W,φW ) ≤ I(V, φU ⊗ φW ) holds, where

φU ⊗ φW is a form on V defined below.

Assume that φU and φW are both bilinear or both conjugate-symmetric sesquilinear

forms. Define the form φU ⊗ φW on V by

(φU ⊗ φW )(u⊗ w, u′ ⊗ w′) := φU (u, u′)φW (w,w′)

for all u⊗w and u′⊗w′ in a tensor product basis of V (see Section 3.1), extended bilinearly

if φU and φW are bilinear, and sesquilinearly if φU and φW are sesquilinear. Then φU⊗φWis bilinear or sesquilinear if φU and φW are both bilinear or both sesquilinear, respectively,

and nondegenerate if and only if both φU and φW are nondegenerate. Hence φU ⊗ φW is

unitary if φU and φW are both unitary. For the bilinear case, φU ⊗ φW is alternating if at

least one of φU and φW is alternating, and φU ⊗ φW is symmetric if both φU and φW are

symmetric.

Table 3.3.4 summarises the types of forms φU ⊗φW that arise according to the various

possibilities for φU and φW , and it presents the possible inclusions I(U, φU )⊗ I(W,φW ) ≤I(V, φU ⊗φW ). The details can be found in [31, Section 4.4] and [43, Section 3.10.5]. The

symbol Oǫ denotes any of O, O+ or O−.

We can extend the above to tensor products of an arbitrary number of subspaces. If

V = U1 ⊗ · · · ⊗ Ut and φi is a nondegenerate form on Ui for each i, which are all bilinear

or all sesquilinear, define φ1 ⊗ · · · ⊗ φt by

(⊗t

i=1φi) (

⊗ti=1ui,⊗t

i=1wi

)=

t∏

i=1

φi(ui, wi)

as ⊗ti=1ui and ⊗t

i=1wi vary over a tensor product basis of V , extended bilinearly if the φi

are bilinear, and sesquilinearly if they are sesquilinear. Then ⊗ti=1φi is a nondegenerate

bilinear (respectively, sesquilinear) form on V . If the spaces (Ui, φi) are all isometric, then

3.4. THE EXCEPTIONAL GROUP G2(q) AND ITS GEOMETRY 51

we can extend the results of Table 3.3.4 to the following (see [31, 43]):

⊗ti=1Sp(m, q) <

Sp(mt, q) if qt odd;

O+(mt, q) else

⊗ti=1O

ǫ(m, q) <

O(mt, q) if qm is odd;

O−(mt, q) if ǫ = − and t is odd;

O+(mt, q) else

⊗ti=1U(m, q) < U(mt, q)

3.4. The exceptional group G2(q) and its geometry

Throughout this section U = Fnq , where q is a power of 2 and n = 6.

Recall from Section 3.3 that there is a symplectic form f defined on U . It is known

[14, 43] that the geometry of (U, f) admits a structure H(q), called a generalised hexagon

with parameters (q, q), which is a point-line incidence structure whose incidence graph has

diameter 6 and girth 12, such that each line contains q + 1 points and each point lies in

q + 1 lines. The points of H(q) are the one-dimensional subspaces of U , and the lines are

the totally isotropic two-dimensional subspaces of U . The group G2(q) is a subgroup of

Sp(6, q) and acts on H(q).

Notation. Denote by P the point set of H(q) (equivalently, the set of one-dimensional

subspaces of U); by L the line set of H(q); by L′ the set of totally isotropic two-dimensional

subspaces of U which are not in L; and by N the set of nondegenerate two-dimensional

subspaces of U .

Observe that L∪L′ ∪N comprises the set of all two-dimensional subspaces of U , and

we have |P| = |L| = (q6−1)/(q−1), |L′| = q2(q6−1)/(q−1), and |N | = q4(q6−1)/(q2−1),

see [14, Section 5]. The following result describes the action of G2(q) on these sets and

is proved in [14, Lemmas 5.1, 5.2 and 5.4] and [13, Lemma 3.1] for parts (1) and (2),

respectively.

Lemma 3.4.1. Let q be a power of 2 and let P, L, L′ and N be as defined above.

(1) The group G2(q) acts transitively on each of the sets P, L, L′ and N .

(2) The action of G2(q) on P has rank 4 with subdegrees 1, q(q + 1), q3(q + 1) and

q5.

Following the notation of [14], for a fixed point X in P, let ∆i(X) (i = 1, 2, 3) denote

the set of points Y ∈ P which are at distance i from X in the point graph of H(q). Clearly,

∆1(X) = Y ∈ P | 〈X,Y 〉 ∈ L.


The proof of Lemma 5.2 in [14] gives a description of each of the sets ∆2(X) and ∆3(X),

which we state below as a lemma.

Lemma 3.4.2. Let X ∈ P be fixed, and let ∆2(X) and ∆3(X) be as defined above.

Then

∆2(X) =Y ∈ P | 〈X,Y 〉 ∈ L′

and

∆3(X) = Y ∈ P | 〈X,Y 〉 ∈ N .

The action of G2(q) on the three-dimensional totally isotropic subspaces of U is also

described in [14]. We give this in the next lemma.

Theorem 3.4.3. The group G2(q) has two orbits on maximal totally isotropic sub-

spaces of U (of dimension 3), with representatives W3 and W ′3 and lengths (q6−1)/(q−1)

and q3(q3 + 1), respectively. Furthermore:

(1) there exists a unique point X of H(q) in W3 such that W3 is the union of all lines

in L that pass through X; and

(2) no two points in W ′3 lie in the same line of H(q). In other words, all two-

dimensional subspaces of W ′3 belong to L′.

Proof. The first part is [14, Lemma 5.3]. Let U3 be a totally isotropic three-

dimensional subspace of U . If there is a two-dimensional subspace U2 of U3 such that

U2 ∈ L, then (1) holds by line 5 of the proof of [14, Lemma 5.3]. Otherwise (2) holds.

3.4.1. Alternative definition of G2(q). The group G2(q) can also be defined as

the automorphism group of the octonion algebra O. Following [43, Section 4.4.3], O

can be defined as the algebra over Fq with basis x1, . . . , x8 and multiplication given by

Table 3.4.1 (blank entries are 0O). Note that x4 + x5 = 1, and for each x ∈ O write

x1 x2 x3 x4 x5 x6 x7 x8x1 . . . . x1 x2 x3 x4x2 . . x1 x2 . . x5 x6x3 . x1 . x3 . x5 . x7x4 x1 . . x4 . x6 x7 .

x5 . x2 x3 . x5 . . x8x6 x2 . x4 . x6 . x8 .

x7 x3 x4 . . x7 x8 . .

x8 x5 x6 x7 x8 . . . .

Table 3.4.1. Octonion algebra

3.5. THE TRANSITIVE FINITE LINEAR GROUPS 53

x := x+ 〈x4 + x5〉. The bilinear form B′ defined on O by

B′(xi, xj) =

1 if i+ j = 9;

0 else

induces an alternating bilinear form B on U0 := 〈x4 + x5〉⊥/〈x4 + x5〉 defined by

B(xi, xj) = B′(xi, xj),

with respect to which x1, x2, x3, x6, x7, x8 form a symplectic basis. The points of the

generalised hexagon H(q) are the one-dimensional subspaces in U0, and the lines of H(q)

are the two-dimensional subspaces 〈x, y〉 where the product of x and y in O is 0O [43,

Section 4.3.8]. Hence, for instance, 〈x1, x2〉 ∈ L while 〈x2, x3〉 ∈ L′.

From the second definition we can obtain explicit descriptions of some elements of

G2(q). Those that we use in our proofs in Subsection 4.3.3 are

r : (x1, x2, x3, x4, x5, x6, x7, x8) 7→ (x1, x3, x2, x4, x5, x7, x6, x8);

s : (x1, x2, x3, x4, x5, x6, x7, x8) 7→ (x2, x1, x6, x5, x4, x3, x8, x7);

and, for any λ ∈ Fq,

E(λ) : x3 7→ x3 + λx2, x7 7→ x7 + λx6,

xi 7→ xi for all other i;

F (λ) : x2 7→ x2 + λx1, x4 7→ x4 + λx3,

x5 7→ x5 + λx3, x8 7→ x8 + λx7,

x6 7→ x6 + λx4 + λx5 + λ2x3,

xi 7→ xi for all other i.

We also use the subgroup

T :=diag(λ, µ, λµ−1, 1, 1, λ−1µ, µ−1, λ−1) λ, µ ∈ F#

q

of GL(8, q). Observe that s, T , and the maps F (λ) for all λ ∈ Fq stabilise 〈x1, x2〉 (and

consequently 〈x1, x2〉), while r, T , and all the E(λ) stabilise 〈x2, x3〉 (and 〈x2, x3〉). This

fact will be useful later.

3.5. The transitive finite linear groups

By Theorem 4.1.1, the quotient-complete graphs which have at least three nontrivial

complete normal quotients arise from transitive finite linear groups. All these groups were

determined by C. Hering in [25], and are presented in Theorem 3.5.1 below.

Theorem 3.5.1. [34, Appendix 1] Let U be a vector space of dimension d over Fp,

where p is prime, and let H ≤ GL(d, p) where H acts transitively on U#. Then H is one


of the types given in Table 3.5.1, or, setting q = pd/n for some divisor n of d, H belongs

to one of the following classes:

(1) H ≤ ΓL(1, q), n = 1;

(2) H D SL(n, q), n ≥ 2;

(3) H D Sp(n, q), d and n even;

(4) H DG2(q), n = 6 and p = 2.

p d H

1 5, 7, 11, 23 2 H ≤ NGL(d,p)(Q8)

2 11, 19, 29, 59 2 H D SL(2, 5)

3 3 4 SL(2, 5) EH ≤ ΓL(2, 9)

4 3 4 H ≤ NGL(4,3)(D8 Q8)

5 2 4 Alt (6)

6 2 4 Alt (7)

7 3 6 SL(2, 13)

Table 3.5.1. Sporadic transitive finite linear groups

3.6. Aschbacher’s classification

The subgroups of the finite classical groups are classified by Aschbacher’s Theorem.

In this result eight classes of subgroups are identified, and it is asserted that an arbitrary

subgroup either belongs to one of these eight types or is almost simple and subject to

certain conditions. Theorem 3.6.1 presents the original statement of this result, which

concerns almost simple classical groups. The classes C1 to C8 are described in Subsection

3.6.1. The maximal subgroups of ΓL(n, q) and ΓSp(n, q) are given in [31], and these are

listed in Theorems 3.6.2 and 3.6.3, respectively.

Theorem 3.6.1 (Aschbacher’s Theorem). [6, 30] Let G0 ≤ PΓL(n, q) be a simple

classical group and G0 ≤ G ≤ PΓL(n, q). If H < G such that H does not contain G0,

then either H is contained in one of the classes C1, . . . , C8, or the following hold:

(1) T ≤ H ≤ Aut (T ) for some nonabelian simple group T (i.e., H is almost simple).

(2) If L is the preimage of T in GL(n, q), then the representation of L is absolutely

irreducible and cannot be realised over a proper subfield of Fq.

(3) If L fixes a form on V (n, q) then G0 is the group PSL(n, q), PSp(n, q), PSU(n, q),

PΩ(n, q), or PΩ±(n, q) corresponding to the form.

3.6.1. Description of the classes Ci. Assume throughout that V = Fnq and φ is a

left-linear form on V which is one of the following: the trivial form, a symplectic form, a

unitary form, or a nondegenerate quadratic form. For each i the description given refers

to the members of Ci.

3.6. ASCHBACHER’S CLASSIFICATION 55

Class C1: Reducible subgroups. Stabilisers of nontrivial proper subspaces U of V , where

U is one of the following: a nondegenerate subspace not isometric to U⊥; a nonsingular,

totally isotropic one-dimensional subspace (which arises if φ is quadratic and p = 2); or a

totally singular subspace.

Class C2: Imprimitive subgroups. Stabilisers of direct sum decompositions V = ⊕ti=1Ui,

where t ≥ 2, Ui = Fmq for each i, n = mt, and one of the following holds:

(i) The Ui’s are pairwise orthogonal and isometric.

(ii) The form φ is quadratic and p is odd, t = 2, dim (Ui) is odd, and U1 and U2 are

orthogonal and similar.

(iii) The dimension n is even, t = 2, U1 and U2 are totally singular of dimension n/2.

If φ is symplectic and n = 4 then p is odd.

Class C3: Superfield subgroups. Stabilisers of vector space structures V = Fn/rqr over an

extension field Fqr of Fq, where r varies over the prime divisors of n.

Class C4: Tensor product subgroups. Stabilisers of tensor product decompositions

V = U ⊗W , where U = Fkq and W = Fm

q with forms φU and φW , respectively, such that

φ = φU ⊗ φW , (U, φU ) and (W,φW ) are non-isometric, n = km, and k 6= m.

Class C5: Subfield subgroups. Stabilisers of vector space structures FqV0, where V0 =

Fnq1/r

and Fq1/r varies over the subfields of Fq of prime index r.

Class C6: Normalisers of symplectic-type subgroups. Groups with extraspecial normal

subgroups R, where R varies over the groups in Table 3.6.1 such that r is a prime not

equal to p and n = rt, R acts irreducibly on V , and one of the following holds:

(i) The group R is of type 1 or type 2, and q = pe where e is the smallest integer

such that pe ≡ 1 (mod |Z(R)|). The form φ is trivial if e is odd, and is unitary

if e is even.

(ii) The form φ is symplectic, q = p, and R is of type 4.

(iii) The form φ is quadratic of plus type, q = p, and R is of type 3.

Class C7: Wreathed tensor product subgroups. Stabilisers of tensor product decompo-

sitions V = ⊗ti=1Ui, where t ≥ 2, Ui = Fm

q with a form φi for each i, n = mt, φ = ⊗ti=1φi,

and the spaces (Ui, φi) are all isometric.

Class C8: Classical subgroups. Groups ΓI(n, q, φ0), with I given in Table 3.3.1 corre-

sponding to φ0, as φ0 varies over the Fq-forms on V subject to one of the following:

(i) The form φ is trivial and φ0 is quadratic with p odd, symplectic, or unitary.

(ii) The form φ is symplectic, p = 2, and φ0 is quadratic with associated bilinear

form φ.

In addition to these eight classes, we define the class C9 to consist of all subgroups

which do not belong to C1 ∪ . . . ∪ C8.


r R T

Type 1 odd R0 · · · R0︸︷︷︸t

, R0 := r1+2+ Sp(2t, r)

Type 2 2 Z4 Q8 · · · Q8︸︷︷︸t

Sp(2t, 2)

Type 3 2 D8 · · · D8︸︷︷︸t

O+(2t, 2)

Type 4 2 D8 · · · D8︸︷︷︸t−1

Q8 O−(2t, 2)

Table 3.6.1. Normalisers of symplectic-type r-groups

3.6.2. Maximal subgroups of ΓL(n, q) and ΓSp(n, q). In the following let τ denote

the Frobenius automorphism of Fq, with the action defined in (3.2.2).

Theorem 3.6.2. IfM is a maximal subgroup of ΓL(n, q) not containing SL(n, q), then

M is one of the following groups:

(C1)([qm(n−m)

].(GL(m, q)×GL(n−m, q))

)⋊Aut (Fq), where 1 < m < n;

(C2) (GL(m, q) ≀ Sym(t))⋊Aut (Fq), where mt = n;

(C3) ΓL(m, qr), where r is prime and mr = n;

(C4) (GL(k, q)⊗GL(m, q))⋊Aut (Fq), where km = n and k 6= m, and the action of τ

is defined with respect to a tensor product basis of Fkq ⊗ Fm

q ;

(C5)(GL(n, q1/r) Zq−1

)⋊Aut (Fq), where n ≥ 2, q is an rth power and r is prime;

(C6) ((Zq−1 R).T )⋊Aut (Fq), where n = rt with r prime, q is the smallest power of

p such that q ≡ 1 (mod r), and R and T are as given in Table 3.6.1;

(C7) (GL(m, q) ≀⊗ Sym(t)) ⋊ Aut (Fq), where mt = n and the action of τ is defined

with respect to a tensor product basis of ⊗ti=1F

mq ;

(C8) ΓO(n, q) or ΓO±(n, q) with q odd, ΓSp(n, q), or ΓU(n, q);

(C9) the preimage of an almost simple group H ≤ PΓL(n, q) satisfying conditions (1)

and (2) of Theorem 3.6.1.

Remark. In case (C1), the symbol[qm(n−m)

]denotes the subgroup of all matrices

with block structure

(Im 0

C In−m

), where C is an arbitrary (n−m)×m matrix over Fq.

Theorem 3.6.3. If M is a maximal subgroup of ΓSp(n, q), then M is one of the

following groups:

(C1.1) ([q − 1].(Sp(m, q)× Sp(n−m, q))⋊Aut (Fq));

(C1.2)([qm/2+mn−3m2/2

].(GL(m, q)×GSp(n− 2m, q))

)⋊Aut (Fq);

(C2.1)((Sp(m, q)t.[q − 1].Sym (t))

)⋊Aut (Fq), where m = n/t;

(C2.2) (GL(m, q).[2]) ⋊Aut (Fq), where m = n/2;

3.6. ASCHBACHER’S CLASSIFICATION 57

(C3.1) (Sp(m, qr).[q − 1])⋊Aut (Fq), where r is prime and m = n/r;

(C3.2) ΓU(m, q2), where m = n/2 and q is odd;

(C4) (GSp(k, q)×GOǫ(m, q))⋊Aut (Fq), where q is odd and m ≥ 3;

(C5)(GSp(n, q1/r) Zq−1

)⋊Aut (Fq)

(C6) (Zq−1 R) .O−(2t, 2), where q ≥ 3 and is prime, and R is of type 4 in Table 3.6.1;

(C7) (GSp(m, q) ≀⊗ Sym(t))⋊Aut (Fq), where qt is odd;

(C8) ΓO±(n, q), where q is even;

(C9) the preimage of an almost simple group H ≤ PΓL(n, q) satisfying conditions (1)

and (2) of Theorem 3.6.1, with L symplectic.

Remark. Assume that V = 〈x1, y1〉 ⊥ . . . ⊥ 〈xn/2, yn/2〉, where (xi, yi) is a hyperbolic

pair for each i, and F#q = 〈µ〉. Then in Theorem 3.6.3, the symbol [q − 1] denotes the

group 〈δµ〉, where

δµ : xi 7→ µxi, yi 7→ yi

for all xi and yi. In case (C1.2) the group[qm/2+mn−3m2/2

]is generated by all maps αi,j(µ)

and βi,j(µ), with 1 ≤ i ≤ m < j ≤ n/2, where

αi,j(µ) : yi 7→ yi + µyj, xj 7→ xj − µxi;

βi,j(µ) : yi 7→ yi + µxj , yj 7→ yj + µxi

and αi,j(µ), βi,j(µ) fix all other basis vectors. Finally, in case (C4), GOǫ can be any of

GO,GO+, or GO−.

CHAPTER 4

Quotient-complete symmetric graphs

4.1. Overview and main results

In this chapter we explore the structure of quotient-complete graphs in the case where

there are at least 3 distinct nontrivial complete normal quotients. That is to say, we

consider Case (1) of Theorem 1 with N = 1. Recall from Definition 2.4.8 that Γ is G-

quotient-complete if it has at least one proper nontrivial G-normal quotient, and if each

of its proper nontrivial G-normal quotients is either a complete graph or an empty graph.

Our technical version of Theorem 2, given below, describes the structure of G when a

G-symmetric, G-quotient complete graph has at least three nontrivial complete G-normal

quotients.

Theorem 4.1.1. Let Γ be a graph and G ≤ Aut (Γ). The graph Γ is connected,

G-symmetric and G-quotient-complete with at least three distinct, nontrivial, complete

G-normal quotients if and only if Γ ∼= Cay(V, S) and G ∼= TV ⋊G0, where:

(1) V = U ⊕ U , with U = Fdp for some prime p and integer d,

(2) G0 = (h, h) | h ∈ H ≤ GL(V ) for some H ≤ GL(U) which is transitive on U#,

and

(3) S is a G0-orbit in V# with S = −S and 〈S〉 = V .

In particular, either H ≤ ΓL(1, q) (where q = pd), or H and S are as in Tables 4.1.1 or

4.1.2. Furthermore, Γ has exactly∣∣CGL(U)(H)

∣∣+2 nontrivial complete G-normal quotients,

each of which has order |U |.

The rest of the chapter is organised as follows: In Section 4.2 we analyse the gen-

eral structure of quotient-complete, symmetric graphs with at least 3 nontrivial complete

normal quotients, and prove Theorem 2 and parts (1), (2) and (3) of Theorem 4.1.1. In

Section 4.3 we consider the different cases which arise from (1), (2) and (3) of Theorem

4.1.1, corresponding to each possible transitive linear group H with H ΓL(1, pd), and

we determine the entries of Tables 4.1.1 and 4.1.2. The diameter 2 graphs in Tables 4.1.1

and 4.1.2 are identified in Section 4.4. Finally, we consider the case where H ≤ ΓL(1, pd)

in Section 4.5 — we do not treat this case completely, but rather we consider the subcases

where H belongs to certain infinite families of subgroups of ΓL(1, pd), and construct the

graphs with the above properties which arise from these.

59

60 4. QUOTIENT-COMPLETE SYMMETRIC GRAPHS

4.1.1. Tables. For Tables 4.1.1 and 4.1.2 we have G ∼= TV ⋊G0 and Γ ∼= Cay(V, S),

with G, V , G0 and S as in Theorem 4.1.1 (1) – (3). The integer n is a divisor of d and

q = pd/n such that H can be viewed as a subgroup of ΓL(n, q). The graphs marked with a

“†” have diameter 2. In Table 4.1.1, Vλ, Vind and α(H) are as defined in (4.3.1) to (3.2.1),

respectively, while A(λ), C(λ), Sλ, SL, and SL′ are as in (4.3.3) to (4.3.7), respectively.

Table 4.1.2 lists the conjugacy class representatives of the possible groups H and the

valency of the graphs Γ, which are grouped according to isomorphism class. For instance,

in line 5 of Table 4.1.2, the group H = NGL(2,5)(Q8) has 42 orbits of length 96 which

yield connected Cayley graphs, and all these graphs are isomorphic; in lines 42 and 43

the group H has 9 orbits of length 640, and the corresponding graphs are divided into

two isomorphism classes of sizes 3 and 6. Unless otherwise stated, distinct lines of Table

4.1.2 correspond to distinct isomorphism classes of graphs. Wherever it is known, we use

the symbol “⊂” to denote subgraphs; i.e., “Γ(i) ⊂ Γ(j)” means that the graphs in line i

are subgraphs of graphs in line j. We do not check exhaustively for all such relationships,

and wherever indicated, Γ(i) ⊂ Γ(j) follows from the fact that the H-orbit corresponding

to Γ(i) is contained in the H-orbit corresponding to Γ(j). Note also that A(1), A(3) and

A(4) are maximal subgroups of NGL(4,3)(D8 Q8). All entries of Table 4.1.2 were obtained

using Magma [1].

(n, q) H S val (Γ)†1 n ≥ 3 H D SL(n, q) Vind (qn − 1) (qn − q)

†2

⋃λ′∈A(λ) V

#λ′ , λ /∈ Fix(α(H)) |A(λ)| (qn − 1)

†3 n even H D Sp(n, q) S0 q

(q6 − 1

) (q4 − 1

)†4

⋃λ′∈C(λ) Sλ′ , λ ∈ F#

q |C(λ)| q5(q6 − 1

)

†5

⋃λ′∈A(λ) V

#λ′ , λ /∈ Fix(α(H)) |A(λ)| (qn − 1)

†6 n = 6, q even H DG2(q) SL q

(q6 − 1

) (q2 − 1

)†7 SL′ q3

(q6 − 1

) (q2 − 1

)†8

⋃λ′∈C(λ) Sλ′ , λ ∈ F#

q |C(λ)| q5(q6 − 1

)

†9

⋃λ′∈A(λ) V

#λ′ , λ /∈ Fix(α(H)) |A(λ)| (qn − 1)

Table 4.1.1. H as in Theorem 3.5.1 (2), (3), (4)

4.2. Examples and general structure

We first look at the general structure of quotient-complete symmetric graphs. Example

2.4.1 gives an infinite family of quotient-complete symmetric graphs with exactly one

nontrivial complete normal quotient, while Examples 2.4.2 and 2.4.3 give infinite families of

examples with exactly two nontrivial complete normal quotients. It should be emphasised

that the property of quotient-completeness is dependent on the choice of the group G. For

instance, recall from Example 2.4.2 that the full automorphism group of Σ = Kn ×Kn is

H = Sym(n) ≀ Z2. However, Σ does not have an H-normal complete quotient.

4.2. EXAMPLES AND GENERAL STRUCTURE 61

i (n, q) H = H(i) val (Γ(i)) Class size Comments

1 (2, 5) H(1) < H(4), index 4 24 20

2 H(2) < H(4), index 4 24 20 H(2) ≇ H(1)

3 H(3) < H(4), index 2 48 10†4 NGL(2,5)(Q8) 96 5

5 (2, 7) H(5) < H(7), index 3 48 42

6 H(6) < H(7), index 3 48 42 H(6) ≇ H(5)†7 NGL(2,7)(Q8) 144 14

8 (2, 11) H(8) < H(9), index 2 120 110

9 NGL(2,11)(Q8) 240 55

10 (2, 23) NGL(2,23)(Q8) 528 506

11 (2, 11) H(11) < H(13), index 5 120 110 H(11) ≇ H(8)

12 H(12) < H(13), index 5 120 110 H(12), H(8) conjugate under

GL(2, 11); Γ(12) ∼= Γ(8)

13 NGL(2,11)(SL(2, 5)) 600 22

14 (2, 19) NGL(2,19)(SL(2, 5)) 1080 114

15 (2, 29) H(15) < H(16), index 2 840 812

16 NGL(2,29)(SL(2, 5)) 1680 406

17 (2, 59) NGL(2,59)(SL(2, 5)) 3480 3422

Table 4.1.2. H as in Table 3.5.1

The rest of this section considers quotient-complete, symmetric graphs with at least

three nontrivial complete normal quotients. Two families of such graphs, having k non-

trivial complete normal quotients with k greater than or equal to some given prime power,

are given in the next example.

Example 4.2.1. Let U = Fnq for some prime power q and integer n, c = qn, and

V = U⊕U . In each case below, Γ = Cay(V, S) and G = TV ⋊G0, where G0 = (h, h) | h ∈H ≤ GL(U), and H and S are as given.

Suppose first that n = 1 and c = q ≥ 3. Take H = GL(1, q) = GL(U) (noting

that H ∼= Zq−1) and S =(u, λu) | u ∈ U#

for some λ ∈ F#

q . Then S is an orbit

of G0, so Γ is G-arc-transitive. The graph Γ is disconnected since 〈S〉 6= V and each

connected component is isomorphic toKq. By Lemma 3.2.2 the minimal normal subgroups

of G are precisely the groups TW , where W is 0U ⊕ U or (u, ηu) | u ∈ U for some

η ∈ Fq, and exactly one of these minimal normal subgroups is T〈S〉. We have ΓT〈S〉∼= Kq,

while ΓTW∼= Kq for the other q minimal normal subgroups TW . Since distinct minimal

normal subgroups yield distinct normal quotients, Γ has exactly q+1 nontrivial G-normal

quotients. So Γ is G-quotient-complete with q distinct, nontrivial, complete G-normal

quotients.

Now suppose that n = 2, and take H = GL(2, q) and S = (u, v) | u, v linearly

independent in U. Again S is a G0-orbit so Γ is G-arc-transitive; this time 〈S〉 = V so

Γ is connected. As in (1) the minimal normal subgroups of G are the groups TW with


i (n, q) H val (Γ(i)) Class size Comments

18 (2, 9) A(1) < NΓL(2,9)(SL(2, 5)), 80 6 Γ(18) ⊂ Γ(31)

index 6; A(1) GL(2, 9)†19 160 24 Γ(19) ⊂ Γ(33)

20 160 12 Γ(20) ⊂ Γ(26) ⊂ Γ(33);

21 A(2) < A(4), index 2; 80 6 Γ(21) ∼= Γ(27) ⊂ Γ(31)

A(2) GL(2, 9) Γ(21) ⊂ Γ(42)

22 240 8 Γ(22) ∼= Γ(44) ⊂ Γ(47);

Γ(22) ⊂ Γ(26),Γ(49)†23 240 6 Γ(23) ⊂ Γ(29) ⊂ Γ(33);

Γ(23) ⊂ Γ(49)†24 240 4 Γ(24) ⊂ Γ(30) ⊂ Γ(33);

Γ(24) ⊂ Γ(47),Γ(49)

25 240 6 Γ(25) ⊂ Γ(33),Γ(42)

26 A(3) < NΓL(2,9)(SL(2, 5)), 480 12 Γ(26) ∼= Γ(28) ∼= Γ(32)

index 2; A(3) ≤ GL(2, 9) Γ(26) ⊂ Γ(33)

27 A(4) < NΓL(2,9)(SL(2, 5)), 80 6 Γ(27) ∼= Γ(21)

index 2; A(4) GL(2, 9)†28 480 4 Γ(28) ∼= Γ(26) ∼= Γ(32)

†29 480 6 Γ(29) ⊂ Γ(33)

†30 480 2 Γ(30) ⊂ Γ(33)

†31 NΓL(2,9)(SL(2, 5)) 160 3

†32 480 4 Γ(32) ∼= Γ(26) ∼= Γ(28)

†33 960 4

34 (4, 3) B(1) < NGL(4,3)(D8 Q8), B(1), A(2) conjugate

index 16 under GL(4, 3)

35 B(2) < B(3), index 2 80 6 Γ(35) ∼= Γ(38) ⊂ Γ(44)

⊂ Γ(47)

36 160 12 Γ(36) ⊂ Γ(39) ⊂ Γ(42)

37 160 24 Γ(37) ⊂ Γ(40) ⊂ Γ(46)

⊂ Γ(49)

38 B(3) < B(5), index 6 80 6 Γ(38) ⊂ Γ(44) ⊂ Γ(47)†39 320 6 Γ(39) ⊂ Γ(42)

40 320 12 Γ(40) ⊂ Γ(46) ⊂ Γ(49)†41 B(4) < NGL(4,3)(D8 Q8), 160 3 Γ(41) ⊂ Γ(47)

index 6†42 640 3 Γ(42) ∼= Γ(45) ∼= Γ(48)

†43 640 6 Γ(43) ⊂ Γ(49)

44 B(5) < NGL(4,3)(D8 Q8), 240 2 Γ(44) ∼= Γ(22) ⊂ Γ(26),Γ(49);

index 2 Γ(44) ⊂ Γ(47)†45 640 3 Γ(45) ∼= Γ(42) ∼= Γ(48)

†46 1920 2 Γ(46) ⊂ Γ(49)

†47 NGL(4,3)(D8 Q8) 480 1

†48 640 3 Γ(48) ∼= Γ(42) ∼= Γ(45)

†49 3840 1

Table 4.1.2 (cont.). H as in Table 3.5.1

W = 0U ⊕ U or (u, ηu) | u ∈ U for η ∈ Fq. For each minimal normal subgroup TW ,

the graph ΓTWis connected and G0 acts transitively on V (ΓTW

) \ W, so ΓTWis the


i (n, q) H val (Γ(i)) Class size Comments

†50 (4,2) Alt (6) 120 1 Γ(38) ⊂ Γ(40)

†51 90 1 Γ(39) ⊂ Γ(40)

†52 Alt (7) 210 1

53 (6,3) SL(2, 13) 728 6

54 2184 6

55 2184 2 possibly isomorphic to Γ(56)


57 2184 6

58 2184 24

59 2184 12

60 2184 12

61 2184 24 possibly isomorphic to Γ(62), Γ(63)



64 2184 24

65 2184 12

66 2184 12

67 2184 12



70 2184 8

71 2184 12

72 2184 8

Table 4.1.2 (cont.). H as in Table 3.5.1

complete graph Kc. Therefore Γ is G-quotient-complete with q + 1 =√c + 1 distinct,

nontrivial complete G-normal quotients.

The remainder of this section is devoted to the proof of Theorem 2 and part of Theorem

4.1.1. Recall that if ΓN is a complete G-normal quotient of Γ, the group GΓN acts 2-

transitively on V (ΓN ). For the rest of the section we assume that the following hypothesis

holds.

Hypothesis 4.2.2. The graph Γ is G-arc-transitive and G-quotient-complete with

exactly k ≥ 3 distinct nontrivial, complete G-normal quotients.

Lemma 4.2.3. Let Γ be a graph and let G ≤ Aut (Γ), such that Γ and G satisfy

Hypothesis 4.2.2. Let M and N be nontrivial normal subgroups of G which are intransitive

on V (Γ), such that ΓM and ΓN are complete graphs and ΓM 6= ΓN . Then the following

hold:

(1) If 1 6= K ⊳G and K ≤M , then ΓK = ΓM .

(2) M ∩N = 1 and MN is transitive on V (Γ).

(3) If M is a minimal normal subgroup of G, then M ∼=MΓN = soc(GΓN ).


Proof. Let 1 6= K⊳G with K ≤M . Since ΓM is complete, the quotient ΓK is not an

empty graph, so by Definition 2.4.8 the graph ΓK is complete. Hence GΓK is 2-transitive

and is thus primitive. Since MΓK ⊳GΓK , it follows that either MΓK is transitive or MΓK

lies in the kernel of the action of G on V (ΓK). If MΓK is transitive then MΓ must also be

transitive, contrary to the assumption. So MΓK must lie in the kernel of the action of G

on V (ΓK), and since K ≤M it follows that the K-orbits and M -orbits in V (Γ) coincide.

In particular, ΓK = ΓM , which proves (1).

Suppose thatM ∩N 6= 1. ThenM ∩N is a nontrivial normal subgroup of G contained

in both M and N , and by (1) we have ΓM = ΓM∩N = ΓN , a contradiction. Therefore

M ∩N = 1. Now MN EG and M,N ≤ MN . If MN is intransitive on V (Γ) then again

by (1) we have ΓM = ΓMN = ΓN , a contradiction. So MN is transitive on V (Γ), which

proves (2).

Suppose that M is minimal normal in G. Let N denote the kernel of the action of G

on V (ΓN ), so that MΓN = MN/N . Since M ∩ N = 1 by (2), it follows that MΓN ∼= M

and MN ∼= M × N . Observe that each subgroup R satisfying N ≤ R ≤ MN has the

form R = (R∩M)N . If MΓN is not minimal normal in G, there exists 1 6= LEGΓN with

L < MΓN . Then L = L′/N for some L′ E G with N < L′ < MN . By the observation

above we have L′ = (L′ ∩M)N . Hence 1 6= L′ ∩M ⊳G and L′ ∩M 6=M , contrary to the

minimality of M . So MΓN must be minimal normal in GΓN , and hence M = soc(GΓN ) by

Theorem 1.3.3. Therefore (3) holds.

Proposition 4.2.4. Let Γ be a graph and let G ≤ Aut (Γ), such that Γ and G satisfy

Hypothesis 4.2.2. Let L, M and N be vertex-intransitive minimal normal subgroups of G,

such that ΓL, ΓM and ΓN are complete and pairwise distinct. Then the following hold:

(1) L ∼=M ∼= N and L, M and N are elementary abelian;

(2) |L| = |M | = |N | =: c and ΓL∼= ΓM

∼= ΓN∼= Kc;

(3) |V (Γ)| = c2; and

(4) soc(G) =M ×N and acts regularly on V (Γ).

Proof. The first statement follows easily from Lemma 4.2.3 (3) since we then have

M ∼=MΓL = soc(GΓL) = NΓL ∼= N, and similarly L ∼=M and L ∼= N . By Theorem 1.3.3,

soc(GΓL) is either regular and elementary abelian, or is nonregular, nonabelian and simple.

Suppose that soc(GΓL) is nonregular, nonabelian and simple, and let T := soc(GΓL). Since

MΓL = NΓL = T as observed above, it follows that (MN)ΓL = T and MN ∼= T × T . So

the kernel K of the action of MN on V (ΓL) is a normal subgroup of MN with K ∼= T .

The only such normal subgroups of MN are M and N . Each is impossible since MΓL and

NΓL are both nontrivial. Therefore T is elementary abelian, and hence so are L, M and

N . This proves (1).

Let c := |T | = |M | = |N |. By Lemma 4.2.3 (3), the groups MΓN , MΓL and NΓM

are minimal normal subgroups of GΓN , GΓL and GΓM , respectively, and are abelian and


regular by Theorem 1.3.3. Thus |V (ΓN )| =∣∣MΓN

∣∣ = |M | = c, and similarly |V (ΓM )| =|V (ΓL)| = c. It follows from Hypothesis 4.2.2 that ΓL

∼= ΓM∼= ΓN

∼= Kc, which proves

(2).

Now let α ∈ V (Γ). Since MΓN is regular, we have (M × N)αN = N , and likewise

(M ×N)αM = M . Hence (M ×N)α ≤ (M ×N)αM ∩ (M ×N)αN = M ∩N = 1, and so

M ×N is semiregular. Moreover, M ×N is transitive by Lemma 4.2.3 (2), so M ×N is

regular. Therefore |V (Γ)| = |M ×N | = c2, and (3) holds.

Finally, observe that the group M × N is self-centralising since it is both transitive

and elementary abelian. Any minimal normal subgroup of G is then contained in M ×N ,

so M ×N = soc(G). Therefore (4) holds.

Proposition 4.2.5. Let Γ be a graph and let G ≤ Aut (Γ), such that Γ and G satisfy

Hypothesis 4.2.2. Then there exists an integer d and prime p such that G ≤ AGL(V ) and

Γ ∼= Cay(V, S), where V = U ⊕ U , U = Fdp, and S is a G0-orbit in V # with S = −S.

Furthermore:

(1) G ∼= TV ⋊ G0, where G0 = (h, h) | h ∈ H ≤ GL(V ) for some H ≤ GL(U)

which is transitive on U#.

(2) If Γ is not connected then Γ ∼= c.Kc where c = pd, and the component containing

0V is Cay(W,W#) where W is a G0-invariant diagonal subspace distinct from

(u, u) | u ∈ U.

Proof. Let K := soc(G). By Proposition 4.2.4 (4) the group KΓ is regular, so by

Theorem 2.2.4 we have Γ ∼= Cay(K,S) for some Cayley subset S of V — that is, (using

additive notation) S ⊆ K# with S = −S. By Theorem 1.1.8 we have G ≤ K ⋊Aut (K),

and since Γ is G-arc-transitive, S is a G0-orbit. Furthermore, by Proposition 4.2.4 (1)

and (4), we have K ∼= T × T for some elementary abelian group T , so K can be identified

with V = Fdp ⊕ Fd

p for some prime p and integer d with pd = |T |. Under this identificationAut (K) corresponds to GL(V ), and if we further identify V with TV , the group G then

corresponds to a subgroup of AGL(V ). In particular, G ∼= TV ⋊G0 where G0 ≤ GL(V ).

To complete the proof of (1) it remains to show that G0 has the given form. Let

U = Fdp. Observe that under the identification above the groups TU⊕0U and T0U⊕U

correspond to the subgroups M and N in Proposition 4.2.4 (4), and hence are minimal

normal in G. Write M = TU⊕0U and N = T0U⊕U , and note that K = M × N by

Proposition 4.2.4 (4). Since k ≥ 3 there exists another minimal normal subgroup L of

G which is distinct from M and N . By Lemma 3.2.2 the corresponding subspace VL is

G0-invariant and diagonal; without loss of generality assume that VL = (u, u) | u ∈ U.Then also by Lemma 3.2.2 we have G0 = (h, h) | h ∈ H for some H ≤ GL(U) which

acts irreducibly on U#. Since GΓL ∼= L.GΓL0 and is arc-transitive, it follows that GΓL

0 is

transitive on V #L . Hence H is transitive on U#, and (1) holds.


Suppose now that Γ is not connected. Then 〈S〉 6= V . Thus by Lemma 3.2.3 and the

previous paragraph, 〈S〉# = S and 〈S〉 is a G0-invariant diagonal subspace distinct from

VL. (Indeed, S is not U ⊕0U, 0U ⊕U or VL since ΓM , ΓN and ΓL are assumed to be

complete.) If we write W = 〈S〉 then Γ ∼= Cay(V,W#). Clearly the connected component

containing 0V is Cay(W,W#), which is a complete graph of order |W | = |U | = pd. Since

|V (Γ)| = p2d and Γ is G-arc-transitive, it has pd connected components which are all

isomorphic to Cay(W,W#). Therefore Γ ∼= c.Kc where c = pd. This proves (2).

Corollary 4.2.6. Let Γ be a graph and let G ≤ Aut (Γ), such that Γ and G satisfy

Hypothesis 4.2.2, and let H and c = pd be as in Proposition 4.2.5. Then∣∣CGL(U)(H)

∣∣ =c1/ℓ − 1 and k = c1/ℓ + δ for some integer ℓ ≥ 1, where

δ =

1 if Γ is connected,

0 if Γ is not connected.

Proof. Since H acts irreducibly on U#, it follows from Schur’s Lemma (see, for

instance, [27, Lemma IX.1.10]) that∣∣CGL(U)(H)

∣∣ = pm − 1 for some divisor m of d, so∣∣CGL(U)(H)

∣∣ = c1/ℓ − 1 where ℓ = d/m. Recall from Lemma 3.2.2 that the minimal

normal subgroups of G are the subgroups N ≤ TV where VN is U ⊕0U, 0U ⊕U , or a

diagonal subspace (u, uϕ) | u ∈ U, ϕ ∈ CGL(U)(H). By Lemma 4.2.3 (1) each nontrivial

normal quotient of Γ is a quotient graph relative to a vertex-intransitive minimal normal

subgroup N ; conversely, it is clear from Lemma 3.2.2 that all minimal normal subgroups of

G are vertex-intransitive, and distinct minimal normal subgroups yield distinct nontrivial

normal quotients. So there is a one-to-one correspondence between the set of minimal

normal subgroups of G and the set of nontrivial G-normal quotients of Γ, and Γ has

exactly∣∣CGL(U)(H)

∣∣ + 2 = c1/ℓ + 1 distinct nontrivial G-normal quotients. Therefore

k ≤ c1/ℓ + 1.

If Γ is connected then so is ΓN for all minimal normal subgroups N of G, so by

Definition 2.4.8 the graphs ΓN are all complete and k = c1/ℓ + 1 in this case. Suppose

that Γ is not connected. Then by Proposition 4.2.5 we have Γ ∼= c.Kc, and in particular

Γ ∼= Cay(V, S), where T〈S〉 is minimal normal in G. The connected components of Γ are

the cosets of 〈S〉 in V . If N = T〈S〉 then ΓN∼= Kc; for any other minimal normal subgroup

N 6= T〈S〉 the subspace VN intersects each coset of 〈S〉 nontrivially, so ΓN∼= Kc. Therefore

k = c1/ℓ for disconnected Γ, as asserted.

Lemma 4.2.7. Let Γ = Cay(V, S) and G = TV ⋊G0, where

(1) V = U ⊕ U with U = Fnq for some prime power q and integer n,

(2) G0 = (h, h) | h ∈ H ≤ GL(V ) for some H ≤ GL(U) which is transitive on U#,

and

(3) S is a G0-orbit in V# with S = −S and 〈S〉 = V .


Then Γ is a connected graph satisfying Hypothesis 4.2.2 with k =∣∣CGL(U)(H)

∣∣ + 2, and

each nontrivial G-normal quotient of Γ is a complete graph of order qn.

Proof. It follows from condition (3) that Γ is undirected, connected, and G-arc-

transitive. The group H is irreducible since it is transitive on U#, so U ⊕ 0U and

0U ⊕ U are minimal G0-invariant subspaces, and TU⊕0U and T0U⊕U are minimal

normal subgroups. By Lemma 3.2.2 the other minimal normal subgroups of G are precisely

the subgroups TW with W = (u, uϕ) | u ∈ U, for all ϕ ∈ CGL(U)(H). As was observed

in the proof of Corollary 4.2.6 above, Lemmas 3.2.2 and 4.2.3 (1) imply that the set of

minimal normal subgroups of G is in one-to-one correspondence with the set of nontrivial

G-normal quotients of Γ. Since Γ is connected all nontrivial normal quotients are complete

graphs, so k =∣∣CGL(U)(H)

∣∣ + 2.

Let N be a minimal normal subgroup of G. The group GΓN is transitive on V (ΓN )

since G is transitive on V (Γ) by assumption. We claim that the stabiliser(GΓN

)N

of N

in the action of G on V (ΓN ) is transitive on V (ΓN )\N. Indeed, the subspace VN is G0-

invariant by Lemma 3.2.2, so (G0)ΓN ≤

(GΓN

)N. Moreover, if M is minimal normal in G

with M 6= N , the elements of VM constitute a complete set of coset representatives for VN

in V . It follows from Lemma 3.2.2 and the transitivity of H that G0 is transitive on V #M ,

which implies that (G0)ΓN is transitive on V (ΓN ) \ N. Therefore

(GΓN

)N

is transitive

on V (ΓN ) \ N as claimed, and hence GΓN acts 2-transitively on V (ΓN ). Since ΓN is

connected for all minimal normal subgroups N it then follows that all nontrivial normal

quotients ΓN are complete. Therefore Γ is G-quotient-complete with k =∣∣CGL(U)(H)

∣∣ +2.

Proof of Theorem 2. The first part follows immediately from Proposition 4.2.5 (1)

with c = pd, so it remains only to show that either k = c and Γ ∼= c.Kc, or k ≤ √c+ 1.

Recall from Proposition 4.2.5 that Γ ∼= Cay(V, S) where V = Fdp⊕Fd

p and S ⊆ V # is a

G0-orbit with S = −S. It follows from Corollary 4.2.6 that there is an integer ℓ ≥ 1 with

k = c1/ℓ + 1 if Γ is connected and k = c1/ℓ otherwise. If ℓ ≥ 2 then k ≤ √c + 1, so we

may assume that ℓ = 1 and hence k = c or k = c+ 1 according as Γ is connected or not.

Suppose that k = c+ 1. Then Γ is connected, and all nontrivial G-normal quotients of Γ

are complete. So k is equal to the number of minimal normal subgroups of G, which by

Lemma 3.2.2 is less than or equal to the number of G0-invariant subspaces of V#, which

in turn is less than or equal to the number of G0-orbits in V #. Since |V | = c2 and H

is transitive on U#, there are at most c + 1 G0-orbits in V # and each orbit has length

at least c − 1. Thus G0 has exactly c + 1 orbits in V #, each of length c − 1, and hence

k is equal to the number of G0-orbits in V #. This implies that each G0-orbit generates

U ⊕ 0U, 0U ⊕ U , or a G0-invariant diagonal subspace. So 〈S〉 must be one of these

subspaces and thus Γ is not connected, a contradiction. Therefore k = c and Γ is not

connected, and by Proposition 4.2.5 (2) we have Γ ∼= c.Kc, which completes the proof.


4.3. Connected quotient-complete symmetric graphs

In this section we determine most of the connected graphs which satisfy Hypothesis

4.2.2. In particular, we identify which of the graphs with form as given in Proposition

4.2.5 and Lemma 4.2.7 are connected, provided that H ΓL(1, q) (see Theorem 3.5.1).

It follows from Proposition 4.2.5 and Lemma 4.2.7 that a connected graph Γ and group

G ≤ Aut (Γ) satisfy Hypothesis 4.2.2 if and only if Γ and G are as described in Lemma

4.2.7 (1) – (3). So to get the desired connected graphs Γ we only need to determine the

G0-orbits S ⊆ V # such that S = −S and 〈S〉 = V .

Notation. Throughout this section U , V , G, G0 and H are as in Lemma 4.2.7 (1)

and (2). Set

V∞ := 0U ⊕ U,

and for each λ ∈ Fq,

Vλ := (u, λu) | u ∈ U. (4.3.1)

Thus, in particular, V0 = U ⊕ 0U.

In cases (2) – (4) of Theorem 3.5.1, there exists a homomorphism σ : H → Aut (Fq),

and it will be shown that∣∣CGL(U)(H)

∣∣ = |Fix(σ(H))| − 1. So if c = qn, Corollary 4.2.6

gives

k = c1/ℓ + δ =∣∣CGL(U)(H)

∣∣+ 1 + δ = |Fix(σ(H))| + δ ≤ q + 1.

4.3.1. SL(n, q)EH ≤ ΓL(n, q), n ≥ 3. Recall that if n ≥ 3 then SL(n, q) is transitive

on the set

Vind := (u,w) | u,w ∈ U linearly independent, (4.3.2)

whereas SL(2, q) is not. Moreover SL(2, q) = Sp(2, q), and this case is considered in

Subsection 4.3.2 along with the other symplectic groups. Thus from now on we assume

that n ≥ 3.

Recall from Section 3.2 that ΓL(n, q) = GL(n, q)⋊Aut (Fq), so the group H consists of

elements h = ρ(h)σ(h) where ρ(h) ∈ GL(n, q) and σ(h) ∈ Aut (Fq). Note that SL(n, q) ≤ρ(H) ≤ GL(n, q) and σ(H) ≤ Aut (Fq). So G ≥ K := (h, h) | h ∈ SL(n, q), which has

orbits Vind, V#∞ , and V #

λ for all λ ∈ Fq. Hence the G0-orbits in V # are Vind, V#∞ , and

⋃λ′∈A(λ) V

#λ′ , where

A(λ) := λσ(H). (4.3.3)

It follows from Lemma 3.2.3 that the only G0-orbits S with 〈S〉 = V are Vind and⋃

λ′∈A(λ) V#λ′ for all λ /∈ Fix(σ(H)), and there is a one-to-one correspondence between

the set of all remaining orbits and the set of minimal normal subgroups of G. We thus

have the following:

4.3. CONNECTED QUOTIENT-COMPLETE SYMMETRIC GRAPHS 69

Proposition 4.3.1. Let Γ be a graph and G = TV ⋊G0, where V , TV and G0 are as

in Lemma 4.2.7 (1) and (2) with n ≥ 3 and SL(n, q)EH ≤ ΓL(n, q). Then Γ is connected,

G-symmetric and G-quotient-complete if and only if Γ ∼= Cay(V, S) where S is one of the

following:

(1) Vind;

(2)⋃

λ′∈A(λ) V#λ′ for some λ /∈ Fix(σ(H)).

Furthermore, such a graph Γ has exactly k = |Fix(σ(H))|+1 nontrivial complete G-normal

quotients.

Proof. This follows immediately from Proposition 4.2.5, Lemma 4.2.7, and the dis-

cussion above.

Observe that for all Γ in Proposition 4.3.1, we have k ≤ q+1 ≤ qn/3 +1, which is less

than the upper bound√qn + 1 of Theorem 2.

4.3.2. Sp(n, q)EH ≤ ΓL(n, q), n even. Let f be the corresponding symplectic form

on U . Recall that g ∈ ΓL(U) normalises Sp(n, q) if and only if there exist γ(g) ∈ F#q and

σ(g) ∈ Aut (Fq), both independent of u and w, such that

f(ug, wg) = γ(g)f(u,w)σ(g) ∀ u,w ∈ U.

Moreover, σ(g) is the same as that defined in (3.2.1). For each λ ∈ Fq define

C(λ) :=γ(h)λσ(h) | h ∈ H

. (4.3.4)

We shall abuse notation slightly and define

Sλ := (u,w) ∈ Vind | f(u,w) = λ, (4.3.5)

where A(λ) is as in (4.3.3) for each λ and Vind is as in (4.3.2). Note that the sets Sλ in

this case are different from the Sλ defined in (3.3.1). Clearly⋃

λ∈FqSλ = Vind, and it is

known that S0 is nonempty if and only if n ≥ 4.

Lemma 4.3.2. The G0-orbits in V # are V #∞ ,

⋃λ′∈A(λ) V

#λ′ , and

⋃λ′∈C(λ) Sλ′ for all

λ ∈ Fq.

Proof. Let K = (h, h) | h ∈ Sp(n, q) and (u,w) ∈ V #. If u,w is linearly

dependent in U then (u,w)K is either V #∞ or V #

λ for some λ ∈ Fq. Suppose that u,wis linearly independent and let α ∈ Fq satisfy α = f(u,w). Then clearly (u,w)K ⊆ Sα,

and it remains to show that Sα ⊆ (u,w)K . Let (u′, w′) ∈ Sα. Then there is a linear

map g : 〈u,w〉 → 〈u′, w′〉 such that ug = u′ and wg = w′, and g is an isometry from

〈u,w〉 to 〈u′, w′〉. Thus by Witt’s Lemma g extends to an isometry g′ on U . That is,

(u′, w′) = (ug′, wg′) where g′ ∈ Sp(n, q) ≤ H, so (u′, w′) ∈ (u,w)K . Hence Sα ⊆ (u,w)K ,

and thus Sα is a K-orbit.


It follows immediately that V #∞ ,⋃

λ′∈A(λ) V#λ′ , and

⋃λ′∈C(λ) Sλ′ for all λ ∈ Fq are the

G0-orbits in V#.

Proposition 4.3.3. Let Γ be a graph and G = TV ⋊G0, where V , TV and G0 are as in

Lemma 4.2.7 (1) and (2), with n even and Sp(n, q)EH ≤ ΓL(n, q). Then Γ is connected,

G-symmetric and G-quotient-complete if and only if Γ ∼= Cay(V, S), where S is one of the

following:

(1)⋃

λ′∈A(λ) V#λ′ for some λ /∈ Fix(σ(H));

(2)⋃

λ′∈C(λ) Sλ′ for any λ ∈ Fq.


quotients. In particular, k =√qn + 1 if and only if n = 2 and σ(H) = 1.

Proof. This follows immediately from Proposition 4.2.5, Lemma 4.2.7, and the dis-

cussion above.

4.3.3. H DG2(q), n = 6 and q even. Recall from Section 3.4 that G2(q) ≤ Sp(6, q).

Let P, L, L′ and N be as defined in Section 3.4, and let Sλ be as in (4.3.5) (with n = 6

and q even) for all λ ∈ Fq. Define

SL := (u,w) | u,w ∈ U ; 〈u,w〉 ∈ L (4.3.6)

and

SL′ := (u,w) | u,w ∈ U ; 〈u,w〉 ∈ L′, (4.3.7)

and for λ ∈ Fq let A(λ) be as in (4.3.3) and C(λ) be as in (4.3.4). Recall that G2(q) ≤Sp(6, q), and is transitive on L, L′ and N by Lemma 3.4.1 (1), so it follows that SL, SL′

and Sλ for any λ ∈ F#q are K-invariant for K := (h, h) | h ∈ G2(q). In fact we show

that:

Lemma 4.3.4. The K-orbits in V # are V #∞ , V #

0 , V #λ and Sλ for each λ ∈ F#

q , SL,

and SL′. Hence the G0-orbits in V # are V #∞ , V #

0 , SL, SL′,⋃

λ′∈A(λ) V#λ′ and

⋃λ′∈C(λ) Sλ′

for all λ ∈ F#q .

To prove Lemma 4.3.4 we need to determine the group induced on a two-dimensional

subspace W of U by the setwise stabiliser in G2(q) of W . We denote this group by

G2(q)WW.

Lemma 4.3.5. If W is a two-dimensional subspace of U then G2(q)WW ≥ SL(2, q).

Moreover, if W is nondegenerate then G2(q)WW = SL(2, q).

Proof. The first statement is a direct consequence of Lemma 3.4.1, and knowledge

of |L|, |L′| and |N |. The second statement follows from the fact that for nondegenerate

W we have G2(q)WW ≤ Sp(2, q) = SL(2, q) (since G2(q) ≤ Sp(6, q)).


Lemma 4.3.6. If W is a totally isotropic two-dimensional subspace of U then

G2(q)WW = GL(2, q).

Proof. Let 〈x1, . . . , x8〉 be a basis for the octonion algebra with multiplication as

given in Table 3.4.1, and let E(λ), F (λ), T , r and s be the elements of G2(q) as defined in

Section 3.4.1. Recall that 〈x1, x2〉 ∈ L and 〈x2, x3〉 ∈ L′. To simplify notation we identify

each coset xi of 〈x4 + x5〉 with its representative xi. As we observed earlier, the stabiliser

of 〈x1, x2〉 contains the subgroup T , the map s, and all elements F (λ). Relative to the

ordered basis x1, x2 (and acting on row vectors), the maps s, F (λ), and the elements of

T acting on 〈x1, x2〉 induce the matrices

(0 1

1 0

),

(1 λ

0 1

), and

(λ′ 0

0 µ

),

respectively, for all λ ∈ Fq and λ′, µ ∈ F#q . Hence 〈T, s, F (λ) | λ ∈ Fq〉 induces the group

GL(2, q) on 〈x1, x2〉. Similarly, the stabiliser of 〈x2, x3〉 contains T , r, and all elements

E(λ), which respectively induce the matrices

(µ 0

0 λ′µ−1

),

(0 1

1 0

), and

(1 0

λ 1

)

for all λ′, µ ∈ F#q and all λ ∈ F#

q , on 〈x2, x3〉 relative to the ordered basis x2, x3. So

〈T, r,E(λ) | λ ∈ Fq〉 also induces GL(2, q) on 〈x2, x3〉.

Proof of Lemma 4.3.4. Let (u,w) ∈ V #. If u,w is linearly dependent then

(u,w)K is V #∞ or V #

λ for some λ ∈ Fq. Suppose that u,w is linearly independent.

If 〈u,w〉 is totally isotropic then G2(q)〈u,w〉〈u,w〉 = GL(2, q) by Lemma 4.3.6, so G2(q)〈u,w〉

is transitive on the set of ordered bases of 〈u,w〉. Since G2(q) is transitive on L and L′ it

follows that K is transitive on SL and SL′. If 〈u,w〉 is nondegenerate then G2(q)〈u,w〉〈u,w〉 =

SL(2, q) = Sp(2, q), so that the orbitals of G2(q)〈u,w〉 in 〈u,w〉 are the sets

(u′, w′) ∈ Sλ | u′, w′ ∈ 〈u,w〉

for each λ ∈ F#q . Again, since G2(q) is transitive on the set of all nondegenerate two-

dimensional subspaces, we have (u,w)K = Sλ where λ is the value of the symplectic form

at (u,w). Hence the K-orbits in V # are V #∞ , V #

0 , V #λ and Sλ for each λ ∈ F#

q , SL, and

SL′ . It follows that the G0-orbits are V#∞ , V #

0 , SL, SL′,⋃

λ′∈A(λ) V#λ′ and

⋃λ′∈C(λ) Sλ′ for

all λ ∈ F#q .

Again by Lemma 3.2.3, the orbits V #∞ and V #

λ (for all λ ∈ Fix(σ(H))) correspond to

disconnected graphs.

Proposition 4.3.7. Let Γ be a graph and G = TV ⋊G0, where V , TV , and G0 are as

in Lemma 4.2.7 (1) and (2), with n = 6, q even, and G2(q) E H. Then Γ is connected,


G-symmetric and G-quotient-complete if and only if Γ ∼= Cay(V, S), where S is one of the

following:

(1) SL;

(2) SL′;

(3)⋃

λ′∈C(λ) Sλ′ for any λ ∈ F#q ;

(4)⋃

λ′∈A(λ) V#λ′ for some λ /∈ Fix(σ(H)).


quotients.

Proof. This follows immediately from Proposition 4.2.5 and Lemmas 4.2.7 and 4.3.4.

4.3.4. Exceptional cases. Assume that H is as given in Table 3.5.1, with n = d and

q = p. Since H ≤ GL(n, q) for each case, the sets V #∞ and V #

λ for all λ ∈ Fq are G0-orbits.

These orbits correspond to disconnected graphs by Lemma 3.2.3, while all other orbits,

which are subsets of Vind, give rise to connected graphs Γ. The orbits in Vind were all

obtained using Magma [1].

For the case n = 2 (i.e., H as in lines 1, 2 and 3 of Table 3.5.1 or lines 1-17 of Table

4.1.2), we prove in Lemma 4.3.8 that all connected graphs arising from each H have the

same valency and belong to one isomorphism class.

Lemma 4.3.8. Let G0 = (h, h) | h ∈ H where H ≤ GL(2, q) is one of the groups

in lines 1, 2 and 3 of Table 3.5.1, U = F2q, and V = U ⊕ U . Then the connected graphs

Cay(V, S) correspond to G0-orbits S ⊆ Vind, and belong to one isomorphism class. More-

over, all of these graphs have valency |H|.

To prove Lemma 4.3.8 we need the following. Recall that the space V = U ⊕ U can

be viewed as a tensor product U ⊗U via the linear transformation f : V → U ⊗U , where

(u,0U ) 7→ e1 ⊗ u and (0U , u) 7→ e2 ⊗ u for all u ∈ U (with e1 = (1, 0) and e2 = (0, 1), the

standard basis vectors in U). Each (x, y) ∈ V then corresponds to e1⊗x+e2⊗y ∈ U ⊗U .

Likewise the group G0 = diag(H×H) can be viewed as the tensor product 〈I2〉⊗H, where

I2 is the 2×2 identity matrix over Fq and each (h, h) ∈ G0 corresponds to I2⊗h ∈ 〈I2〉⊗H.

Observe that for each (x, y) ∈ V and h ∈ H, the element (x, y)(h,h) =(xh, yh

)in U ⊗ U

corresponds to

e1 ⊗ xh + e2 ⊗ yh = (e1 ⊗ x+ e2 ⊗ y)I2⊗h

in U ⊗ U , so the original action of diag(H × H) on U ⊕ U is equivalent to the natural

action of 〈I2〉 ⊗H on U ⊗ U . Furthermore, note that GL(2, q)⊗ 〈I2〉 ≤ NGL(V )(G0).

Proof of Lemma 4.3.8. It was shown in the first paragraph of this section that

the graph Cay(V, S) is connected if and only if S ⊆ Vind. To prove that the connected

graphs belong to one isomorphism class we show that NGL(V )(G0) is transitive on Vind.


Let (u,w), (x, y) ∈ Vind. Then u,w and x, y are ordered bases of U ; let A = (aij)

be the change-of-basis matrix from u,w to x, y. (That is, x = a11u + a21w and

y = a12u+ a22w.) Then in U ⊗ U we have

(e1 ⊗ u+ e2 ⊗ w)A⊗I2 = eA1 ⊗ u+ eA2 ⊗ w

= (a11e1 + a12e2)⊗ u+ (a21e1 + a22e2)⊗ w

= e1 ⊗ (a11u+ a21w) + e2 ⊗ (a21u+ a22w)

= e1 ⊗ x+ e2 ⊗ y

Recall from the remarks preceding the proof that A ⊗ I2 ∈ NGL(V )(G0), so the group

NGL(V )(G0) is transitive on Vind. It follows that the graphs Cay(V, S) are all isomorphic

for all G0-orbits S ⊆ Vind.

Now let (u,w) ∈ Vind and (h, h) ∈ StabG0((u,w)). Then uh = u and wh = w. Since

u,w is a basis of U then h must be the identity in GL(U). Therefore (h, h) is the

identity in GL(V ), and hence G0 acts semiregularly on Vind. Thus |S| = |G0| = |H| forall G0-orbits S in Vind.

The connected graphs arising from transitive groups H ≤ NΓL(2,9)(SL(2, 5)) are pre-

sented in lines 18-33 of Table 4.1.2. We have A(3) = NGL(2,9)(SL(2, 5)) ∼= Z SL(2, 5),

where Z = Z(GL(2, 9)) ∼= Z8, and NΓL(2,9)(SL(2, 5)) ∼= A(3) ⋊ Aut (F9). The graphs

Γ(31) correspond to Cay(V, S) with S = V #λ ∪ V #

λ3 and λ ∈ F9 \ F3. The group A(5)

has two maximal subgroups which are transitive and not contained in GL(2, 9), namely

A(4) = (Z0 SL(2, 5)) ⋊ Aut (F9) and A(1) = (Z K) ⋊ Aut (F9), where Z4∼= Z0 ≤ Z

and K = A(1) ∩ SL(2, 5) ∼= Z10 ⋊ Z2. The group A(2) is isomorphic to the subgroup

SL(2, 5) ⋊Aut (F9).

The group B(1) in Table 4.1.2 is conjugate to A(2), and thus gives the same graphs

as A(2); we do not list these graphs in the table. The group B(3) has order 320 and is

isomorphic to B(4) ∩ B(5). The groups B(1), B(4) and B(5) are maximal in the group

NGL(4,3)(D8 Q8).

For the rest of Table 4.1.2, isomorphisms (and non-isomorphisms) are determined

using Magma [1]. Except where indicated, each line of the table represents a distinct

isomorphism class of graphs. For each H the isomorphisms are induced by elements of

NGL(V )(G0), except for those in line 33: in this case NGL(V )(G0) divides the four graphs of

valency 960 into two classes of sizes 3 and 1; it is determined by Magma that these merge

into one isomorphism class. For lines 53 – 72 the isomorphisms are induced by elements

of 〈g ⊗ h | g ∈ GL(2, 3), h ∈ H〉 ≤ NGL(V )(G0).

The results in Section 4.3 complete the proof of Theorem 4.1.1.

Proof of Theorem 4.1.1. This follows from Proposition 4.2.5, Lemma 4.2.7, and

the results in this section (Propositions 4.3.1, 4.3.3 and 4.3.7 and the above).


4.4. Quotient-complete symmetric graphs with diameter 2

In this section we identify which of the graphs in Section 4.3 have diameter 2. Those

for the exceptional cases (namely, the graphs in Table 4.1.2) were found using Magma [1]

and are indicated by a “†” in the first column of Table 4.1.2. From now on we consider

the graphs in Table 4.1.1. Recall that Cay(V, S) has diameter 2 if and only if S 6= V # and

V ⊆ S ∪ (S + S) (equivalently, V \ S ⊆ S + S) where S + S := x+ y | x, y ∈ S. Since

the S that appear in Table 4.1.1 all satisfy S 6= V # and S = −S, we only need to verify

that V # \ S ⊆ S + S.

As in Section 4.3 we assume throughout that U , V , G, G0 and H are as in Lemma

4.2.7 (1) and (2). Also let Vλ be as in (4.3.1) and Sλ be as in (4.3.5) for each λ ∈ F#q , Vind

be as in (4.3.2), SL be as in (4.3.6) and SL′ be as in (4.3.7).

Lemma 4.4.1. Let S = V #λ ∪ V #

µ where λ 6= µ. Then V # \ S ⊆ S + S.

Proof. Let (u,w) ∈ V # \ S. Since µ 6= λ the matrix

(1 λ

1 µ

)is invertible, and thus

there exist x, y ∈ U (not both 0U ) which satisfy the equation

(u w) = (x y)

(1 λ

1 µ

).

That is,

(u,w) = (x+ y, λx+ µy) = (x, λx) + (y, µy) ∈ S + S.

Therefore V # \ S ⊆ S + S.

Clearly |A(λ)| ≥ 2 if and only if λ /∈ Fix(σ(H)), so it follows from Lemma 4.4.1 that

the graphs in lines 2, 5 and 9 of Table 4.1.1 have diameter 2.

Lemma 4.4.2. Suppose that n ≥ 2, and let S = Vind. Then V # \ S ⊆ S + S.

Proof. Let (u,w) ∈ V # \ S =⋃

λ∈Fq∪∞ V#λ .

Case 1: Suppose that (u,w) = (u,0U ) ∈ V #0 . Take x ∈ U# such that x, u is linearly

independent in U . Then u − x,−u is also linearly independent, and thus (x, u), (u −x,−u) ∈ S. Hence (u,w) = (x, u) + (u− x,−u) ∈ S + S. The case (u,w) ∈ V #

∞ is proved

similarly.

Case 2: Suppose that (u,w) ∈ V #λ for some λ ∈ F#

q , so that u 6= 0U and w = λu.

Take x ∈ U such that u, x is linearly independent. Then u− x, x and x, λu− x are

also linearly independent, and (u − x, x), (x, λu − x) ∈ S. Hence (u,w) ∈ S + S, which

completes the proof.

Lemma 4.4.3. Suppose that dim (U) is even, and let λ ∈ F#q . Then V #\Sλ ⊆ Sλ+Sλ.

Proof. Let (u,w) ∈ V # \ Sλ.

4.4. QUOTIENT-COMPLETE SYMMETRIC GRAPHS WITH DIAMETER 2 75

Case 1: Suppose that (u,w) = (u,0U ) ∈ V #0 . Since the symplectic form f on U is

nondegenerate there exists x ∈ U such that f(x, u) = λ. Then f(u−x,−u) = f(x, u) = λ,

and (u,w) = (u− x,−u) + (x, u) ∈ Sλ + Sλ. The case where (u,w) ∈ V #∞ is similar.

Case 2: Suppose that (u,w) = (u, αu) ∈ V #α for some α ∈ F#

q . Again by the non-

degeneracy of f we can find x ∈ U such that f(x, u) = 1. Set y = αx + λu. Then

f(x, y) = f(x, λu) = λ and

f(u− x,w − y) = f(u− x, (α− λ)u− αx)

= f(u,−αx) + f(−x, (α− λ)u)

= α− (α− λ)

= λ.

So (x, y), (u − x,w − y) ∈ Sλ and (u,w) = (u− x,w − y) + (x, y) ∈ Sλ + Sλ.

Case 3: Suppose that (u,w) ∈ Sµ for some µ ∈ F#q , µ 6= λ. Set x := u − w and

y := λµ−1u. Then

f(x, y) = f(−w, λµ−1u) = λµ−1µ = λ

and

f(u− x,w − y) = f(w,w − λµ−1u) = λ.

So (x, y), (u − x,w − y) ∈ Sλ and (u,w) ∈ Sλ + Sλ.

Case 4: Suppose that n ≥ 4 and (u,w) ∈ S0. (If n = 2 then S0 is empty, so Cases 1 to

3 suffice to prove that V # \Sλ ⊆ Sλ+Sλ.) Since n ≥ 4 the vector space U can be written

as the orthogonal direct sum of two or more hyperbolic planes L1, . . . , Lr (so 2r = n). We

choose the Li’s such that u ∈ L1. Since f(u,w) = 0 we can write w = γu+ x2 + · · ·+ xr,

where γ ∈ Fq and xi ∈ Li for all i, with xi 6= 0U for at least one i (since u and w are linearly

independent). Without loss of generality suppose that x2 6= 0U , and let y2 ∈ L2 such that

f(x2, y2) = λ. Note that f(u, y2) = f(x2, w) = 0, so that f(u−x2, w−y2) = f(x2, y2) = λ.

Hence (x2, y2), (u − x2, w − y2) ∈ Sλ and (u,w) ∈ Sλ + Sλ.

Lemma 4.4.4. Suppose that n ≥ 4 and is even. Then V # \ S0 ⊆ S0 + S0.

Proof. Let (u,w) ∈ V # \ S0.Case 1: Suppose that (u,w) ∈ V #

0 . We can decompose U into an orthogonal direct

sum of r ≥ 2 hyperbolic planes L1, . . . , Lr with u ∈ L1. Let x ∈ L#2 , so that f(x, u) = 0

and x, u is linearly independent; that is, (x, u) ∈ S0. Then also (u − x,−u) ∈ S0

(indeed, u− x and −u are linearly independent, and f(u− x,−u) = f(x, u) = 0). Hence

(x, u), (u − x,−u) ∈ S0 and (u,w) = (u,0U ) = (u − x,−u) + (x, u) ∈ S0 + S0. The case

where (u,w) ∈ V #∞ is similar.

Case 2: Suppose that (u,w) = (u, λu) ∈ V #λ for some λ ∈ F#

q . As in Case 1 we

can write U as the orthogonal direct sum of r ≥ 2 hyperbolic planes L1, . . . , Lr, with

u ∈ L1. Take y ∈ L#2 and set x = y + u. Then f(x, y) = f(u, y) = f(x, u) = 0 and


f(u − x,w − y) = f(u − x, λu − y) = 0. Furthermore x, y, u − x,w − y are linearly

independent, so (x, y), (u−x,w−y) ∈ S0. Therefore (u,w) = (u−x,w−y)+(x, y) ∈ S0+S0.

Case 3: Suppose that (u,w) ∈ Sλ for some λ ∈ F#q . Since n ≥ 4 there exists y ∈ U#

such that f(u, y) = f(w, y) = 0. Set x = y + u. Then f(x, y) = 0 and f(u − x,w −y) = f(−y,w) = 0, and moreover x, y and u − x,w − y = −y,w − y are linearly

independent. So (x, y), (u − x,w − y) ∈ S0, and (u,w) ∈ S0 + S0.

Lemma 4.4.5. Suppose that q is even and n = 6. Then V # \ SL ⊆ SL + SL.

Proof. Let (u,w) ∈ V # \ SL.Case 1: Suppose that (u,w) ∈ V #

λ for some λ ∈ Fq∪∞. If λ ∈ F#q , let ℓ ∈ L be such

that u ∈ ℓ and take x ∈ ℓ \ 〈u〉. Then clearly u+ x /∈ 〈x〉, so 〈x, λ(u+ x)〉 = 〈x, u+ x〉 = ℓ

and (x, λ(u+ x)), (λx, u+ x) ∈ SL. Hence (u,w) = (u, λu) = (x, λ(u+ x)) + (u+ x, λx) ∈SL + SL. If λ = 0 define ℓ and x similarly as the above. Then u + x /∈ 〈u〉 so that

〈x, u〉 = 〈u + x, u〉 = ℓ, and again (u,w) = (u,0U ) = (x, u) + (u + x, u) ∈ SL + SL. The

case where λ = ∞ is proved similarly, with ℓ ∈ L chosen such that w ∈ ℓ.

Case 2: Suppose that (u,w) ∈ SL′ . Then by Lemma 3.4.2 the points 〈u〉 and 〈w〉of the generalised hexagon H(q) are at distance 2 from each other in the point graph of

H(q). Let 〈x〉 be a point of H(q) which is collinear to both 〈u〉 and 〈w〉. Then clearly

〈u + x, x〉 = 〈u, x〉 ∈ L and 〈w + x, x〉 = 〈w, x〉 ∈ L (see Figure 1). Hence (u,w) =

(u+ x, x) + (x,w + x) ∈ SL + SL.

〈x〉

〈u〉

〈w〉〈w + x〉

〈u+ x〉

Figure 1. Diagram for

Lemma 4.4.5, Case 2

〈u〉〈w〉

〈x〉

〈y〉

ℓ


Lemma 4.4.5, Case 3

Case 3: Suppose that (u,w) ∈ Sλ for some λ ∈ F#q . By Lemma 3.4.2 the points

〈u〉 and 〈w〉 are at distance 3 from each other in the point graph of H(q). Let 〈x〉 be

a point of H(q) which is collinear with 〈w〉 and is at distance 2 from 〈u〉 (see Figure

2). By the axioms of H(q), and in particular the fact that at least 3 lines of H(q) pass

through 〈x〉, there exists ℓ ∈ L through 〈x〉 such that all points in ℓ \ 〈x〉 belong in

∆3(〈u〉) ∩∆2(〈w〉) (i.e., at distance 3 from 〈u〉 and distance 2 from 〈w〉). Let y ∈ ℓ \ 〈x〉be such that y 6= 0U and f(u, y) = f(u,w). Then 〈w, x, y〉 is a maximal totally isotropic

subspace, and it follows from Theorem 3.4.3 that it is the union of all lines in L that

pass through 〈x〉. Now y + w ∈ 〈w, x, y〉, so 〈y + w〉 is collinear with 〈x〉 in H(q). Since

f(u, y+w) = f(u, y) + f(u,w) = 0, Lemma 3.4.2 implies that either 〈y+w〉 ∈ ∆1(〈u〉) or〈y + w〉 ∈ ∆2(〈u〉). We consider each of these cases in turn.

4.4. QUOTIENT-COMPLETE SYMMETRIC GRAPHS WITH DIAMETER 2 77

Case 3.1: Suppose that 〈y+w〉 ∈ ∆1(〈u〉) (see Figure 3). Since it is also collinear with

〈x〉, the 3-dimensional subspace 〈u, x, y +w〉 is totally isotropic. It follows from Theorem

3.4.3 that 〈u, x, y+w〉 consists of all lines through 〈y+w〉. In particular, 〈x+u〉 is collinearwith 〈y + w〉, and so (u,w) = (x+ u, y + w) + (x, y) ∈ SL + SL, as required.

〈u〉〈w〉

〈x+ u〉

〈y + w〉

〈y〉

〈x〉


Lemma 4.4.5, Case 3.1

〈u〉〈w〉

〈x+ u〉

〈z〉

〈z + w〉= 〈y′〉

〈x〉

〈y〉

〈y +w〉


Lemma 4.4.5, Case 3.2

Case 3.2: Suppose that 〈y+w〉 ∈ ∆2(〈u〉). Let 〈z〉 be the point collinear with both 〈u〉and 〈x〉 (see Figure 4). By Theorem 3.4.3 (1), the point 〈x+u〉 is collinear with 〈z〉. Claimthat z ∈ 〈x, y + w〉. Suppose otherwise, and let 〈z′〉 be the point collinear with 〈u〉 and

〈y+w〉. Then 〈z〉 6= 〈z′〉, so that 〈u〉, 〈z〉, 〈x〉, 〈y+w〉, and 〈z′〉 form a 5-cycle in H(q). But

this is impossible by the axioms of H(q). Therefore 〈z〉 = 〈z′〉, and hence z ∈ 〈x, y + w〉,which proves the claim. Take y′ = z+w, which is collinear with 〈x〉 by Theorem 3.4.3 (1).

Then 〈x, y′〉, 〈x+u, y′+w〉 ∈ L, and therefore (u,w) = (x+u, y′+w)+(x, y′) ∈ SL+SL.

Lemma 4.4.6. Suppose that q is even and dim (U) = 6. Then V # \ SL′ ⊆ SL′ + SL′.

Proof. Let (u,w) ∈ V # \ SL′ .

Case 1: Suppose that (u,w) ∈ V #λ for some λ ∈ Fq∪∞. An argument similar to that

in Case 1 of the proof of Lemma 4.4.5, but taking ℓ ∈ L′, shows that (u,w) ∈ SL′ + SL′ .

Case 2: Suppose that (u,w) ∈ SL. Take ℓ ∈ L such that ℓ contains 〈u + w〉 and ℓ 6=〈u,w〉 (see Figure 5). Let 〈x〉 be a point in ℓ distinct from 〈u+w〉. Then 〈u, x〉, 〈w, x〉 ∈ L′

by Lemma 3.4.2, and likewise 〈u+x, x〉, 〈w+x, x〉 ∈ L′. Thus (u,w) = (u+x, x)+(x,w+

x) ∈ SL′ + SL′ .

〈u+ w〉

〈u〉

〈w〉〈x〉

ℓ


Lemma 4.4.6, Case 2

〈u〉〈w〉

〈x〉

〈x+ u〉

〈y +w〉

〈y〉


Lemma 4.4.6, Case 3

Case 3: Suppose that (u,w) ∈ Sλ for some λ ∈ F#q . Then 〈u〉 and 〈w〉 have distance 3

in H(q) by Lemma 3.4.2. Take points 〈x〉 and 〈y〉 such that 〈x〉 6= 〈u〉 and 〈x〉 ∈ ∆1(〈u〉)∩∆3(〈w〉), and 〈y〉 ∈ ∆1(〈w〉) ∩ ∆2(〈u〉) ∩ ∆2(〈x〉) (see Figure 6). Then 〈x〉 ∈ ∆2(〈y〉).Choose x such that f(x,w) = f(u,w). Then f(x + u,w) = 0, so by Lemma 3.4.2 either


〈x + u〉 ∈ ∆1(〈w〉) or 〈x + u〉 ∈ ∆2(〈w〉). Since 〈x + u〉 lies on the line 〈x, u〉, it follows

that 〈x+ u〉 ∈ ∆2(〈w〉). Finally, since y + w ∈ 〈y,w〉, we have 〈y + w〉 ∈ ∆2(〈x+ u〉). So〈x, y〉, 〈x+u, y+w〉 ∈ L′ by Lemma 3.4.2, and (u,w) = (x+u, y+w)+(x, y) ∈ SL′+SL′ .

The diameter 2 graphs in Table 4.1.1 are indicated by a “†” in the first column, and

this follows from Lemmas 4.4.1 to 4.4.6.

4.5. Quotient-complete symmetric graphs arising from H ≤ ΓL(1, q)

Let U , V , G, G0 and H be as in Lemma 4.2.7 (1) and (2), with q = pd (p prime),

n = 1, and H ≤ ΓL(1, q). Let ω be a fixed primitive element of Fq and let ω denote scalar

multiplication by ω. Also, let τ be the Frobenius automorphism of Fq, that is, τ : λ 7→ λp

for all λ ∈ Fq. Then GL(1, q) = 〈ω〉 and ΓL(1, q) = 〈ω, τ〉.The following result, originally by D.A. Foulser, gives a standard generating set for

H, as well as necessary and sufficient conditions (with respect to this generating set) for

H to be transitive on U#.

Lemma 4.5.1. [33, Lemmas 4.4 and 4.7] Let H ≤ ΓL(1, q) = 〈ω, τ〉, where q = pd

(p prime), ω is scalar multiplication by a fixed primitive element ω ∈ Fq, and τ is the

Frobenius automorphism of Fq. Then there exist unique integers a, b and c such that

H =⟨ωa, ωbτ c

⟩and the following hold: a > 0 and a divides q − 1; c > 0 and c divides d;

0 ≤ b < a and b(q − 1)/(pc − 1) ≡ 0 (mod d). Moreover, H is transitive on U# = F#q if

and only if one of the following holds:

(1) a = 1 (so b = 0), or

(2) b > 0, a divides b(pac − 1)/(pc − 1), and a does not divide b(pa′c − 1)/(pc − 1)

whenever 1 < a′ < a.

We do not treat this case completely; rather, we consider the subcases for which either

H ∩GL(1, q) = GL(1, q), or H = 〈ωa, ωτ c〉 where pc ≡ 1 (mod a).

4.5.1. H = 〈ω, τ c〉 where d ≡ 0 (mod c). Let τ and ω be as in Lemma 4.5.1, and

suppose that H is of the type described in Lemma 4.5.1 (1). Then H = 〈ω, τ c〉 for some

divisor c of d. The G0-orbits in V# are the sets V #

∞ and⋃

λ′∈D(λ) V#λ′ for all λ ∈ Fq, where

D(λ) := λσ | σ ∈ 〈τ c〉 = λpci | 0 ≤ i < d/c− 1. (4.5.1)

Then |D(λ)| = 1 if and only if λ ∈ Fix(〈τ c〉), where |Fix(〈τ c〉)| = pc, and there exist

connected graphs if and only if d ≥ 2.

Proposition 4.5.2. Let Γ be graph and G = TV ⋊G0, where V , TV and G0 are as in

Lemma 4.2.7 (1) and (2), with n = 1, q = pd (p prime), and d ≥ 2. Also, suppose that

H = 〈ω, τ c〉, where ω and τ are as in Lemma 4.5.1 and c divides d. Then Γ is connected, G-

symmetric and G-quotient-complete if and only if Γ ∼= Cay(V, S) where S =⋃

λ′∈D(λ) V#λ′

4.5. QUOTIENT-COMPLETE SYMMETRIC GRAPHS ARISING FROM H ≤ ΓL(1, q) 79

for some λ /∈ Fix(〈τ c〉). Furthermore, such a graph Γ has diameter 2 and has exactly pc+1

nontrivial complete G-normal quotients.

Proof. This follows immediately from Proposition 4.2.5, Lemma 4.2.7, Lemma 4.4.1,

and the preceding remarks.

4.5.2. H = 〈ωa, ωτ c〉, where d ≡ 0 (mod c), q ≡ pc ≡ 1 (mod a), and

(q − 1)/(pc − 1) ≡ 0 (mod d). Again we let ω and τ be as in Lemma 4.5.1. Suppose

that H = 〈ωa, ωτ c〉 for some divisor a of q − 1 and divisor c of d, such that pc ≡ 1

(mod a) and (q − 1)/(pc − 1) ≡ 0 (mod d). Then H is a transitive subgroup of ΓL(1, q)

of the type described in Lemma 4.5.1 (2). (Indeed, observe that (pac − 1)/(pc − 1) =

pc(a−1) + pc(a−2) + . . .+ 1 ≡ 0 (mod a) and (pa′c − 1)/(pc − 1) ≡ a′ (mod a) ≡/ 0 (mod a)

for all 1 < a′ < a.) The G0-orbits in V# are V #

∞ and (1, λ)G0 for each λ ∈ U .

Let r := pc. For any positive integer j,

(ωτ c)j : λ 7→ λrjωr(rj−1)/(r−1) for all λ ∈ U,

where U is identified with Fq. Now for any (1, λ) ∈ V ,

(1, λ)G0 =(ωai, λωai

)(ωτc)jall i and all j

,

where for each i and j we have(ωai, λωai

)(ωτc)j=(ωℓ, λr

jωℓ)and ℓ = ai+ r(rj − 1)/(r−

1) ≡ j (mod a). Hence

(1, λ)G0 =(ωai, λωai

),(ωai+1, λrωai+1

), . . . ,

(ωai+m, λr

jωai+m

), . . .

all i and all j; 0 ≤ m < a and m ≡ j (mod a).

Clearly (1, λ)G0 = V #λ if and only if λ ∈ Fix(〈τ c〉).

Proposition 4.5.3. Let Γ be graph and G = TV ⋊G0, where V , TV and G0 are as in

Lemma 4.2.7 (1) and (2), with n = 1, q = pd (p prime), and d ≥ 2. Also, suppose that

H = 〈ωa, ωτ c〉, where ω and τ are as in Lemma 4.5.1, a divides q − 1, c divides d, pc ≡ 1

(mod a), and (q − 1)/(pc − 1) ≡ 0 (mod d). Then Γ is connected, G-symmetric and G-

quotient-complete if and only if Γ ∼= Cay(V, S) where S = (1, λ)G0 for some λ /∈ Fix(〈τ c〉).Furthermore, such a graph Γ has exactly pc + 1 nontrivial complete G-normal quotients.

Proof. This follows immediately from Proposition 4.2.5, Lemma 4.2.7, and the pre-

ceding remarks.

In general,∣∣(1, λ)G0

∣∣ = lcm (a, |D(λ)|)(q − 1)/a

with D(λ) as in (4.5.1), since λrjωai+m = λωai if and only if j is divisible by both a and

|D(λ)|. Moreover, (1, λ)G0 ⊆ ⋃λ′∈D(λ) V

#λ′ where

∣∣∣⋃

λ′∈D(λ) V#λ′

∣∣∣ = |D(λ)|(q − 1), and by

comparing cardinalities we get that (1, λ)G0 =⋃

λ′∈D(λ) V#λ′ if and only if |D(λ)| is coprime

to a. In this case the resulting graphs Cay(V, (1, λ)G0) have diameter 2 and are the same


as the graphs in Proposition 4.5.2. There may be other diameter 2 graphs distinct from

the type described in Proposition 4.5.2, arising from the other G0-orbits, as can be seen

in the following example.

Example 4.5.4. Suppose that q = 34 and let H = 〈ω2, ωτ〉. Then the G0-orbits in

V # are the sets V #∞ and (1, λ)G0 for each λ ∈ U , where

(1, λ)G0 =(ω2i, λω2i

),(ω2i+1, λ3ω2i+1

),(ω2i, λ9ω2i

),(ω2i+1, λ27ω2i+1

)all i

.

Observe that (1, λ)G0 = (1, µ)G0 if and only if µ = λ9, so there are 46 G0-orbits in all.

Also, since d = 4 and c = 1, |D(λ)| ∈ 1, 2, 4 for all λ ∈ Fq (since by its definition

|D(λ)| must divide d/c). So by the preceding remarks (1, λ)G0 =⋃

λ′∈D(λ) V#λ′ if and only

if λ ∈ Fix(〈τ〉), which yield disconnected graphs by Lemma 3.2.3. Hence there are no

connected graphs of the form described in Proposition 4.5.2. Moreover, there are 10 orbits

which have length 80 — namely, V #∞ and (1, λ)G0 where λ ∈ 0, 1, ω10, ω20, . . . , ω70 —

and 36 of length 160. Of the graphs which arise, 24 have diameter 2, and all of these

have valency 160. It was verified using Magma that the diameter 2 graphs belong to one

isomorphism class.

CHAPTER 5

Symmetric vertex-quasiprimitive graphs: affine case

5.1. Overview and main results

We now turn our attention to symmetric graphs that satisfy Case (2) of Theorem 1 with

N = 1; that is, we look at symmetric graphs that admit a vertex-quasiprimitive subgroup

of automorphisms. Recall from Theorem 1.3.8 that there are eight types of quasiprimitive

permutation groups; in this chapter we consider graphs with quasiprimitive automorphism

group of type HA. Graphs that correspond to the other quasiprimitive types are discussed

in Chapter 6.

Lemma 5.1.1 describes the structure of symmetric graphs Γ, together with G ≤Aut (Γ), such that G acts as an affine-type quasiprimitive permutation group on V (Γ).

This result follows from basic properties of affine quasiprimitive permutation groups and

Cayley graphs (see Sections 1.3.4 and 2.2).

Lemma 5.1.1. [39] Let Γ be a graph and let G ≤ Aut (Γ), where G acts quasiprimi-

tively on V (Γ) and is of affine type. Then G ∼= V ⋊G0 ≤ AGL(d, p) and Γ ∼= Cay(V, S) for

some vector space V = Fdp over a prime field Fp, where V is identified with its translation

group and G0 ≤ GL(d, p) is irreducible. Moreover, Γ is G-symmetric with diameter 2 if

and only if S is a G0-orbit of nonzero vectors satisfying −S = S and S ∪ (S + S) = V .

Remark. Recall from the remarks after Lemma 2.2.5 that if diam(Cay(V, S)) = 2

then |V | ≤ |S|2 + 1, where |S| ≤ |G0|. Hence if |V | > |G0|2 + 1, then for any G0-orbit S

in V the graph Cay(V, S) has diameter greater than 2.

We are interested in the case where the irreducible subgroup G0 is maximal in GL(d, p)

with respect to being intransitive on V #. By Aschbacher’s classification of the subgroups

of the finite classical groups (see Section 3.6), the irreducible subgroups of GL(d, p) which

do not contain SL(d, p) are organised into eight classes Ci, 2 ≤ i ≤ 9. The maximal

subgroups in these classes are all intransitive on V #, except for those in the class C3 —

which are of the form ΓL(n, q), where qn = pd and d/n is prime — as well as the C8-subgroup GSp(d, p). The irreducible maximal subgroups of ΓL(n, q) and GSp(d, p) which

do not contain SL(n, q) and Sp(d, p), respectively, are again organised into classes C2 to C9.In ΓL(n, q), the maximal Ci-subgroups which are transitive on V # are those in the class

C3, which have the form ΓL(m, qn/m) with n/m prime, and the C8-subgroup ΓSp(n, q).

81

82 5. SYMMETRIC VERTEX-QUASIPRIMITIVE GRAPHS: AFFINE CASE

In GSp(d, p), all maximal Ci-subgroups for i 6= 3 are intransitive on V #; a maximal C3-subgroup either contains Sp(n, q), in which case it is contained in ΓSp(n, q), or has the

form GU(d/2, p2), in which case it is contained in ΓU(d/2, p2), which in turn is intransitive

on V # and is a maximal C8-subgroup of ΓL(d/2, p2). Thus the maximal C3-subgroups ofGSp(d, p) can be dealt with by considering maximal intransitive subgroups of ΓL(n, q) and

ΓSp(n, q) of type Ci, i 6= 3, for all n and q.

In view of the above, we assume for the rest of this chapter that the following hypothesis

holds:

Hypothesis 5.1.2. Let V = Fdp with p prime and d ≥ 2, and let q = pd/n for some

divisor n of d (possibly n composite or n = d). The space V is viewed as Fnq , τ denotes the

Frobenius automorphism of Fq, B is a fixed Fq-basis of V , and τ acts on V as in (3.2.2)

with respect to B. Define G := V ⋊G0 ≤ AGL(d, p) and L := G0 ∩GL(n, q), where G0 is

a maximal Ci-subgroup of H for 2 ≤ i ≤ 9, i 6= 3, and one of the following holds:

(1) H = ΓL(n, q) = GL(n, q)⋊ 〈τ〉 and G0 Sp(n, q), or

(2) H = ΓSp(n, q) = GSp(n, q) ⋊ 〈τ〉, the group of semisimilarities of a symplectic

form on V .

All irreducible subgroups of GL(d, p) which are maximal with respect to being intran-

sitive on V # thus occur as subcases of the groups considered in Hypothesis 5.1.2. (Indeed,

G0 is maximal intransitive if n = d or if d/n is prime.) Our goal is to identify the diameter

2 Cayley graphs Cay(V, S), if any, that arise from the G0-orbits S in V #. We thus have

two main concerns: first, to determine the G0-orbits S in V #, and second, to identify

which of these orbits satisfy S = −S and S ∪ (S + S) = V . In the cases where we are not

able to do one or the other of these, we obtain bounds on certain parameters to reduce

the number of unresolved cases.

For the rest of this chapter we prove Theorem 4, which is done by considering separately

each of the Aschbacher classes C2, C4, C5, C6, C7 and C8.

5.2. Class C8

In this case the space V has a form φ, which is symplectic, unitary, or nondegenerate

quadratic if H = ΓL(n, q), and is nondegenerate quadratic if H = ΓSp(n, q) with q even.

The group G0 is the semisimilarity group ΓI(n, q) of (V, φ) where I is as in Table 3.3.1.

Since the symplectic group is transitive on V #, we only consider the unitary and orthogonal

cases here.

Recall from Theorem 3.3.3 that the orbits of the isometry group of V are the sets Sλ

defined in (3.3.1), where λ ∈ Imφ with φ as defined in (3.3.2). Since −In ∈ I(n, q) for all I

in Table 3.3.1, it follows that −Sλ = Sλ for all possible λ. So to prove that Cay(V, S), for

some G0-orbit S, has diameter 2, we only need to show that V \S ⊆ S+S. In most cases

this is done by showing that for any v ∈ V \ S there exists a w ∈ S such that v − w ∈ S.

5.2. CLASS C8 83

As in Section 3.3, let Fq :=

α2 | α ∈ F#

q

and F⊠

q := F#q \ F

q . (So F⊠q 6= ∅ only if q

is odd.) For θ ∈ ,⊠,# let

Sθ :=⋃

λ∈Fθq

Sλ. (5.2.1)

If q is a square (as in the unitary case), let q0 :=√q and let Fq0 denote the subfield of Fq

of index 2. Also let Tr : Fq → Fq0 denote the trace map, that is, Tr(α) = α+ αq0 for all

α ∈ Fq.

Remark. As was pointed out in Section 3.3, we use the notation U(n, q) to denote the

unitary group on Fnq , instead of the standard U(n, q0). Also GI(n, q) denotes the similarity

group of (Fnq , φ).

Proposition 5.2.1. Let V = Fnq , φ be a unitary or nondegenerate quadratic form on

V , and G0 = GI(n, q) with I as in Table 3.3.1. Let S0 be as in (3.3.1) and S, S⊠ and S#

be as in (5.2.1).

(1) If φ is unitary, then the G0-orbits in V # are S0 and S#.

(2) If φ is nondegenerate quadratic, then the G0-orbits in V # are as follows:

(i) S# if n = 1;

(ii) S0 and S# if n is even;

(iii) S0, S and S⊠ if n is odd and n ≥ 3.

Proof. It follows from Theorem 3.3.3 that S0 is a G0-orbit for all cases (that is,

provided that S0 6= ∅), so we only need to determine the G0-orbits in S#.

Suppose first that φ = f is unitary. Let v ∈ S#; clearly, vG0 ⊆ S#. For any u ∈ S#

set α := f(u, u)f(v, v)−1. Then α ∈ Fq0, so α = βq0+1 for some β ∈ Fq. Hence f(u, u) =

βq0+1f(v, v) = f(βv, βv), so by Theorem 3.3.3 we have u = (βv)g for some g ∈ U(n, q).

Then u = vβg, where βg ∈ GU(n, q). Therefore vG0 = S#, as required.

Suppose now that φ = Q is a nondegenerate quadratic form. Observe that 〈v〉# ⊆ vG0

for any v ∈ V since G0 contains all nonzero scalar matrices. Therefore V # = vG0 if n = 1,

which proves (i).

From now on assume that n ≥ 2. It follows from Theorem 3.3.3 that the set S0 is a

G0-orbit. Let v ∈ S. Then for any w ∈ S we have Q(w) = Q(αv) for some α ∈ F#q ,

so w ∈ (αv)I(n,q) by Theorem 3.3.3, and it follows that w ∈ vG0 . So S ⊆ vG0 , and by

a similar argument S⊠ is also contained in one G0-orbit. If q is even then F⊠q = ∅, so

S = S# is a G0-orbit, which proves one case of (ii) (recall that if Q is nondegenerate

and q is even, then n is even). Suppose that q is odd. Then either S ∪ S⊠ = S# is a

G0-orbit, or S and S⊠ are distinct G0-orbits. Let u ∈ S and w ∈ S⊠. Then U := 〈u〉and W := 〈w〉 are non-isometric spaces, and thus D(Q|U ) 6= D(Q|W ). It follows from

Theorem 3.3.6 (1) that also D(Q|U⊥) 6= D(Q|W⊥), so U⊥ and W⊥ are non-isometric. We

have two cases.


Case 1: Suppose that n is even. Then U⊥ and W⊥ have odd dimension and are thus

similar. Let h : U⊥ → W⊥ be a similarity, and let λ ∈ F#q satisfy Q(uh0) = λQ(u0) for

all u0 ∈ U⊥. From Theorem 3.3.6 (2) applied to U⊥, we deduce that λd−1 ∈ F⊠q , and

hence λ ∈ F⊠q . So there exists w0 ∈ W such that Q(w0) = λQ(u), and the linear map

g : U → W , where u 7→ w0, is a similarity with Q(xg) = λQ(x) for all x ∈ U . Writing

V = U ⊥ U⊥, let ρ ∈ GL(V ) be defined by x + y 7→ xg + yh for all x ∈ U and y ∈ U⊥.

Then ρ ∈ M with uρ ∈ W , which shows that S ∪ S⊠ is a G0-orbit. This completes the

proof of (ii).

Case 2: Suppose that n is odd. Then dim(U⊥) = dim

(W⊥) is even, and there is no

similarity of V which sends U⊥ to W⊥. It follows that there is no g ∈ M with ug = w.

Therefore S and S⊠ are distinct G0-orbits, which proves (iii).

Recall that if Q is a nondegenerate quadratic form of minus type and n = 2, then V

is anisotropic. In this case S0 = ∅, so by Proposition 5.2.1 (2.ii), G0 is transitive on V #.

Lemma 5.2.2. Let V = Fnq and φ be a unitary or nondegenerate quadratic form on

V . Let S0 and S# be as in (3.3.1) and (5.2.1), respectively. Then S# ⊆ S0 + S0 if φ is

unitary and n ≥ 2, or if φ is nondegenerate quadratic and n ≥ 3.

Proof. Let v ∈ S#. Then by Corollary 3.3.5 there exists u ∈ 〈v〉⊥ with φ(u) = −φ(v).If φ is unitary set w := β(u+ v), where β := αφ(v)−1 and α ∈ Fq such that Tr(α) = φ(v).

If φ is quadratic and q is odd set w := 12(u+v). Then in both cases we have w, v−w ∈ S0,

so v ∈ S0 + S0 and therefore S# ⊆ S0 + S0. If φ is quadratic and q is even then n must

also be even, so in particular n ≥ 4. Let fφ be the associated bilinear form of φ and let

u ∈ V with fφ(u, v) = φ(v), so that W := 〈u, v〉 is non-degenerate with dim(W⊥) ≥ 2.

Note that φ = φ, hence Imφ|W⊥ = Fq by Lemma 3.3.4. In particular there exists x ∈W⊥

with φ(x) = φ(u + v). Setting w := u+ v + x, we again get w, v − w ∈ S0, and therefore

S# ⊆ S0 + S0. This completes the proof.

Lemma 5.2.3. Let V = Fnq , φ be a unitary or nondegenerate quadratic form on V , φ

as in (3.3.2) and Sλ as in (3.3.1). Then S0 ⊆ Sµ+Sµ for any µ ∈ (Imφ)#, if φ is unitary

and n ≥ 2, or if φ is quadratic and n ≥ 4.

Proof. Let v ∈ S0. Suppose first that either φ is unitary and n ≥ 3, or φ is nonde-

generate quadratic with n ≥ 4. Then by Corollary 3.3.5, for any µ ∈ (Imφ)# there exists

w ∈ Sµ ∩ 〈v〉⊥. It is easy to verify that φ(v − w) = φ(w), so v −w ∈ Sµ and v ∈ Sµ + Sµ.

Therefore S0 ⊆ Sµ + Sµ.

If φ = f is unitary and n = 2 then 〈v〉⊥ = 〈v〉 for any v ∈ S0. We claim that there

exists u ∈ S0 such that f(u, v) = 1. Indeed, take x ∈ V \ 〈v〉. Then f(v, x) 6= 0. If x ∈ S0

define u′ := x; if x /∈ S0 let u′ := αv + f(x, x)−1x where α ∈ Fq with Tr(α) = −f(x, x).Then in both cases u′ ∈ S0 and f(u′, v) 6= 0, and we take u to be the suitable scalar

5.2. CLASS C8 85

multiple of u′. This proves the claim. Let w := βu+ γv, where β, γ ∈ Fq with Tr(β) = 0

and Tr(βq0γ) = µ. Then w, v−w ∈ Sµ, and thus v ∈ Sµ+Sµ. Therefore S0 ⊆ Sµ+Sµ.

The diameter 2 Cayley graphs arising from ΓU(V ) can now be easily deduced from

the three results above.

Proposition 5.2.4. Let Γ be a graph and G ≤ Aut (Γ) such that G satisfies Hypothesis

5.1.2 with G0 = ΓU(n, q). Then Γ is G-symmetric with diameter 2 if and only if n ≥ 2

and Γ ∼= Cay(V, S), where V = Fnq and S ∈ S0, S#.

Proof. By Lemma 5.1.1 and Proposition 5.2.1 we only need to prove that Cay(V, S)

has diameter 2 if and only if n ≥ 2. If n = 1 then V is anisotropic, so GU(n, q) is

transitive on V # by Proposition 5.2.1 (1) and Cay(V, S) is a complete graph. If n ≥ 2

then V # \ S0 = S# and V # \ S# = S0 by Proposition 5.2.1 (1). Hence the result follows

from Lemma 5.2.2 if S = S0, and from Lemma 5.2.3 if S = S#.

Proposition 5.2.6 gives the diameter 2 Cayley graphs that arise from the general or-

thogonal groups. Its proof uses the following technical lemma.

Lemma 5.2.5. Suppose that q ≥ 5 and is odd.

(1) For any α ∈ F#q , there exist β ∈ F

q and γ ∈ F⊠q such that α = β + γ.

(2) For any α ∈ F#q , there exist β, γ ∈ F#

q \ αFq such that α = β − γ.

Proof. Since q ≥ 5, there exist β0 ∈ Fq and γ0 ∈ F⊠

q with β0 + γ0 6= 0. For any

α ∈ F#q we have α = δβ0 + δγ0, where δ := α(β0 + γ0)

−1. Clearly δβ0 /∈ δγ0Fq , and we

can take β, γ = δβ0, δγ0. This proves (1).It follows from (1) that for any β′ ∈ F⊠

q , there exist γ′ ∈ F⊠q and δ′ ∈ F

q such that

β′ = γ′ + δ′; multiplying both sides by (δ′)−1, we get that for some β0, γ0 ∈ F⊠q we have

β0 = γ0 + 1. For any α ∈ F#q set γ := γ−1

0 α. Then α = αβ0 − αγ0, with αβ0, αγ0 /∈ αFq ,

so we can take β = αβ0 and γ = αγ0. Thus we have proved (2).


5.1.2 with G0 = ΓO(n, q) or G0 = ΓOǫ(n, q) (ǫ = ±). Then Γ is G-symmetric with

diameter 2 if and only if Γ ∼= Cay(V, S) with V = Fnq and the conditions listed in Table

5.2.1 hold.

Proof. By Lemma 5.1.1 and Proposition 5.2.1 it remains to prove that Cay(V, S)

has diameter 2 if and only if the conditions in Table 5.2.1 hold. Suppose first that n = 2.

Then by Proposition 5.2.1 (2) the G0-orbits in V # are S0 and S#. If φ is of minus type

then V is anisotropic, so S# = V # and there is no graph of diameter 2. If φ is of plus type

then V has a basis x, y which is a hyperbolic pair, and we have S0 = 〈x〉# ∪ 〈y〉# and

S# =αx+ βy | α, β ∈ F#

q

. Then V # \S0 = S# ⊆ S0+S0, so diam(Cay(V, S0)) = 2. If


n q φ S

1 2 2 plus-type S02 2 q > 2 plus-type S0, S#3 n > 2, even all q plus-/minus-type S0, S#4 3 3 S05 3 q > 3 S0, S, S⊠6 n > 3, odd all q S0, S, S⊠

Table 5.2.1

q = 2 then S# = x+ y, so Cay(V, S#) is disconnected and we get the first line of Table

5.2.1. If q > 2, then e = (αx + y) + ((1 − α)x − y) ∈ S# + S# where α ∈ F#q \ 1, and

similarly f ∈ S# + S#. Thus V \ S# = S0 ⊆ S# + S# and diam(Cay(V, S#)) = 2, which

yields one case of line 2 of Table 5.2.1. Now suppose that n > 2 with n even. Then the

G0-orbits in V # are again S0 and S#, and it follows from Lemmas 5.2.2 and 5.2.3 that

Cay(V, S0) and Cay(V, S#) both have diameter 2. This gives lines 2 and 3 of Table 5.2.1.

From now on assume that n is odd. Then the G0-orbits in V# are S0, S and S⊠. It

again follows from Lemma 5.2.2 that diam(Cay(V, S0)) = 2. For Cay(V, Sθ), θ ∈ ,⊠,we consider three cases.

Case 1: Suppose that n = 3 and q = 3. Then F3 = 1 and F⊠

3 = 2. Let v ∈ S0.

If also v ∈ Sθ + Sθ, then w, v − w ∈ Sθ for some w ∈ V , so that φ(w) = φ(v − w) and

B(v,w) = 0. So w ∈ 〈v〉⊥, and thus Imφ|〈v〉⊥ = φ(w), 0 by Corollary 3.3.5 (3). Since

there exists u ∈ S0 with Imφ|〈u〉⊥ 6= φ(w), 0, we conclude that S0 * Sθ + Sθ. Hence

diam(Cay(V, Sθ)) 6= 2, which proves line 4 of Table 5.2.1.

Case 2: Suppose that n = 3 and q > 3, and let v ∈ V # \S. If v ∈ S0, take u ∈ V such

that u, v is a hyperbolic pair. Then Imφ|〈u,v〉 = Fq by Lemma 3.3.4, and hence there

exists w′ ∈ Sθ ∩ 〈u, v〉. It follows from Lemma 5.2.5 (1) that we can find β ∈ F#q \ Fθ

q such

that φ(w′) − β ∈ Fθq; set w := βu + (φ(w′)β−1)v. Then w, v − w ∈ Fθ

q and v ∈ Sθ + Sθ.

So S0 ⊆ Sθ + Sθ. If v ∈ Sθ′ := S# \ Sθ, then Imφ|〈v〉⊥ = Fq by Lemma 3.3.4. By Lemma

5.2.5 (2) there are elements γ, δ ∈ Fθq such that φ(v) = δ − γ, and hence φ(v) + γ ∈ Fθ

q.

Take w ∈ 〈v〉⊥ such that φ(w) = γ; then w, v − w ∈ Sθ and Sθ′ ⊆ Sθ + Sθ. Therefore

diam(Cay(V, Sθ)) = 2 for θ ∈ ,⊠, which proves line 5 of Table 5.2.1.

Case 3: Suppose that n > 3. Then S0 ⊆ Sθ+Sθ by Lemma 5.2.3. If v ∈ Sθ′ := S# \Sθthen Imφ|〈v〉⊥ = Fq by Corollary 3.3.5 (1). If q = 3 take u ∈ 〈v〉⊥ with φ(u) = φ(v), and

set w := −(u + v). Then w, v − w ∈ Fθq and Sθ′ ⊆ Sθ + Sθ, so diam(Cay(V, Sθ)) = 2. If

q > 3 then by Lemma 5.2.5 (2) there is a γ ∈ Fθq such that φ(v) + γ ∈ Fθ

q. Take w ∈ 〈v〉⊥

with φ(w) = γ. Then w, v − w ∈ Fθq, so again Sθ′ ⊆ Sθ + Sθ and diam(Cay(V, Sθ)) = 2.

This proves line 6 of Table 5.2.1.

5.3. CLASS C2 87

5.3. Class C2

In this case V = ⊕ti=1Ui, where Ui = Fm

q for each i, mt = n, m ≥ 2 and t ≥ 2. Assume

that B =⋃t

i=1 Bi, where each Bi is a basis for Ui. We write the elements of V as t-tuples

over Fmq ; under this identification the τ -action is equivalent to the natural componentwise

action.

Assume first that H = ΓL(n, q). Then by Theorem 3.6.2 we have

G0 = (GL(m, q) ≀ Sym(t))⋊ 〈τ〉, (5.3.1)

and so

L = G0 ∩GL(n, q) = GL(m, q) ≀ Sym (t) . (5.3.2)

Lemma 5.3.1. Let G0 be as in (5.3.1). Then the G0-orbits in V# are the sets Xs for

each s ∈ 1, . . . , t, where

Xs := (u1, . . . , ut) ∈ V # | exactly s coordinates nonzero. (5.3.3)

Proof. Let v = (u1, . . . , ut) ∈ Xs. Then clearly vG0 ⊆ Xs. For any w ∈ Xs,

say, w = (x1, . . . , xt), take π ∈ Sym (t) such that ui′ = 0 if and only if xi = 0, where

i′ := iπ−1

. Then there are elements g1, . . . , gt ∈ GL(m, q) such that ugi′i′ = xi for all i.

Hence v(g1,...,gt)π = w, so Xs ⊆ vL ⊆ vG0 . Therefore Xs is a G0-orbit for each s, and

X1, . . . ,Xt are all the G0-orbits in V#.


5.1.2 with G0 as in (5.3.1). Then Γ is G-symmetric with diameter 2 if and only if Γ ∼=Cay(V,Xs), where Xs is as in (5.3.3), such that qm > 2 and s ≥ t/2.

Proof. In view of Lemmas 5.1.1 and 5.3.1, we only need to show that V = Xs ∪(Xs +Xs) under the given conditions.

Suppose first that qm = 2. Then V = ⊕ti=1F2. Fix s ∈ 1, . . . , t and let u,w ∈ Xs.

Then u+w ∈ Xr, where r is the number of positions in which u and w differ, so r is even.

That is, Xr0 * Xs+Xs whenever r0 is odd, and if there exists an odd r0 ∈ 1, . . . , t with

r0 6= s then diam(Γ) > 2 by Lemma 5.1.1. In particular we have diam(Γ) > 2 for all t ≥ 3,

and if t = 2 then diam(Γ) = 2 if and only if S = X1. In the last case, however, we have

G0 = C2 × C2 and G is not quasiprimitive.

Now suppose that qm > 2. If s < t/2 then Xt * Xs +Xs, so diam(Γ) > 2. Assume

from now on that s ≥ t/2, and take u = (u1, . . . , ut) ∈ Xs. For each r < s, take

w = (w1, . . . , wt) ∈ Xs with wi = −ui in t − r positions and wi 6= −ui in the rest, and

for each r > s take w ∈ Xs with ui 6= wi = 0 for 1 ≤ i ≤ r − s, wi 6= ui = 0 for

r − s + 1 ≤ i ≤ s, and ui 6= −wi (both nonzero) for s + 1 ≤ i ≤ r. Then u + w ∈ Xr in

each case, so Xr ⊆ Xs +Xs for all r 6= s. Therefore diam(Γ) = 2.


We now consider the case where H = ΓSp(n, q) with n ≥ 4. Recall from Section 3.6.1

that we have two subcases:

(C2.1) The dimensionm is even and Ui is a symplectic space for each i, and the subspaces

Ui are pairwise orthogonal. Then by Theorem 3.6.3,

G0 = (g1, . . . , gt)πσ | π ∈ Sym (t) , σ ∈ 〈τ〉, gi ∈ GSp(m, q),

λ(gi) = λ(g1)∼= (Sp(m, q)t.[q − 1].Sym (t))⋊ 〈τ〉, (5.3.4)

where λ : GSp(n, q) → F#q is as defined in (3.3.3), so that

L = G0 ∩GL(n, q) ∼= Sp(m, q)t.[q − 1].Sym (t) .

(C2.2) The dimension m = n/2 so that t = 2, and both subspaces Ui are totally singular

with dimension n/2. Also q is odd if n = 4. By Theorem 3.6.3 we have

G0 =(g, g−⊤

)πσ π ∈ Sym(t) , σ ∈ 〈τ〉, g ∈ GL(m, q)

∼= (GL(m, q).[2]) ⋊ 〈τ〉, (5.3.5)

where g⊤ denotes the transpose of g, and g−⊤ = (g⊤)−1. Then

L = G0 ∩GL(n, q) ∼= GL(m, q).[2]. (5.3.6)

Lemma 5.3.3. For each s ∈ 1, . . . , t let Xs be as in (5.3.3). The G0-orbits in V #

are

(1) the sets Xs for each s ∈ 1, . . . , t if case (C2.1) holds and G0 is as in (5.3.4);

(2) the sets X1 and⋃

σ∈〈τ〉Wβσ for all β ∈ Fq, if case (C2.2) holds and G0 is as in

(5.3.5), where

Wβ := (w1, xβ)L, (5.3.7)

L is as in (5.3.6) and w1 := (1, 0, . . . , 0) ∈ Fmq , and xβ ∈ (Fm

q )# with first

component β.

Proof. The proof of part (1) is similar to that of Lemma 5.3.1, and uses the tran-

sitivity of Sp(m, q) on U#i , so we only need to prove part (2). Assume that case (C2.2)

holds. Then L = K.Sym (2), where K :=(g, g−⊤) g ∈ GL(m, q)

. It is easy to see that

U1 ⊕ 0 and 0 ⊕ U2 are K-orbits, so X1 = (U1 ⊗ 0) ∪ (0 ⊕ U2) is a G0-orbit. Let

(u, v) ∈ X2, and for any β ∈ Fq define

wβ :=

(β, 0, . . . , 0) if β 6= 0,

(0, 1, 0, . . . , 0) if β = 0.(5.3.8)

Since w1 ∈ uGL(m,q) we can assume that u = w1. Suppose that v = (β, v2, . . . , vm).

5.3. CLASS C2 89

Claim 1: (w1, y) ∈ (w1, v)K if and only if y = (β, y2, . . . , ym) for some y2, . . . , ym ∈ Fq.

Indeed, (w1, y) ∈ (w1, v)K if and only if y = vh

−⊤for some h ∈ StabGL(m,q)(w1). Now

wh1 = w1 if and only if the matrix of h−⊤ has the form

1 C

0... D

0

where C is a 1 × (m − 1) matrix over Fq and D ∈ GL(m − 1, q). Clearly, the orbit of v

under the subgrouph−⊤ h ∈ StabGL(m,q)(w1)

is the set of all nonzero vectors in Fm

q

with first component β. Therefore Claim 1 holds.

Claim 2: (w1, v)L = (w1, v)

K . By Claim 1 we can assume that v = wβ. If β 6= 0 let

g :=

β 0 · · · 0

0... Im−1

0

.

If β = 0 let g :=

(0 1

1 0

)if m = 2, and

g :=

0 1

1 0 0

0 Im−2

if m > 2. Then g ∈ GL(m, q) for all cases, and wg1 = wg⊤

1 = v. Hence (wg1 , v

g−⊤) = (v,w1),

so that (v,w1) ∈ (w1, v)K . Therefore (w1, v)

L = (w1, v)K ∪ (v,w1)

K = (w1, v)K , which

proves Claim 2.

It follows from Claims 1 and 2 that each set Wβ is an L-orbit (and moreover Wβ =

Wβ′ if and only if β = β′). It follows from the definition of the τ -action on V # that

(w1, v)G0 =

⋃σ∈〈τ〉Wβσ . This completes the proof of part (2).


5.1.2 with H = ΓSp(n, q) and i = 2. Then Γ is G-symmetric with diameter 2 if and only

if Γ ∼= Cay(V, S), where

(1) if case (C2.1) holds, then qm > 2, G0 is as in (5.3.4), S = Xs, and s ≥ t/2;

(2) if case (C2.2) holds with q = 2, then G0 is as in (5.3.5) and S = Wβ for any

β ∈ Fq;

(3) if case (C2.2) holds with qm > 2, then G0 is as in (5.3.5), and S = X1 or

S =⋃

σ∈〈τ〉Wβσ for some β ∈ Fq;

with Xs as in (5.3.3) and Wβ as (5.3.7).


Proof. The graph of (1) is precisely that of Proposition 5.3.2, and the fact that it

is G-symmetric follows from Lemma 5.1.1. So assume that case (C2.2) holds. By Lemma

5.1.1 we only need to show that V = S ∪ (S+S) unless S = X1 and q = 2. It follows from

Proposition 5.3.2 that Cay(V,X1) has diameter 2 (with G quasiprimitive) if and only if

qm > 2. Thus we may assume that S =⋃

σ∈〈τ〉Wβσ for some β ∈ Fq. It remains to prove

that V = S ∪ (S + S).

Let wβ be as in (5.3.8) and γ ∈ Fq, with γ 6= β. Define

g0 :=

(1 1

0 1

)and h0 :=

(0 −1

−1 γ0

),

where γ0 := 1 − β−1γ if β 6= 0 and γ0 := 0 if β = 0. If m = 2 let g := g0 and h := h0; if

m ≥ 3 define g and h by

g :=

(g0 0

0 Im−2

)

and

h :=

0 · · · 0

h0 1 · · · 1

0 Im−2

.

Then g, h ∈ GL(m, q) for all m ≥ 2, and wg1 + wh

1 = w1. Recall that q is odd if m = 2, so

we can take x ∈ (Fmq )# where

x :=

wβ if β 6= 0;

(0,−γ/2) if β = 0 and m = 2;

(0, 0, 1, 0, . . . , 0) if β = 0 and m ≥ 3.

Then for all cases y := xg−⊤

+xh−⊤

has first component γ. Hence, applying Lemma 5.3.3,

we have Wγ = (w1, y)G0 ⊆ Wβ +Wβ for any γ 6= β. Since also 0 ∪X1 ⊆ Wβ +Wβ, it

follows that V =Wβ ∪ (Wβ +Wβ). Therefore V = S ∪ (S+S), which completes the proof

of parts (2) and (3).

5.4. Class C4

In this case V = U ⊗W = Fkq ⊗ Fm

q with k,m ≥ 2, and B is a tensor product basis of

V (see Section 3.1), say,

B = ui ⊗ wj | 1 ≤ i ≤ k, 1 ≤ j ≤ m,

where BU := u1, . . . , uk and BW := w1, . . . , wm are fixed bases of U and W , respec-

tively. We choose τ to fix each of the vectors ui⊗wj. Then for any simple vector u⊗w ∈ V ,

we have (u⊗ w)τ = uτ ⊗ wτ , and for any v =∑r

i=1 (ai ⊗ bi) ∈ V ,

vτ =

r∑

i=1

aτi ⊗ bτi .

5.4. CLASS C4 91

Recall that k 6= m in the description given in Section 3.6.1; however, all of the results in

this section also hold for k = m, so we do not assume that k and m are distinct. In this

way the results yield useful information for the C7 case.

Recall from Section 3.1 that the tensor weight wt (v) of v ∈ V #, in the decomposition

V = Fkq ⊗ Fm

q , is the least number s such that v can be written as the sum of s simple

vectors in Fkq ⊗Fm

q . It follows from Lemma 3.1.1 that wt (v) ≤ min k,m for any v ∈ V #,

and that for each s ∈ 1, . . . ,min k,m there is a vector v ∈ V # with weight s.

Lemma 5.4.1. For any v ∈ V # we have wt (vσ) = wt (v) for each σ ∈ 〈τ〉.

Proof. Let v =∑wt(v)

i=1 ai ⊗ bi ∈ V #. It follows from the above that

vσ =

wt(v)∑

i=1

aσi ⊗ bσi

for any σ ∈ 〈τ〉, so wt (vσ) ≤ wt (v). Likewise wt(vσ

−1)≤ wt (v), so wt (v) ≤ wt (vσ).

Therefore wt (vσ) = wt (v).

Assume that H = ΓL(n, q). By Theorem 3.6.2,

G0 = (GL(k, q) ⊗GL(m, q))⋊ 〈τ〉 (5.4.1)

and

L = G0 ∩GL(n, q) = GL(k, q) ⊗GL(m, q). (5.4.2)

Lemma 5.4.2. Let G0 be as in (5.4.1). Then the G0-orbits in V # are the sets Ys for

each s ∈ 1, . . . ,min k,m, where

Ys := v ∈ V # | wt (v) = s. (5.4.3)

Proof. Let v ∈ Ys, say, v =∑s

i=1 ai ⊗ bi. It follows from Lemmas 3.1.1 and 5.4.1

that vG0 ⊆ Ys. Let v′ =

∑si=1 a

′i ⊗ b′i ∈ Ys. It follows from Lemma 3.1.1 that we can find

g ∈ GL(k, q) and h ∈ GL(m, q) for which agi = a′i and bgi = b′i for all i. So vg⊗h = v′, and

thus Ys ⊆ vL ⊆ vG0 . Therefore Ys is an orbit of G0 for all s ∈ 1, . . . ,min k,m, andthe sets Ys are all the G0-orbits in V

#.


5.1.2 with G0 as in (5.4.1), where k and m may be equal. Then Γ is G-symmetric with

diameter 2 if and only if Γ ∼= Cay(V, Ys), where s ≥ 12min k,m and Ys is as in (5.4.3).

Proof. As in the proof of Proposition 5.3.2 we only need to show that V \Ys ⊆ Ys+Ys

under the given conditions. Set µ := min k,m. If s < µ/2 then it follows from Lemma

3.1.1 that Yµ * Ys + Ys, so diam(Γ) > 2 in this case. Assume from now on that s ≥ µ/2.

Let v =∑s

i=1 ui ⊗ wi ∈ Ys, and r ∈ 1, . . . , µ with r 6= s. If r < s take z =∑s

i=1 xi ⊗ wi,

where xi = ui+1−ui for 1 ≤ i ≤ r− 1, xr = u1−ur, and xi = −ui for r+1 ≤ i ≤ s. Then


z ∈ Ys and v + z ∈ Yr by Lemma 3.1.1, so Yr ⊆ Ys + Ys. If r > s, extend u1, . . . , us and

w1, . . . , ws to linearly independent sets u1, . . . , ur ⊂ Fkq and w1, . . . , wr ⊆ Fm

q . Take

z =∑r

i=r−s+1 xi ⊗ wi with the vectors xi chosen as follows: if r = 2s, then xi = ui for

s+ 1 ≤ i ≤ r; if r = 2s− 1 then xs = ur − us, xi = ui for s+ 1 ≤ i ≤ r − 1, and xr = us;

and if r /∈ 2s, 2s − 1, then xi = ui+1 − ui for r − s + 1 ≤ i ≤ s − 1, xs = ur−s+1 − us

and xi = ui for s + 1 ≤ i ≤ r. Again by Lemma 3.1.1 we have z ∈ Ys and v + z ∈ Yr, so

Yr ⊆ Ys + Ys. Therefore Yr ⊆ Ys + Ys for all r 6= s, and thus diam(Γ) = 2.

Now assume that H = ΓSp(n, q). In this case k is even, m ≥ 3, q is odd, and

φ = φU ⊗ φW (see Subsection 3.3.2), where φU is a symplectic form on U and φW is a

nondegenerate symmetric bilinear form on W . By Theorem 3.6.3

G0 = (GSp(k, q)⊗GOǫ(m, q))⋊ 〈τ〉, (5.4.4)

and

L = GSp(k, q) ⊗GOǫ(m, q).

Recall from Proposition 5.2.1 that the GOǫ(m, q)-orbits inW# are S0 and S# if m is even,

and S0, S and S⊠ if m is odd, with S0 as defined in (3.3.1) and S#, S and S⊠ as in

(5.2.1). For each s ∈ 1, . . . ,min k,m let Ys be as in (5.4.3).

If v =∑s

i=1 ai ⊗ bi ∈ Ys, it is easy to see that

vG0 =

s∑

i=1

a′i ⊗ b′i ai ∈ U#, b′i ∈ bGOǫ(m,q)i

.

If s = 1 then the set Y1 of simple vectors splits into the G0-orbits Yθ1 , where θ ∈ 0,# if

m is even and θ ∈ 0,,⊠ if m is odd, and

Y θ1 :=

a⊗ b | a ∈ U#, b ∈ Sθ

.

If s > 1 suppose that exactly r of the vectors bi belong in S# for some r, 0 ≤ r ≤ s; if

m is odd suppose further that exactly r belong in S and r⊠ in S⊠. If m is even then

vG0 ⊂ Y rs , where

Y rs :=

s∑

i=1

a′i ⊗ b′i ∈ Ys exactly r of the vectors b′i are in S#

,

and if m is even then vG0 ⊂ Y r,r⊠s , where

Y r,r⊠s :=

s∑

i=1

a′i ⊗ b′i ∈ Ys exactly rθ of the vectors b′i are in Sθ

.

Note that the sets Y rs and Y

r,r⊠s above are, in general, not G0-orbits. For instance, if

s = 2, the weight-2 vectors a1 ⊗ b1 + a2 ⊗ b2, a′1 ⊗ b′1 + a′2 ⊗ b′2 ∈ Y 0

2 (or Y 0,02 if m is even),

such that b1 ⊥ b2 and b′1 ⊥/ b′2, belong to different G0-orbits.


5.1.2 with G0 as in (5.4.4), where k and m may be equal. If Γ is G-symmetric with

5.5. CLASS C5 93

diameter 2, then Γ ∼= Cay(V, S) where S = vG0 for some v ∈ Ys, where Ys is as in (5.4.3)

and s ≥ 12min k,m.

Proof. This follows immediately from the discussion above and Proposition 5.4.3.

5.5. Class C5

In this case n ≥ 2, d/n is composite with a prime divisor r, and V has a fixed ordered

basis

B := (v1, . . . , vn).

Let q0 := q1/r and let Fq0 denote the subfield of Fq of index r. Let V0 be the Fq0-span of

B. Then V0 is a vector space over Fq0 that is contained in V , but V0 is not an Fq-subspace

of V .

Regard the field Fq as a vector space of dimension r over Fq0 , and for any a ∈ 1, . . . , r,define

K(a) :=

Fq if a = r,

Fq0 otherwise.(5.5.1)

The significance of K(a) is given by Lemma 5.5.1.

Lemma 5.5.1. Let Fq0 be a proper nontrivial subfield of Fq with prime index r, and

let D be a nontrivial Fq0-subspace of Fq with dimFq0(D) = a. Then

λ ∈ Fq | λD = D = K(a),

where K(a) is as defined in (5.5.1).

Proof. Let K := λ ∈ Fq | λD = D. Clearly, Fq0 ⊆ K since D is a vector space over

Fq0 . It is easy to see that K# is a subgroup of the multiplicative group F#q . In general,

for any distinct λ, µ ∈ K, we have (λ − µ)D ⊆ λD − µD ⊆ D + D = D, and hence

(λ− µ)D = D as D contains no zero divisors. So K is a group under addition in Fq, and

thus K is a subfield of Fq. In particular, Fq0 ≤ K ≤ Fq, and since Fq0 has prime index in

Fq, the only possibilities for K are Fq0 and Fq. We get K = Fq if and only if Fq ⊆ D, that

is, D = Fq, in which case a = r. Therefore K = K(a), as asserted.

For a ∈ 1, . . . , r define

η(a) :=

[r

a

]

q0∣∣∣F#q : K(a)#

∣∣∣, (5.5.2)

where [r

a

]

q0

:=a−1∏

i=0

qr0 − qi0qa0 − qi0

,

the number of a-dimensional subspaces of Frq0 . Lemma 5.5.2 gives some elementary obser-

vations about η, whose significance will be apparent in Corollary 5.5.6.


Lemma 5.5.2. Let a ∈ 1, . . . , r and let D denote the set of Fq0-subspaces of Fq

with dimension a. Let K(a) and η(a) be as in (5.5.1) and (5.5.2), respectively. Then for

D ∈ D, the sets [D] = λD | λ ∈ F#q partition D. Moreover, |[D]| = |F#

q : K(a)#|, andthe number of distinct parts [D] in D is η(a).

Proof. It follows immediately from the definition of [D] that [D] | D ∈ D is a

partition of D. For any D ∈ D and λ, µ ∈ F#q , note that λD = µD if and only if λ−1µD =

D, and, equivalently, λ−1µ ∈ K(a) by Lemma 5.5.1. Therefore |[D]| = |F#q : K(a)#|, thus

proving the first assertion. Hence the number of classes [D] in D is |D|/|F#q : K(a)#|,

which completes the proof since

|D| =[r

a

]

q0

.

To any v =∑n

i=1 αivi ∈ V we can associate the Fq0-subspace

Dv := 〈α1, . . . , αn〉Fq0. (5.5.3)

Set

c(v) := dimFq0(Dv), (5.5.4)

and note that c(v) ≤ r (since Dv is an Fq0-subspace of Fq) and c(v) ≤ n (since Dv is

n-generated). For any λ ∈ Fq it is clear that Dλv = λDv, so c(λv) = c(v). For any

σ ∈ Aut (Fq) we have Dvσ = 〈ασ1 , . . . , α

σn〉Fq0

; since σ|Fq0∈ Aut (Fq0), it follows that

(Dv)σ := δσ | δ ∈ Dv = Dvσ and |Dvσ | = |(Dv)

σ| = |Dv |. Hence, in particular,

c(vσ) = c(v). As in Lemma 5.5.2, let

[Dv] := λDv | λ ∈ F#q ,

and observe that Du ∈ [Dvσ ] if and only if Du = λDvσ =(λσ

−1

Dv

)σfor some λ ∈ F#

q .

Hence (Du)σ−1

= Duσ−1 = λσ

−1

Dv , so that Duσ−1 ∈ [Dv ]. Thus [Dvσ ] = [Dv ]

σ.

5.5.1. Case H = ΓL(n, q). By Theorem 3.6.2

G0 = (GL(n, q0) Zq−1)⋊ 〈τ〉

and L = GL(n, q0) Zq−1.

The main result in this subsection, which relies on the value of the parameter c(v), is

the following:


5.1.2 with H = ΓL(n, q) and i = 5. Then Γ is connected and G-symmetric if and only if

Γ ∼= Cay(V, vG0) for some v ∈ V #. Moreover, if Dv and c(v) are as in (5.5.3) and (5.5.4),

respectively, then the following hold.

(1) If c(v) = r or c(v) = r − 1 then diam(Γ) = 2.

5.5. CLASS C5 95

(2) If c(v) = 1 then diam(Γ) = min n, r. In particular diam(Γ) = 2 if and only if

n = 2 or r = 2.

(3) If 2 ≤ c(v) < 12min n, r then diam(Γ) > 2.

(4) Let η be as defined in (5.5.2), s be the largest divisor of d/n with s ≤ η(c(v)), and

k1(q0) :=

18s/17 if q0 = 2;

s− 5/4 if q0 > 2.

If 3 ≤ n < r and n/2 ≤ c(v) < (r(n− 2) + k1(q0))/(2n), then diam(Γ) > 2.

The cases not covered by Proposition 5.5.3 are discussed briefly at the end of the

section. The proof of Proposition 5.5.3 is given after Lemma 5.5.7, and relies on several

intermediate results. We begin by describing the GL(n, q0)-orbits in terms of the subspaces

Dv, which in turn leads to a description of the G0-orbits in V#.

Lemma 5.5.4. For any v ∈ V # let Dv and c(v) be as in (5.5.3) and (5.5.4), respec-

tively, and let U denote the set of all Fq0-independent c(v)-tuples in V0. Then for any fixed

Fq0-basis β1, . . . , βc(v) of Dv,

vGL(n,q0) =

c(v)∑

i=1

βiui (u1, . . . , uc(v)) ∈ U

=u ∈ V # Du = Dv

.

Proof. Suppose that v =∑n

i=1 αivi. Define

U :=u ∈ V # Du = Dv

(5.5.5)

and

W :=

c(v)∑

i=1

βiui (u1, . . . , uc(v)) ∈ U

. (5.5.6)

Claim 1: vGL(n,q0) ⊆ U . Let g ∈ GL(n, q0) with matrix [gjk] with respect to B. Thenvg =

∑nk=1 α

′kvk, where α

′k =

∑nj=1 αjgjk ∈ Dv for each k. Hence Dvg ≤ Dv. Since v and

g are arbitrary, we also have Dv ≤ Dvg . So Dvg = Dv , and therefore vGL(n,q0) ⊆ U .

Claim 2: U ⊆W . Let u =∑n

j=1 α′jvj ∈ X. Writing α′

j =∑c(v)

i=1 βiγij for each j, where

all γij ∈ Fq0 , we get u =∑c(v)

i=1 βiui, with ui =∑n

j=1 γijvj ∈ V0 for all i. It remains to show

that the set u := u1, . . . , uc(v) is Fq0-independent. Indeed, let u′1, . . . , u′b be a maximal

Fq0-independent subset of u, and extend this to an ordered Fq0-basis B′ := (u′1, . . . , u′d)

of V0. Then u =∑b

k=1 β′kuk for some β′1, . . . , β

′b ∈ Fq, and if g ∈ GL(n, q0) is the change

of basis matrix from B′ to B, then ug =∑b

k=1 β′kvk. So Du = Dug by Claim 1, and thus

b ≤ c(v) = dimFq0(Du) = dimFq0

(Dug) ≤ b. Hence b = c(v) and u is Fq0-independent.

Therefore U ⊆W .

Claim 3: W ⊆ vGL(n,q0). It is easy to see thatW is contained in one orbit of GL(n, q0),

and it follows from Claims 1 and 2 that v ∈W . So W ⊆ vGL(n,q0), as claimed.


Thus we have vGL(n,q0) = U =W by Claims 1 – 3.

Proposition 5.5.5. For any v ∈ V # let Dv and c(v) be as in (5.5.3) and (5.5.4),

respectively, and let U be the set of all Fq0-independent c(v)-tuples in V0. Then for any

fixed Fq0-basis β1, . . . , βc(v) of Dv we have

vL =

λ

c(v)∑

i=1

βiui (u1, . . . , uc(v)) ∈ U , λ ∈ F#q

=u ∈ V # Du = λDv, λ ∈ F#

q

and

vG0 =

λ

c(v)∑

i=1

βσi ui (u1, . . . , uc(v)) ∈ U , λ ∈ F#q , σ ∈ 〈τ〉

=u ∈ V # Du = λ(Dv)

σ, λ ∈ F#q , σ ∈ 〈τ〉

.

Proof. Let U ′ := u ∈ V # | Du = λDv for some λ ∈ F#q . It follows from Lemma

5.5.4 that

vL =⋃

λ∈F#q

λU ⊆ U ′,

with U as in (5.5.6). For any w ∈ U ′ with, say, Dw = µDv for µ ∈ F#q , it is easy to show

that µ−1w ∈ U . Hence w ∈ µU ⊆ vL, and therefore vL = U ′. It follows that

vG0 =⋃

σ∈〈τ〉uσ | u ∈ vL ⊆W ′,

where W ′ := u ∈ V # | Du = λ(Dv)σ, λ ∈ F#

q , σ ∈ 〈τ〉. For any w ∈ W with Dw =

µ(Dv)ρ for µ ∈ F#

q and ρ ∈ 〈τ〉, we have w ∈ (vρ)L ⊆ vG0 . Therefore vG0 = W ′, and the

rest follows from Lemma 5.5.4.

Corollary 5.5.6. Let v ∈ V #, and let K, η, Dv and c(v) be as defined in (5.5.1),

(5.5.2), (5.5.3) and (5.5.4), respectively.

(1) For a ∈ 1, . . . ,min n, r, the number of orbits vL with c(v) = a is η(a).

(2)∣∣vL∣∣ =

[n

c(v)

]

q0

|GL(c(v), q0)|∣∣∣F#

q : K(c(v))#∣∣∣

(3)∣∣vG0

∣∣ = s∣∣vL∣∣ for some divisor s of d/n with s ≤ η(c(v)).

Proof. It follows from Proposition 5.5.5 that the map vL 7→ [Dv] := λDv | λ ∈ F#q

is a one-to-one correspondence between the set of L-orbits and the set of classes [D] of

Fq0-subspaces of Fq. Therefore, by Lemma 5.5.2, there are exactly η(a) orbits vL with

c(v) = a, which proves part (1). Also by Proposition 5.5.5, we have |vL| = |U||[Dv ]|,where U is the set of Fq0-independent c(v)-tuples in V0. So

|U| =[

n

c(v)

]

q0

|GL(c(v), q0)|,

5.5. CLASS C5 97

and by Lemma 5.5.2, |[Dv ]| = |F#q : K(c(v))#|. This proves part (2). Applying part (5)

of Theorem 1.1.2 we get∣∣vG0

∣∣ = s∣∣vL∣∣ for some s dividing |G0 : L| = |Aut (Fq) | = d/n.

Also s ≤ η(c(v)) since c(vσ) = c(v), which proves part (3).

Lemma 5.5.7. Let Γ = Cay(V, vG0) for some v ∈ V #, and let c(v) be as in (5.5.4).

Let w ∈ V .

(1) If w ∈ vG0 + vG0 then c(w) ≤ 2c(v).

(2) If Dw < Dv then w ∈ vG0 + vG0 .

Proof. Let U and W denote the sets of Fq0-independent c(v)- and c(w)-tuples, re-

spectively, in V .

Suppose first that w = x + y for some x, y ∈ vG0 . Then by Proposition 5.5.5 we can

write x and y as x =∑c(v)

i=1 λβρi xi and y =

∑c(v)i=1 µβ

σi yi for some scalars λ, µ ∈ F#

q , maps

ρ, σ ∈ Aut (Fq), and c(v)-tuples(x1, . . . , xc(v)

),(y1, . . . , yc(v)

)∈ U . Hence

Dw = Dx+y ⊆ 〈λβρ1 , . . . , λβρc(v), µβσ1 , . . . , µβσc(v)〉Fq0,

and therefore c(w) = c(x+ y) ≤ 2c(v). This proves part (1).

To prove part (2), observe that Lemma 5.5.4 implies that we can write v and w as

v =∑c(v)

i=1 γiui and w =∑c(w)

i=1 δizi for some(u1, . . . , uc(v)

)∈ U and

(z1, . . . , zc(w)

)∈ W,

and for some fixed Fq0-bases γi, . . . , γc(v) and δ1, . . . , δc(w) of Dv and Dw, respectively.

Since Dw < Dv then c(w) < c(v), and we can extend δ1, . . . , δc(w) to an Fq0-basis

δ1, . . . , δc(v) of Dv, and(z1, . . . , zc(w)

)to(z1, . . . , zc(v)

)∈ U . Set x :=

∑c(v)i=1 δizi and

y :=∑c(v)

i=1 δiyi, where yi := zi+1 − zi if 1 ≤ i ≤ c(w)− 1, yc(w) := z1 − zc(w), and yi := −ziif c(w)+1 ≤ i ≤ c(v). Then

(y1, . . . , yc(v)

)∈ U and Dx = Dy = Dv , so by Lemma 5.5.4 we

have x, y ∈ vGL(n,q0) ⊆ vG0 . Therefore x+ y ∈ vG0 + vG0 . Now Dw = Dx+y, so applying

Lemma 5.5.4 again we get w ∈ (x+ y)GL(n,q0) ⊆ vG0 + vG0 . Thus (2) holds.

Proof of Proposition 5.5.3. Suppose that r − 1 ≤ c(v) ≤ r. Observe that η(r −1) = η(r) = 1, so for either value of c(v) we have vL = u ∈ V | c(u) = c(v), whichin turn implies that vG0 = vL. If c(v) = r then Dv = Fq, and clearly Dw < Dv for any

w ∈ V #\vG0 . So w ∈ vG0+vG0 by part (2) of Lemma 5.5.7, and thus V #\vG0 ⊆ vG0+vG0 .

Therefore diam(Γ) = 2. Now suppose that c(v) = r − 1, and let w ∈ V # \ vG0 . If

c(w) < r − 1 then it follows from part (1) of Corollary 5.5.6 that Dw < λDv = Dλv for

some λ ∈ F#q . Thus w ∈ (λv)G0 + (λv)G0 = vG0 + vG0 by Lemma 5.5.4. If c(w) = r let

x :=∑r−1

i=1 αivi and y :=∑r−2

i=1 βivi + γvr, where α1, . . . , αr−1 is an Fq0-basis of Dv,

γ ∈ F#q \Dv , and

βi :=

αi+1 − αi if 1 ≤ i ≤ r − 3;

α1 − αr−2 if i = r − 2.


Then c(x) = c(y) = r − 1 and c(x+ y) = r, so x, y ∈ vG0 and w ∈ (x+ y)G0 ⊆ vG0 + vG0 .

Therefore V # \ vG0 ⊆ vG0 + vG0 , and again we have diam(Γ) = 2. This completes the

proof of part (1).

If c(v) = 1 then we get the special case vL = vG0 = (FqV0)#. Let distΓ(0V , w) denote

the distance in Γ between the vertices 0V and w; we claim that distΓ(0V , w) = c(w)

for any w ∈ V . Let ℓ(w) := distΓ(0V , w). Then w ∈ Y by Proposition 5.5.5, where

Y is as in (5.5.6), so w can be written as a sum of c(w) elements of (FqV0)# and thus

ℓ(w) ≤ c(w). On the other hand w =∑ℓ(w)

i=1 λiui, where λi ∈ F#q and ui ∈ V #

0 for all i.

Writing each ui as ui =∑n

j=1 µi,jwj where µi,j ∈ Fq0 for all i, j, we get w =∑n

j=1 λ′jwj

where λ′j =∑ℓ(w)

j=1 λiµi,j for each j. Hence Dw ≤ 〈λ1, . . . , λℓ(w)〉Fq0, so that c(w) ≤ ℓ(w).

Therefore ℓ(w) = c(w), as claimed. It follows immediately that diam(Γ) = min n, r, andthat diam(Γ) = 2 if and only if n = 2 or r = 2. This proves (2).

Suppose that diam(Γ) = 2. Then c(w) ≤ 2c(v) for any w ∈ V # by part (1) of

Lemma 5.5.7, and in particular 2c(v) ≥ min n, r since there clearly exists u ∈ V # with

c(u) = min n, r. Hence c(v) ≤ 12min n, r implies that diam(Γ) > 2, and part (3) holds.

Finally, let a := c(v), S := vG0 , and η(a) as in (5.5.2). By Corollary 5.5.6 we have

|S| ≤[n

a

]

q0

|GL(a, q0)|∣∣∣F#

q : F#q0

∣∣∣ s,

where s is the largest divisor of d/n with s ≤ η(a). Hence

|S|2 + 1 < q2an0

∣∣∣F#q : F#

q0

∣∣∣2s2.

Observe that s < qst0 for all s ≥ 1, where t = 917 if q0 = 2, and t = 1

2 if q0 ≥ 3. Also, for

q0 ≥ 3, we have q0 − 1 > q5/80 , so that

∣∣∣F#q : F#

q0

∣∣∣ < qr−5/80 . With these bounds we obtain

|S|2 + 1 < q2(an+r)+k1(q0)0 ,

where k1(q0) is as defined in (4). It is easy to verify that if a < (r(n − 2) − k1(q0))/(2n)

then 2(an + r) + k1(q0) < rn, so |S|2 + 1 < |V |, and thus diam(Γ) > 2 by Lemma 5.1.1.

This proves part (4).

Remark. Some small cases covered by Proposition 5.5.3 are summarised in Table

5.5.1. The cases left unresolved by Proposition 5.5.3 are the following:

(1) 5 ≤ r ≤ n, r/2 ≤ c(v) ≤ r − 2;

(2) 2 = n ≤ r − 2, c(v) = 2;

(3) 3 ≤ n < r, max n/2, (r(n − 2)− k1(q0))/(2n) ≤ c(v) ≤ r − 2.

Let a := c(v) < r, S = vG0 , and s as in Proposition 5.5.3 (4). Then s ≥ 1,∣∣∣F#

q : F#q0

∣∣∣ >qr−20 and

[n

a

]

q0

|GL(a, q0)| > q2a(n−1)0 ,

5.5. CLASS C5 99

r n c(v) Conclusion about Γ = Cay(V, vG0)

2 ≥ 2 1 diam(Γ) = 2 by Proposition 5.5.3 (2)

2 diam(Γ) = 2 by Proposition 5.5.3 (1)

3 2 1 diam(Γ) = 2 by Proposition 5.5.3 (2)










2 diam(Γ) > 2 by Proposition 5.5.3 (3)



Table 5.5.1

so

|G0|2 + 1 ≥([

n

a

]

q0

|GL(a, q0)|∣∣∣F#

q : F#q0

∣∣∣ s)2

+ 1

> q2a(n−1)+2(r−2)0 .

It is easy to show that if condition (1) or (2) holds then 2(a(n−1)+ r−2) > rn, and thus

|G0|2 + 1 > |V |. This, unfortunately, does not lead to any conclusion about diam(Γ).

5.5.2. Case H = ΓSp(n, q). By Theorem 3.6.3,

G0 = (GSp(n, q0) Zq−1)⋊ 〈τ〉

and L = GSp(n, q0) Zq−1. The main result in this section is parallel to part (4) of

Proposition 5.5.3.


5.1.2 with H = ΓSp(n, q) and i = 5. Then Γ is connected and G-symmetric if and only if

Γ ∼= Cay(V, vG0) for some v ∈ V #. Moreover, if s := d/n and c(v) is as defined in (5.5.4),

and if

t :=

9/17 if q0 = 2,

1/2 if q0 > 2

then the following hold:

(1) If c(v) < 12min n, r then diam(Γ) > 2.

(2) If 3 ≤ n ≤ r, c(v) ≥ n/2 and r > (n2 + n+ 2st)/(n − 2), then diam(Γ) > 2.

Proof. In view of Lemma 5.1.1 and part (1) of Proposition 5.5.3, we only need to

prove statement (2).


Let S = vG0 . Observe that for any λ ∈ F#q and g ∈ GSp(n, q0), we have λvg = vλg ∈

vGSp(n,q0) if and only if λIn ∈ Zq0−1, the subgroup of scalar matrices in GL(n, q0). Hence

vL =⋃

λ∈F#qλvGSp(n,q0) can be written as a disjoint union vL =

⋃λ∈T λv

GSp(n,q0), where

T is a transversal of F#q0 in F#

q . Thus |vL| ≤ |T ||GSp(n, q0)| = (qr0 − 1)|Sp(n, q0)| and|S| ≤ s|vL|, where s = |G0 : L| = |Aut (Fq) | = d/n. Using the formula for the order of the

symplectic group given in Table 3.3.2, we obtain the bound |Sp(n, q0)| < q(n2+n)/20 . Also,

as in the proof of Proposition 5.5.3 (4), we have s < qst0 for any s, where t = 917 if q0 = 2,

and t = 12 if q0 ≥ 3. Hence

|S|2 + 1 < s2(qr0 − 1)2qn2+n

0 < qn2+n+2r+2st

0 .

If r > (n2+n+2st)/(n−2) then rn > n2+n+2r+2st, so |V | > |S|2+1 and diam(Γ) > 2

by Lemma 5.1.1. Therefore part (2) holds.

5.6. Class C6

In this case dim (V ) = rt where r is a prime different from p, q is the smallest power

of p such that q ≡ 1 (mod |Z(R)|) for some R in Table 3.6.1, and

G0 = (Zq−1 R).T ⋊ 〈τ〉,

with T as in Table 3.6.1. By Theorems 3.6.2 and 3.6.3, if H = ΓL(n, q) then R is of type

1 or 2, and if H = ΓSp(n, q) with q odd then R is of type 4.

Proposition 5.6.1. Let V and G0 be as above, and let Γ := Cay(V, S) for some

G0-orbit S ⊆ V #.

(1) Suppose that r is odd, q ≡ 1 (mod r), and R is of type 1. If diam(Γ) = 2 then

either rt = 3, or 1 ≤ t ≤ 3, r ≤ r0(t), and q < q0(r, t), where r0(t) and q0(r, t)

are given in Table 5.6.1.

(2) Suppose that r = 2, t ≥ 2, q ≡ 1 (mod 4), and R is of type 2. If diam(Γ) = 2

then 2 ≤ t ≤ 6 and q < q0(t), where q0(t) is given in Table 5.6.2.

(3) Suppose that r = 2, t ≥ 2, q is odd, and R is of type 4. If diam(Γ) = 2 then

2 ≤ t ≤ 7 and q < q0(t), where q0(t) is given in Table 5.6.3.

(4) Suppose that r = 2, t = 1, q is odd, and R is of type 2 or 4. Then diam(Γ) = 2

for any S.

Proof. If q = pℓ and R is of type 1 or 2, then |G0| = ℓ(q − 1)r2t|Sp(2t, r)|. Recall

from Section 3.3 that |Sp(2t, r)| = rt2∏t

i=1 (r2i − 1), so that

|G0|2 + 1 < ℓ2(q − 1)2r2t2+4t

t∏

i=1

r4i = ℓ2(q − 1)2r4t2+6t.

Suppose first that r is odd and R is of type 1. We have q ≡ 1 (mod r), so q ≥ 4. Note

that ℓ ≤ p9ℓ/17 for all ℓ ≥ 1 if p ≥ 3, and for all ℓ ≥ 2 if p = 2, so ℓ ≤ q9/17 for all q ≥ 4.

5.6. CLASS C6 101

t 1 2 3

r0(t) 27 7 4

q0(3, t) - 170 12

q0(5, t) 3823 8 -

q0(7, t) 138 - -

q0(11, t) 21 - -

q0(13, t) 13 - -

q0(17, t) 8 - -

q0(19, t) 7 - -

q0(23, t) 5 - -

Table 5.6.1. Bounds for r and q when R is Type 1

t 2 3 4 5 6

q0(t) 32767 645 86 22 8

Table 5.6.2. Bounds for q when R is Type 2 and t ≥ 2

t 2 3 4 5 6 7

q0(t) 1919 149 38 13 6 4

Table 5.6.3. Bounds for q when R is Type 4 and t ≥ 2

We then have

|G0|2 + 1 < q18/17q2r4t2+6t = q52/17r4t

2+6t.

Fix q and let φ(r, t) := ln (q52/17r4t2+6t

)− ln (qrt). Using elementary calculus we can show

that φ(r, t) < 0 for all r ≥ 3 whenever t ≥ 4, whereas for each t ∈ 1, 2, 3 there exists

some r0(t) such that φ(r, t) < 0 for all r ≥ r0(t). The values of r0(t) are given in Table

5.6.1. Clearly, φ(r, t) < 0 implies that |G0|2 + 1 < |V |, and consequently (by Lemma

5.1.1), diam(Γ) > 2 for all Cayley graphs Γ arising from the G0-orbits. So if diam(Γ) = 2

for some Γ then it must be that 1 ≤ t ≤ 3 and r < r0(t).

Now fix t ∈ 1, 2, 3 and r < r0(t), and define π(q) := lnµ2 − ln (|V | − 1), where

µ := q9/17(q − 1)|Sp(2t, r)|. Note that π is a decreasing function of q and µ > |G0|. If

rt = 3 (i.e., (r, t) = (3, 1)) then π(q) > 0 for all q ≥ 4, which gives no information on the

diameter of the Cayley graphs that arise. If rt > 3 then for each pair (r, t) there is a value

q0(r, t) such that π(q) < 0 for all q ≥ q0(r, t); the values q0(r, t) are again in Table 5.6.1.

Since π(q) < 0 again implies that |G0|2 + 1 < |V |, it follows that if diam(Γ) = 2 for some

Γ then 1 ≤ t ≤ 3, r < r0(t), and q < q0(r, t). This proves part (1).

To prove (2), suppose that r = 2, t ≥ 2, and R is of type 2. In this case q ≡ 1

(mod 4) so q ≥ 5 and ℓ ≤ 2. Hence q1/2 > 2 and 4(q − 1) < q3/2 for all such q. We have


|G0| ≤ (q − 1)22t+1|Sp(2t, r)|, so

|G0|2 + 1 < (q − 1)224t2+6t+2 = 16(q − 1)224t

2+6t−2 < q2t2+3t+2.

Define ρ(t) := 2t2 + 3t + 2 − 2t. Again, using elementary calculus, we can show that

ρ(t) < 0 for all t ≥ 7. So for these cases |G0|2+1 < |V |, and by Lemma 5.1.1 diam(Γ) > 2

for all possible Γ. Fix t ∈ 2, . . . , 6 and let σ(q) := ln ν2 − ln (|V | − 1), where ν :=

(q−1)22t+1|Sp(2t, 2)|. Note that ν > |G0|. It can be shown that σ is a decreasing function

of q, and that for each t there is a value q0(t), given in Table 5.6.2, such that σ(q) < 0 for

all q ≥ q0(t). Hence, for these t and q, we again get |G0|2 + 1 < |V | so that diam(Γ) > 2

for all Γ. Therefore if diam(Γ) = 2 for some Γ we must have 2 ≤ t ≤ 6 and q < q0(t).

Thus part (2) holds.

For (3), suppose that r = 2, t ≥ 2, and R is of type 4. Then q is odd, and since

|Z(R)| = 2 then ℓ = 1. We have |G0| ≤ (q − 1)22t|O−(2t, 2)|, where |O−(2t, 2)| =

2t(t−1)+1(2t + 1)∏t−1

i=1 (22i − 1). Hence

|G0|2 + 1 < (q − 1)224t24t2−2t+4 = 16(q − 1)224t

2+2t.

Since q ≥ 3 we have 4(q − 1) < q2 and q2/3 > 2, so

|G0|2 + 1 < q23(4t2+2t+3).

Set ρ(t) := 23 (4t

2 + 2t + 3) − 2t. As in the previous cases we can show that ρ(t) < 0 for

all t ≥ 8, so |G0|2 + 1 < |V |, and diam(Γ) > 2 for all possible Γ, whenever t ≥ 8. Fix

t ∈ 2, . . . , 7 and define σ(q) := ln ν2 − ln (|V | − 1), where ν := (q − 1)22t|O−(2t, 2)|.Then ν > |G0|. For all t the function σ(q) is decreasing, and for each t there is a value

q0(t), given in Table 5.6.3, such that σ(q) < 0 for all q ≥ q0(t). Hence for each t we have

diam(Γ) > 2 whenever q ≥ q0(t). That is, if diam(Γ) = 2 for some Γ then 2 ≤ t ≤ 7 and

q < q0(t). This proves part (3).

Finally, define the matrices a, b, c ∈ GL(V ) by

a :=

(0 1

−1 0

), b :=

(α 0

0 α

), and c :=

(β δ

δ −β

),

where α, β, γ ∈ Fq such that α2 = −1 and β2 + γ2 = −1. Then a representation of R in

GL(2, q) is 〈a, b, c〉 if R is type 2, and 〈a, c〉 if R is type 4 (see [31, pp. 153-154]). Since

R is irreducible on V in both cases, any R-orbit vR in V # contains a basis v1, v2 of V ,

and vG0 contains 〈v1〉# ∪ 〈v2〉#. Clearly V # ⊆ 〈v1〉# + 〈v2〉#. Therefore V ⊆ vG0 + vG0 ,

and thus diam(Γ) = 2. This proves (4), and completes the proof of the proposition.

5.7. Class C7

In this case V = ⊗ti=1Ui with Ui = Fm

q for all i, m ≥ 2, t ≥ 2, and d = mt. Assume

that B is a tensor product basis of V (see Section 3.1), with

B :=⊗t

i=1ui,j 1 ≤ j ≤ m.

5.7. CLASS C7 103

As in Section 5.4, it is not difficult to show that for any v =∑r

i=1

(⊗t

j=1vi,j

)∈ V # we

have

vτ =

r∑

i=1

(⊗t

j=1vτi,j

),

where τ acts on each Ui with respect to the basis ui,j | 1 ≤ j ≤ m.Recall from Section 3.1 that the tensor weight wt (v) of v ∈ V #, in the decomposition

V = ⊗ti=1Ui, is the least s such that v is the sum of s simple vectors in ⊗t

i=1Ui.

Lemma 5.7.1. For any v ∈ V #, g ∈ ⊗ti=1GL(m, q) and σ ∈ 〈τ〉, we have wt (vg) =

wt (v) and wt (vσ) = wt (v).

Proof. Let v =∑wt(v)

i=1

(⊗t

j=1vi,j

). It follows from the above that

vσ =

wt(v)∑

i=1

(⊗t

j=1vσi,j

)

for any σ ∈ 〈τ〉, so wt (vσ) ≤ wt (v). Since also wt(vσ

−1)

≤ wt (v), we then have

wt (vσ) = wt (v). The proof that wt (vg) = wt (v) is similar.

5.7.1. Case H = ΓL(n, q). By Theorem 3.6.2

G0 = (GL(m, q) ≀⊗ Sym (t))⋊ 〈τ〉. (5.7.1)

If t = 2 then the results in Section 5.4 hold, with m = n, and we obtain the examples

in Proposition 5.4.3.

Corollary 5.7.2. Let V = ⊗ti=1F

mq and let G0 be as in (5.7.1) with m ≥ 2 and t = 2.

Then the G0-orbits in V # are the sets Ys for each s ∈ 1, . . . ,m, where Ys is defined as

in Lemma (5.4.2). Moreover, for any G0-orbit S ⊆ V #, the graph Cay(V, S) has diameter

2 if and only if S = Ys for some s ≥ m/2.

Proof. This follows immediately from Lemma 5.4.2 and Proposition 5.4.3.

If t ≥ 3 it is not easy to describe the G0-orbits in V#. We do know, from Lemma 5.7.1,

that for any v ∈ V #, vG0 consists of vectors having the same tensor weight as v. This in

itself is not very helpful, since describing the G0-orbits involves determining the weight

of an arbitrary vector, which is not easy in general. Clearly, if Cay(V, S) has diameter

2, then S must consist of vectors whose weight is at least 12ω, where ω is the maximum

weight that occurs in V . The exact value of ω is difficult to determine; all we have so far

are the results in Section 3.1 (see Lemmas 3.1.6 and 3.1.7), which show that

m ≤ ω ≤ mt−3(m2 −

⌊m2

⌋).

Using Lemma 5.1.1, we get the following bounds which significantly reduce the cases

that have to be considered.


Proposition 5.7.3. Let Γ be a graph and let G ≤ Aut (Γ), such that G satisfies

Hypothesis 5.1.2 with G0 as in (5.7.1), m ≥ 2 and t ≥ 3. Then Γ is connected and G-

symmetric if and only if Γ ∼= Cay(V, vG0) for some v ∈ V #. Moreover, if diam(Γ) = 2

then either:

(1) m = 2 and t ∈ 3, 4, 5; or(2) t = 3 and m ∈ 3, 4, 5.

Proof. Since (αg1)⊗g2⊗· · ·⊗gt = g1⊗· · ·⊗(αgi)⊗· · ·⊗gt for all g1, . . . , gt ∈ GL(m, q)

and i ∈ 1, . . . , t, it follows that

|G0| ≤ |GL(m, q)|tt!ℓ/(q − 1)t−1

where q = pℓ. Noting that t < qt for all q ≥ 2 and t ≥ 3, and ℓ < q1/2 for q ≥ 3 (ℓ = 1 if

q = 2), we get |G0| ≤ qm2t+(t2+f(t))/2, where

f(t) :=

1 if q = 2;

t+ 2 otherwise.

Hence

|G0|2 + 1 < q2m2t+t2+f(t)+1.

Let φ(m, t) := 2m2t+ t2 + f(t) + 1−mt. Using elementary calculus it can be shown that

φ(m, t) < 0 whenever t ≥ 7 and m ≥ 2, and whenever m ≥ m0 for each value of t given

in Table 5.7.1. So for these (m, t) we have |G0|2 + 1 < |V |, and thus diam(Γ) > 2 for all

φ(m, t) m0

φ(m, 3) 7

φ(m, 4) 4

φ(m, 5) 3

φ(m, 6) 3

Table 5.7.1. Values for m0

Cayley graphs Γ arising from G0-orbits in V#. For the remaining (m, t) fix m and t and

let ρ(q) := lnµ − 12 ln (q

mt−1), where µ = |GL(m, q)|tt!q/(q − 1)t−1. Note that µ > |G0|and qm

t−1 < |V | − 1, so ρ(q) < 0 implies that |G0|2 + 1 < |V |. It can be shown that

ρ(q) < 0 for (m, t) ∈ (2, 6), (3, 4), (6, 3), so again diam(Γ) > 2 for all Cayley grpahs Γ

arising from the G0-orbits in V#, for these pairs (m, t). These leaves us with (m, t) where

m = 2 and 3 ≤ t ≤ 5, or t = 3 and 3 ≤ m ≤ 5. This completes the proof.

5.7.2. Case H = ΓSp(n, q). By Theorem 3.6.3, both q and t are odd and

G0 = (GSp(m, q) ≀⊗ Sym(t))⋊ 〈τ〉. (5.7.2)

Hence q, t ≥ 3.

5.7. CLASS C7 105


5.1.2 with G0 as in (5.7.2), m ≥ 2 and t ≥ 3. Then Γ is connected and G-symmetric if

and only if Γ ∼= Cay(V, vG0) for some v ∈ V #. Moreover, if diam(Γ) = 2 then either:

(1) m = 2 and t ∈ 3, 4, 5; or(2) t = 3 and m = 4.

Proof. Recall that |GSp(m, q)| = (q − 1)|Sp(m, q)| < (q − 1)qm2+m/2. We have

t < qt/2 for all q ≥ 3 and t ≥ 3, which implies that if q = pℓ then ℓ < q1/2 for all q ≥ 3,

and t! < q(t−1)(t+2)/4 . It follows that

|G0| ≤ |GSp(m, q)|tt!ℓ/(q − 1)t−1 < (q − 1)q(m2t+mt+1)/2+(t−1)(t−2)/4 .

We then have

|G0|2 + 1 < qt2/2+(m2+m−3/2)t+5.

Set φ(m, t) := t2/2 + (m2 + m − 3/2)t + 5 − mt. As in Proposition 5.7.3 we can show,

using calculus, that φ(m, t) < 0 whenever t ≥ 6 and m ≥ 2. Also, for each t ∈ 3, 4, 5there exists a value m0, given in Table 5.7.2, such that φ(m, t) < 0 for all m ≥ m0. Thus

|G0|2 +1 < |V | (and diam(Γ) > 2) for all m and t except possibly if m = 2 and 3 ≤ t ≤ 5,

or t = 3 and m = 4 (recall that m must be even). This completes the proof.

φ(m, t) m0

φ(m, 3) 5

φ(m, 4) 3

φ(m, 5) 3

Table 5.7.2. Values for m0

CHAPTER 6

Other quasiprimitive types

In this chapter we consider briefly graphs Γ with vertex-quasiprimitive automorphism

group G, where G has nonabelian socle and is maximal in Sym(V (Γ)) such that G is not

2-transitive. It can be deduced from Theorem 1.3.8 that one of the following holds:

(1) G = T d.(Out (T )× Sym(d)), d ≥ 2, for some nonabelian simple group T , acting

on V (Γ) = T d−1 with the diagonal action defined in (1.3.3) and (1.3.4);

(2) G = Sym (Λ) ≀ Sym (m), |Λ| ≥ 2 and m ≥ 2, acting on V (Γ) = Λm with the

product action defined in (1.3.1); or

(3) G is an almost simple group.

Assume throughout that Γ is connected and G-symmetric. Recall from Theorem 2.1.2

that Γ is then an orbital graph for G, and for any ω ∈ V (Γ), the set Γ(ω) of neighbours

of ω is an orbit of Gω. As usual, we are interested in the diameter 2 graphs. Our goal is

to prove Theorems 5 and 6.

6.1. Diagonal type subgroups

If G is a maximal subgroup of diagonal type then G = T d.(Out (T ) × Sym(d)) for

some d ≥ 2 and nonabelian simple group T , with the diagonal action (1.3.3) and (1.3.4)

on V (Γ) = T d−1. Fix

ω := (1T , . . . , 1T ) ∈ T d−1.

Recall that in the diagonal action the subgroup T d−1 acts regularly on itself, so Γ ∼=Cay(T d−1, S) by Theorem 2.2.4, where S is an orbit ofGω = Aut (T )×Sym(d) in T d−1\ωwith S−1 = S.

6.1.1. The case d = 2. If d = 2 then T d−1 = T , and if, say, S = tGω , then by the

definition of the diagonal action we get S = tAut(T ) ∪ (t−1)Aut(T ). Thus S = S−1 and is

a union of conjugacy classes of T . We get diameter 2 if S ∪ S2 = T . Examples exist for

those groups T that satisfy Thompson’s Conjecture, which we state below.

Thompson’s Conjecture. [5] Every finite nonabelian simple group T contains a

conjugacy class C such that C2 = T .

Thompson’s conjecture has been verified for the alternating groups by C. Hsu in [24],

for the sporadic simple groups by J. Neubuser et.al. in [38], and for most of the simple

groups of Lie type by E. Ellers and N. Gordeev in [18]. The remaining open cases are

summarised in [29]. Most of the proofs involve character-theoretic techniques rather than

107

108 6. OTHER QUASIPRIMITIVE TYPES

explicit descriptions of the conjugacy classes C. In the case of the alternating groups,

several papers have investigated the problem of identifying the conjugacy classes C whose

squares cover the whole group; the results in [8] are particularly useful.

R. Guralnick and G. Malle in [23] prove a weaker version of Thompson’s Conjecture.

Theorem 6.1.1. [23, Theorem 1.4] Let T be a finite nonabelian simple group. There

exist conjugacy classes C1 and C2 in T such that C1C2 = T#. Moreover, aside from

T = PSL(2, q) with q = 7 or q = 17, we can assume that each class Ci consists of elements

of order prime to 6.

Thus, if S contains a conjugacy class C that satisfies Thompson’s Conjecture, or if S

contains two conjugacy classes C1 and C2 with C1C2 = T#, then the graph Cay(T, S) has

diameter 2. It remains to determine if there exist diameter 2 graphs Cay(T, S) apart from

these.

Problem 1. Determine other possible S, apart from those that contain conjugacy

classes satisfying Thompson’s Conjecture or Theorem 6.1.1, for which Cay(T, S) has di-

ameter 2.

6.1.2. The case d ≥ 3. Suppose that d > 2. If diam(Γ) = 2, then by the remarks

after Lemma 2.2.5 we have |T |d−1 ≤ 1 + |S|2. Here |S| ≤ |Gω| = |T ||Out (T ) |d!, so

1 + |S|2 < |T |2|Out (T ) |2d2d. This yields

|T |d−3 < |Out (T ) |2d2d.

The following is known.

Theorem 6.1.2. [32] If T is a nonabelian simple group then

|Out (T ) | < log2 |T |.

Substituting this bound into the inequality above yields the following.

Proposition 6.1.3. Let Γ be a G-symmetric graph, where G = T d.(Out (T )×Sym(d))

for some d ≥ 4 and nonabelian simple group T , and G is a quasiprimitive group on

V (Γ) = T d−1 of diagonal type. If diam(Γ) = 2 then the order of T is bounded above by a

function of d; more precisely,

|T |d−3

log2 |T |< d2d.

Proof. This follows immediately from the preceding discussion.

Suppose that d = 3. Then V (Γ) = T 2 and S = (t1, t2)Gω for some (t1, t2) ∈ T 2 \ ω.

Hence S =⋃

(s1, s2)Aut(T ), where (s1, s2) varies over the elements of

(t1, t2)Sym(3) =

(t1, t2), (t2, t1),

(t−11 , t3

),(t3, t

−11

),(t−12 , t−1

3

),(t−13 , t−1

2

),

6.1. DIAGONAL TYPE SUBGROUPS 109

with t3 := t−11 t2. If S = S−1 then

(t−11 , t−1

2

)= (s1, s2)

σ for some (s1, s2) ∈ (t1, t2)Sym(3)

and σ ∈ Aut (T ). Proposition 6.1.4 gives a necessary condition for Γ to have diameter 2.

Proposition 6.1.4. Let Γ be a G-symmetric graph, where G = T 3.(Out (T )×Sym(3))

for some nonabelian simple group T , and G is a quasiprimitive group on V (Γ) = T 2 of

diagonal type. If diam(Γ) = 2 then Γ ∼= Cay(T 2, S) where S is an orbit of Aut (T )×Sym(3)

with S ⊆ (T 2)# and S = S−1, and S does not contain (t, 1T ), (1T , t), or (t, t) for any

t ∈ T .

Proof. By Theorems 2.1.1 and 2.2.4, it remains to prove that S does not contain

(t, 1T ), (1T , t) or (t, t) for any t ∈ T#. Suppose otherwise. Observe that if S contains an

element of one of the three types above, then it must also contain elements of the other

two types, so it is enough to consider the case where S = (t, 1T )Sym(3) for some t ∈ T#.

(Indeed, (t, 1T )Gω = (t, 1T )

Aut(T )∪(1T , t)Aut(T )∪

(t−1, t−1

)Aut(T ).) Let C := tAut(T ). Then

S = (s, 1), (1, s),(s−1, s−1

)| s ∈ C, and thus S = S−1 implies that C = C−1. Hence, if

(x, y) ∈ S2 with x, y /∈ C ∪1T , then (x, y) =(s−11 s−1

2 , s−11 s−1

2

)for some s1, s2 ∈ C. That

is, S2 does not contain elements (x, y) with x, y /∈ C ∪ 1T and x 6= y. So S ∪ S2 6= T 2

and diam(Γ) > 2, a contradiction. Therefore S cannot contain (t, 1T ), (1T , t) or (t, t) for

any t ∈ T .

Diameter 2 graphs do exist when d = 3, and some examples are given below.

Example 6.1.5. Let G = T 3.(Out (T ) × Sym(3)) with T = Alt (5). All connected

G-symmetric graphs Cay(T 2, S), which were found using Magma [1], are given in Table

6.1.1. The graphs in lines 3, 4 and 6 all contain an element of the form (t, 1T ) in the

neighbourhood S of ω, and hence have diameter greater than 2 by Proposition 6.1.4. It is

interesting to note that the other graphs with diameter greater than 2 (i.e., those in lines

1, 2, 5, 7, 8 and 9) all correspond to S = (t1, t2) where t1, t2, and t−11 t2 are all conjugate

under T , while none of the graphs in lines 10 to 16 have this property.

We can generalise Proposition 6.1.4 as follows.

Proposition 6.1.6. Let Γ be a G-symmetric graph, where G = T d.(Out (T )×Sym (d))

for some d ≥ 4 and nonabelian simple group T , and G is a quasiprimitive group on

V (Γ) = T d−1 of diagonal type. If diam(Γ) = 2 then Γ ∼= Cay(T d−1, S), where S is

an orbit of Aut (T ) × Sym(d) satisfying the following: S ⊆ (T d−1)#; S = S−1; and

for all Aut (T )-orbits C in T# and k ≤ 12min d− 1, |T | − |C| − 1, S does not contain

(t1, . . . , tk, 1T , . . . , 1T ) where ti ∈ C for all i.

Proof. By Theorems 2.1.1 and 2.2.4, it remains to prove that S does not contain an

element of the given form. Suppose otherwise. Since S = S−1 it follows that C = C−1.


S = (t1, t2)Gω valency diameter 2

1 ((3 5 4), (3 4 5)) 20 no

2 ((1 3)(2 5), (1 5)(2 3)) 30 no

3 ((1 2)(4 5), 1T ) 45 no

4 ((1 5 3), (1 5 3)) 60 no

5 ((1 3 5 2 4), (1 4 2 5 3)) 72 no

6 (1T , (1 2 4 3 5)) 72 no

7 ((1 5 3 4 2), (1 2 5 4 3)) 120 no

8 ((1 3 2), (1 5 2)) 120 no

9 ((1 2)(4 5), (1 2)(3 4)) 180 no

10 ((1 3)(4 5), (1 5 3 4 2)) 360 yes

11 ((1 5)(2 3), (1 2 3)) 360 yes

12 ((1 2 5 4 3), (1 2)(3 5)) 360 yes

13 ((1 5 2), (1 4 3)) 360 yes

14 ((1 4 3), (1 3 2 5 4)) 360 yes

15 ((1 2 3), (1 5 3 2 4)) 360 yes

16 ((2 5 4), (1 4)(2 3)) 720 yes

Table 6.1.1. G-symmetric graphs for G = T 3.(Out (T )× Sym (3)), T = Alt (5)

Setting ti := 1T for all i with k + 1 ≤ i ≤ d, we have

S =(t−1d′ t1′ , . . . , t

−1d′ t(d−1)′

)σσ ∈ Aut (T ) , π ∈ Sym(d) , i′ := iπ

−1.

The elements of S can be divided into two types as follows.

(i) If 1 ≤ d′ ≤ k then

(t−1d′ t1′ , . . . , t

−1d′ t(d−1)′

)σ

= (t−1d′ t1, . . . , t

−1d′ td′−1, t

−1d′ td′+1, . . . , t

−1d′ tk, t

−1d′ , . . . , t

−1d′︸︷︷︸

d−k

)αρ

for some α ∈ Aut (T ) and ρ ∈ Sym(d).

(ii) If k + 1 ≤ d′ ≤ d then t−1d′ = 1T and

(t−1d′ t1′ , . . . , t

−1d′ t(d−1)′

)σ= (t1, . . . , tk, 1T , . . . , 1T︸︷︷︸

d−k−1

)αρ

for some α ∈ Aut (T ) and ρ ∈ Sym(d).

Let ℓ := min d− 1, |T | − |C| − 1.Claim 1: If ℓ = d − 1, then S2 does not contain an element (s1, . . . , sd−1), where

si /∈ C ∪ 1T for all i and the components si are pairwise distinct. Indeed, if x, y are of

type (i) then xy has at least d−1−2(k−1) = d−2k+1 components of the form (t−1d′ )

φ(t−1d′′ )

τ ,

for some φ, τ ∈ Aut (T ) and d′, d′′ ∈ 1, . . . , k. Since d − 2k + 1 ≥ d − ℓ + 1 = 2, the

product xy has at least two identical components — that is, the components of xy are not

all pairwise distinct. If x, y are of type (ii) then either all components of xy are in C, or

at least one component of xy is 1T . So xy is not of the desired form. If x is type (i) and y

is type (ii) then xy has at least d− 2k components equal to (t−1d′ )

τ , for some τ ∈ Aut (T )

6.2. QUASIPRIMITIVE WREATH PRODUCTS 111

and d′ ∈ 1, . . . , k. Since d− 2k ≥ d− ℓ = 1, xy has at least one component in C ∪ 1T .This proves Claim 1.

Claim 2: If ℓ = |T |−|C|−1 < d−1, then S2 does not contain an element (s1, . . . , sd−1)

such that T \ (C ∪ 1T ) ⊆ s1, . . . , sd−1. Suppose that xy = (s1, . . . , sd−1). If x, y are

of type (i) then xy has at least d − 2k + 1 identical components (equal to (t−1d′ )

φ(t−1d′′ )

τ ,

for some φ, τ ∈ Aut (T ) and d′, d′′ ∈ 1, . . . , k), where d − 2k + 1 ≥ d − ℓ + 1 ≥ 2. So

xy has at most d − (d − ℓ + 1) = ℓ − 1 = |T \ (C ∪ 1T )| − 1 distinct components, and

thus T \ (C ∪ 1T ) * s1, . . . , sd−1. If x, y are both of type (ii), then at least d− k − 1

components are in C ∪ 1T , so there can be at most k components in T \ (C ∪ 1T ),where clearly k < |T \ (C ∪ 1T )|. If x is of type (i) and y is of type (ii), then xy

has at least d − 2k ≥ d − ℓ components equal to (t−1d′ )

τ , for some τ ∈ Aut (T ) and

d′ ∈ 1, . . . , k. That is, xy has at least d − ℓ components in C ∪ 1T . So xy can have

at most ℓ− 1 = |T \ (C ∪ 1T )| − 1 distinct components from T \ (C ∪ 1T ), and thus

T \ (C ∪ 1T ) * s1, . . . , sd−1. This proves the claim.

It follows immediately from claims 1 and 2 that S2∪S 6= T d−1 for any d ≥ 3. Therefore

diam(Γ) 6= 2, a contradiction. Thus S does not contain an element (t1, . . . , tk, 1T , . . . , 1T )

with ti ∈ C for all i and k ≤ ℓ/2, as asserted.

Theorem 5 summarises the preceding results.

Proof of Theorem 5. Part (1) follows from Theorem 2.2.4 and Lemma 2.2.5; part

(2) follows from Proposition 6.1.4; and part (3) follows from Propositions 6.1.3 and 6.1.6.

In general, it is quite difficult to obtain a “workable” sufficient condition for Γ to have

diameter 2, and the search for one is a topic for further research.

Problem 2. Find sufficient conditions in order for Cay(T d−1, S), with d ≥ 3 ,to have

diameter 2.

6.2. Quasiprimitive wreath products

We now consider graphs Γ with automorphism groups of the form U ≀Sym (m), where U

is a quasiprimitive subgroup of Sym(Λ) with m ≥ 2, acting with the product action (1.3.1)

on V (Γ) = Λm. If U is of type HS, SD, or AS, then under some additional conditions, the

group G = U ≀ Sym (m) is quasiprimitive of type HC, CD, or PA, respectively. Moreover,

soc(U) = T k and soc(G) = T km for some k ≥ 1 and nonabelian simple group T . Clearly,

a maximal intransitive subgroup of this form is Sym(Λ) ≀ Sym(m) with |Λ| ≥ 5.

Hypothesis 6.2.1. Let G = U ≀ Sym(m) and Γ be as above. Let

ω := (λ, . . . , λ) ∈ V (Γ)


for some fixed λ ∈ Λ, and let α := (α1, . . . , αm) be a fixed vertex in Γ(ω). For each

i ∈ 1, . . . ,m let ∆i be the orbital graph for U with arc set (λ, αi)U = (λu, αu

i ) | u ∈ U.

Example 6.2.2 gives an infinite family of graphs that admit as symmetric, vertex-

transitive groups of automorphisms wreath products satisfying Hypothesis 6.2.1.

Example 6.2.2. The Hamming graph H(m, q), for some m ≥ 2 and q ≥ 2, is the

graph with vertex set Λm where Λ = 1, . . . , q, and edge set consisting of pairs of m-

tuples that differ in exactly one component. The distance between vertices β and γ is the

number of components in which they differ, which is also called the Hamming distance

dH(β, γ) between β and γ. Hence H(m, q) has diameter m. The full automorphism group

of H(m, q) is Sym(q)≀Sym(m), which is symmetric on H(m, q); this is maximal of product

type whenever q ≥ 5. The graph H(m, q) is isomorphic to the Cartesian product of m

copies of the complete graph Kq (see Section 2.3).

For any fixed ν ∈ 1, . . . ,m, define Hν(m, q) to be the graph with the same ver-

tex set as H(m, q), and whose edges are the pairs β, γ such that dH(β, γ) = ν. The

graph Hν(m, q) is called the distance-ν graph of H(m, q); this is also symmetric with

automorphism group Sym(q) ≀ Sym(m). Note that H1(m, q) = H(m, q).

Lemma 6.2.3. Assume Hypothesis 6.2.1 and let β and γ be distinct vertices of Γ,

with β := (β1, . . . , βm) and γ := (γ1, . . . , γm). Then β ∼Γ γ if and only if the following

conditions hold:

(i) dH(β, γ) = dH(ω,α); and

(ii) there exists π ∈ Sym(m) such that βj ∼∆j′γj whenever βj 6= γj and j′ := jπ

−1

.

Proof. Applying Theorem 2.1.2 and (1.3.1), the vertices β and γ are adjacent if and

only if (β, γ) ∈ (γ, α)G, that is, for some (u1, . . . , um) ∈ Um and π ∈ Sym(m), we have

β = (λu1′ , . . . , λum′ ) and γ =(αu1′

1′ , . . . , αum′

m′

), where i′ := iπ

−1

for all i. Equivalently,

dH(β, γ) = dH(ω,α) and βi ∼∆i′γi whenever βi 6= γi, as required.

Example 6.2.4. Assume Hypothesis 6.2.1 with m = 2 and α1 6= λ = α2. Then

dH(ω,α) = 1 and ∆2 = (λ, λ)U is an empty graph. It follows that two vertices (β1, β2)

and (γ1, γ2) are adjacent in Γ if and only if β1 = γ1 and β2 ∼∆1γ2, or β1 ∼∆1

γ1 and

β2 = γ2. Hence Γ is the Cartesian product ∆1 ∆1 (see Section 2.3). By Lemma 2.3.2

we get diam(Γ) = 2 if and only if ∆1 is a complete graph, in which case Γ ∼= H(2, q) with

q = |Λ|.

An easy necessary condition for Γ satisfying Hypothesis 6.2.1 to have diameter 2 is

given below.

Proposition 6.2.5. Assume Hypothesis 6.2.1. If diam(Γ) = 2, then dH(ω,α) ≥ 12m.

6.2. QUASIPRIMITIVE WREATH PRODUCTS 113

Proof. Suppose that diam(Γ) = 2 and that dH(ω,α) < m/2. Set β := (β0, . . . , β0)

and γ := (γ0, . . . , γ0) where β0 6= γ0. Then dH(β, γ) = m so that β ≁Γ γ by Lemma 6.2.3,

and by our assumption there is a vertex δ := (δ1, . . . , δm) with β ∼Γ δ ∼Γ γ. Applying

Lemma 6.2.3 again we then have dH(β, δ) = dH(ω,α) = dH(δ, γ). Since β0 6= γ0, we have

dH(δ, γ) ≥ m − dH(β, δ) = m− dH(ω,α) > dH(ω,α), a contradiction. So if diam(Γ) = 2

then dH(ω,α) ≥ m/2.

Observe that m− dH(ω,α) gives the number of U -orbital graphs ∆i which are empty.

The graph ∆i is a complete graph for some i if and only if U is a 2-transitive subgroup

of Sym (q) for q = |Λ|, and equivalently, for any j ∈ 1, . . . ,m, either ∆j is empty or

∆j is complete. In this case, two distinct vertices β and γ are adjacent exactly when

dH(β, γ) = dH(ω,α). Thus Γ is the distance-ν graph Hν(m, q) for ν := dH(ω,α).

We are now ready to prove Theorem 6.

Proof of Theorem 6. By the remarks above, Γ is the distance-ν graph Hν(m, q)

of H(m, q) for some ν ∈ 1, . . . ,m. If diam(Γ) = 2 then ν ≥ m/2 by Proposition 6.2.5,

so it remains to prove the converse.

Suppose that ν ≥ m/2. Since Sym(Λ) is clearly 2-transitive it follows that all con-

nected graphs ∆i are complete, and by Lemma 6.2.3 two distinct vertices of Γ are adjacent

if and only if their Hamming distance is ν. Let β′ := (β1, . . . , βm) and γ′ := (γ1, . . . , γm)

be distinct nonadjacent vertices of Γ. Without loss of generality, assume that βi 6= γi

exactly when 1 ≤ i ≤ dH(β, γ). Write µ := dH(β, γ). We have two cases.

Case 1: Suppose that dH(β, γ) < ν. Choose δi ∈ Λ satisfying

δi 6= βi, γi if 1 ≤ i ≤ ν;

δi = βi(= γi) if ν + 1 ≤ i ≤ m.

Let δ := (δ1, . . . , δm). Then dH(β, δ) = dH(γ, δ) = ν, and therefore β ∼Γ δ ∼Γ γ. So

distΓ(β, γ) = 2.

Case 2: Suppose that dH(β, γ) > ν. Observe that µ ≤ m ≤ 2ν, so that ν ≥ ⌊µ/2⌋.Hence ν + ⌊µ/2⌋+1 ≥ 2⌊µ/2⌋+1 ≥ µ, and thus βi = γi for all i ≥ ν + ⌊µ/2⌋+1. Choose

δi ∈ Λ satisfying

δi = βi(6= γi) if 1 ≤ i ≤ ⌊µ/2⌋;

δi = γi(6= βi) if ⌊µ/2⌋ + 1 ≤ i ≤ 2⌊µ/2⌋;

δi 6= βi, γi if 2⌊µ/2⌋ + 1 ≤ i ≤ ν + ⌊µ/2⌋;

δi = βi(= γi) if ν + ⌊µ/2⌋+ 1 ≤ i ≤ m.

Then dH(β, δ) = dH(γ, δ) = ⌊µ/2⌋+ν−⌊µ/2⌋ = ν. So again β ∼Γ δ ∼Γ γ and distΓ(β, γ) =

2. Therefore diam(Γ) = 2.


Remark. Theorem 6 is also true if G = U ≀ Sym(m) for any 2-transitive group U .

In particular, if U = AGL(k, q0) for some k ≥ 2 and prime power q0, then G is maximal

affine-type subgroup belonging to the Aschbacher class C2, and we get Proposition 5.3.2.

6.3. Almost simple subgroups

In this case the unique minimal normal subgroup of G is a nonabelian simple group

T , which acts transitively on V (Γ). A classification of symmetric, diameter 2 graphs

admitting a maximal almost simple, vertex-quasiprimitive group of automorphisms —

which would necessarily depend on the classification of quasiprimitive almost simple groups

— is infeasible at this point. Examples are known to exist: for instance, orbital graphs

for the almost simple quasiprimitive rank 3 graphs of even order (which are all known, see

Section 1.4) are symmetric and diameter 2, and thus fall under this case.

There are many other examples with arbitrarily large rank, some of which we give

below.

Example 6.3.1. Let k ≥ 2 and n ≥ 2k + 1, and let Ω be the set of k-element subsets

of 1, . . . , n. The Kneser graph KG(n, k) is the graph with vertex set Ω and in which

two vertices are adjacent if and only if they are disjoint. In [42] it is proved that KG(n, k)

has diameter ⌈k − 1

n− 2k

⌉+ 1.

In particular, KG(n, k) has diameter 2 if and only if n ≥ 3k − 1. The full automorphism

group of KG(n, k) is Sym(n), which acts on Ω via

αg := xg | x ∈ α ∀ α ∈ Ω, g ∈ Sym(n) .

Then Sym (n) is arc-transitive on KG(n, k). The stabiliser of a point is isomorphic to

Sym (k) × Sym(n− k), which is maximal in Sym(n), and hence the action of Sym(n)

is primitive (and thus quasiprimitive) by Theorem 1.2.1. Furthermore, Sym (n) has rank

k + 1 in this action: indeed, for a fixed ω ∈ Ω, the Gω-orbits in Ω \ ω are the sets

α ∈ Ω | |α ∩ ω| = ℓ for each ℓ ∈ 0, . . . , k.

Although the quasiprimitive case is too difficult to handle, restricting to the primi-

tive case might be reasonable, albeit the possibility of a classification also remains to be

determined.

Problem 3. Classify the symmetric diameter 2 graphs with vertex-primitive auto-

morphism groups that are maximal of almost simple type.

APPENDIX A

Magma codes

We present here some of the Magma algorithms used for computing the entries in

Table 4.1.2, and for obtaining the results described in Examples 4.5.4 and 6.1.5.

A.1. Algorithms for Table 4.2

The algorithm used for computing the entries of Table 4.1.2 is quite straightforward,

and consists of the following steps:

Construct the group G0 ≤ GL(V ).

Get the orbits S of G0 in V #.

Compare V # and S ∪ (S + S).

Isomorphisms of graphs are checked by looking at the action of the normalizer of G0,

or if this doesn’t work, by constructing the corresponding graphs and using Magma’s

IsIsomorphic function.

A.1.1. Lines 1-10.

Q8 := ExtraSpecialGroup(2,1 : Type:= "-");

n := 0;

H := []; HH := [];

for p in [5,7,11,23] do

print "p", "=", p;

U := VectorSpace(GF(p),2); V := DirectSum(U,U);

k := Subgroups(GL(U) : OrderEqual:=8);

c := @ j : j in [1..#k] | IsIsomorphic(k[j]‘subgroup,Q8) @;K := k[c[1]]‘subgroup;

N := Normalizer(GL(U),K);

h := Subgroups(N);

I := @ i : i in [1..#h] | IsTransitive(OrbitImage(h[i]‘subgroup,

(Set(U) diff U!0))) @;O := [];

for i in [1+n..#I+n] do

H[i] := h[I[i-n]]‘subgroup;

HH[i] := sub< GL(V) | DirectSum(h,h) : h in H[i] >;

O[i] := Orbit(HH[i],V![1,0,1,1]);

if #( x+y : x,y in O[i] join Set(O[i]) ) eq #V then

print "H",i, "order", #H[i], ":", #GL(U)/#O[i], "graphs with diameter 2";

elif sub< V | O[i] > eq V then

115

116 A. MAGMA CODES

print "H",i, "order", #H[i], ":", #GL(U)/#O[i], "connected graphs with

diameter greater than two";

else print "H",i, "order", #H[i], ":", "no connected graphs";

end if;

end for;

n := n + #I;

end for;

A.1.2. Lines 11-17. To obtain the entries for lines 11-17 of Table 4.1.2, the same

code as above was used, apart the first eight lines which are replaced with:

for p in [11,19,29,59] do

print "p", "=", p;

U := VectorSpace(GF(p),2); V := DirectSum(U,U);

k := Subgroups(GL(U) : OrderEqual:=120);

c := @ j : j in [1..#k] | IsIsomorphic(k[j]‘subgroup,SL(2,5)) @;

A.1.3. Lines 18-49. The code below computes the entries for lines 18-33; the code

for lines 34-49 is similar.

U := VectorSpace(GF(3),4); V := DirectSum(U,U);

X := SemiLinearGroup(GL(2,9),GF(3));

y := Subgroups(X : OrderEqual:=#GL(2,9));

i := @ j : j in [1..#y] | IsIsomorphic(y[j]‘subgroup,GL(2,9)) @;Y := y[i[1]]‘subgroup;

k := Subgroups(X : OrderEqual:=120);

i := @ j : j in [1..#k] | IsIsomorphic(k[j]‘subgroup,SL(2,5)) @;K := k[i[1]]‘subgroup;

N := Normalizer(X,K); #N;

h := Subgroups(N : OrderMultipleOf:=(#U-1));

I := @ i : i in [1..#h] | IsTransitive(OrbitImage(h[i]‘subgroup,

(Set(U) diff U!0))) @;A := [];

for i in [1..#I] do

A[i] := h[I[i]]‘subgroup;

end for;

AA := []; MA := [];

for i in [1..#A] do

AA[i] := sub< GL(V) | DirectSum(h,h) : h in A[i] >;

MA[i] := Normalizer(GL(V),AA[i]);

end for;

for i in [1..#A] do

print "A",i, "order", #A[i];

S[i] := []; s[i] := [];

o := @ j : j in [1..#Orbits(MA[i])] | #Orbits(MA[i])[j] gt 1 @;

A.1. ALGORITHMS FOR TABLE 4.2 117

for j in [1..#o] do

s[i][j] := @ k : k in [1..#Orbits(AA[i])] | Orbits(AA[i])[k] subset

Orbits(MA[i])[o[j]] @;S[i][j] := Set(Orbits(AA[i])[s[i][j][1]]);

SS := x+y : x,y in S[i][j] ;if (SS join S[i][j]) eq Set(V) then

print "S", i,j, ":", #s[i][j], "of length", #S[i][j], ";",

"diameter 2", ";", "|S+S| =", #SS;

elif sub< V | S[i][j] > eq V then

print "S", i,j, ":", #s[i][j], "of length", #S[i][j], ";", "connected

with diameter > 2", ";", "|S+S| =", #SS;

else print "S", i,j, ":", #s[i][j], "of length", #S[i][j], ";",

"disconnected";

end if;

end for;

end for;

for i in [2..#A] do

for j in [2..#Orbits(MA[i])-1] do

for k in [1..(i-1)] do

for l in [2..#Orbits(MA[k])-1] do

r := # m : m in s[k][l] | Orbits(AA[k])[m] subset S[i][j] ;if r ge 1 then

print "S",i,j, ":", r, x, "S",k,l;

end if;

end for;

end for;

end for;

end for;

A.1.4. Lines 53-72.


h := Subgroups(GL(U) : OrderEqual:=#SL(2,13));

i := @ j : j in [1..#h] | IsIsomorphic(h[j]‘subgroup,SL(2,13)) @;H := h[i[1]]‘subgroup;

HH := sub< GL(V) | DirectSum(h,h) : h in H >;

M := sub< GL(V) | KroneckerProduct(g,h) : g in GL(2,3), h in H >;

O := Orbits(HH);

A := @ # x+y : x,y in O[i] : i in [1..#O] @;B := []; X := [];

for j in [1..#A] do B[j] := o : o in O | # x+y : x,y in o eq A[j] ;X[j] := OrbitImage(M,B[j]);

print j, ":", #B[j], "orbits,", "at most", #Orbits(X[j]), "class(es) with

valency", #o : o in B[j] , ";", "|S+S| =", A[j];

118 A. MAGMA CODES

end for;

A.2. Algorithms for Example 4.5.4

F<w> := GF(3^4);


X := SemiLinearGroup(GL(1,F),GF(3));

s := @ g : g in G | (U!Eltseq(F!1))*g eq U!Eltseq(w) and

(U!Eltseq(w))*g eq U!Eltseq(w^2) @[1];t := @ g : g in G | (U!Eltseq(F!1))*g eq U!Eltseq(F!1) and

(U!Eltseq(w))*g eq U!Eltseq(w^3) @[1];H := sub< GL(U) | s^2, t*s >;

HH := sub< GL(8,3) | DirectSum(h,h) : h in H >; O := Orbits(HH);

for i in [1..#O] do

print i, ":", < Seqelt([u[i] : i in [1..4]],F) : u in U | V!

Insert([(U!Eltseq(F!1))[i] : i in [1..4]],5,4,[u[i] : i in [1..4]]) in

O[i] , #O[i], # x+y : x,y in O[i] >;

if #(O[i] join x+y : x,y in O[i] ) eq #V then

print "diameter two";

else print "not diameter two";

end if;

end for;

A.3. Algorithms for Example 6.1.5

n := 5; T := Alt(n); d := 3;

N := CartesianProduct([T : i in [1..d]]);

diagN := a : a in N | forall i : i in [1..d-1] | a[i] eq a[d] ;V := a : a in N | a[d] eq Id(T) ;G1 := CartesianProduct(Sym(n),Sym(d));

f := map< CartesianProduct(V,G1) -> V | c :-> < (Inverse(c[1][d^c[2][2]])*

c[1][j^c[2][2]])^c[2][1] : j in [1..d] > >;

s := []; O := [];

s[1] := N!< Id(T) : i in [1..d] >;

O[1] := f(Set(CartesianProduct(s[1],G1)));S := O[1]; j := 1;

for i in [1..#V] do

if exists(t) u : u in Set(V) diff S then

j := 1+j;

s[j] := t;

O[j] := f(Set(CartesianProduct(s[j],G1)));S := S join O[j];

end if;

end for;

A.3. ALGORITHMS FOR EXAMPLE 6.1.5 119

H := CartesianProduct(N,Sym(d));

g := map< CartesianProduct(V,H) -> V | c :-> <

Inverse(c[2][1][d^c[2][2]])*Inverse(c[1][d^c[2][2]])*

c[1][j^c[2][2]]*c[2][1][j^c[2][2]] : j in [1..d] > >;

W := < v[1],v[2] > : v in V ;R := []; S := []; SS := [];

for i in [2..#O] do

t1 := s[i][1]; t2 := s[i][2]; t3 := Inverse(t1)*t2;

R[i] := @ < t1, t2 >, < t2, t1 >, < Inverse(t1), Inverse(t1)*t2 >,

< Inverse(t1)*t2, Inverse(t1) >, < Inverse(t2)*t1, Inverse(t2) >,

< Inverse(t2), Inverse(t2)*t1 > @;print i, ":", s[i], " ", "valency =", #O[i], ",", "undirected -",

<Inverse(t1),Inverse(t2)> in S[i], ",", "diameter two -",

#(W diff (SS[i] join S[i])) eq 0;

end for;

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