Finite Transitive Permutation Groups and Finite ... - Semantic Scholar

Finite Transitive Permutation Groups and Finite Vertex-Transitive Graphs Cheryl E. Praeger with the assistance of Cai Heng Li and Alice C. Niemeyer Department of Mathematics The University of Western Australia Nedlands, WA 6907 Australia March 25, 2013

1

Introduction

The theory of vertex-transitive graphs has developed in parallel with the theory of transitive permutation groups. In this chapter we explore some of the ways the two theories have influenced each other. On the one hand each finite transitive permutation group corresponds to several vertex-transitive graphs, namely the generalised orbital graphs which we shall discuss below. On the other hand, each finite vertex-transitive graph gives rise to (usually) several transitive permutation groups, namely the vertex-transitive subgroups of the full automorphism group of the graph. We shall study pairs (Γ, G) where Γ is a finite graph and G is a vertex-transitive subgroup of its automorphism group AutΓ. In doing so we shall be bringing together, and learning from, two mathematical cultures: group theory and graph theory. We shall see the interchange of techniques and ideas between the theory of transitive permutation groups and the theory of vertex-transitive graphs. More specifically, we will look at various ways in which permutation group theory has been used to solve problems about finite vertex-transitive graphs. Sometimes only elementary group theoretic techniques were required, while in other cases quite sophisticated group theory was necessary, occasionally involving the finite simple group classification. In one case, the necessary group theory was not available, but the desire to solve the graph theoretic problem stimulated its development. The problems we shall examine relate to the following areas of finite graph theory: (a) the enumeration of vertex-transitive graphs of small order; (b) distance transitive graphs; 1

(c) s-arc transitive graphs; (d) locally-primitive graphs. In addition (and motivated by the lectures of Brian Alspach at this NATO Advanced Studies Institute) a problem about isomorphisms of finite Cayley graphs, and its solution, will be described. The discussion involves an overview of several parts of permutation group theory, including the O’Nan-Scott Theorem on finite primitive permutation groups, and the structure theory of finite quasiprimitive permutation groups.

2

Transitive groups and generalised orbital graphs

We begin with a short review of some of the basic concepts of permutation groups, including in particular a description of the construction of the family of generalised orbital graphs associated with a transitive permutation group. For a set Ω, the symmetric group Sym(Ω) on Ω is the group of all permutations of Ω, and a permutation group on Ω is simply a subgroup of Sym(Ω). If |Ω| = n and G ≤ Sym(Ω), then G is said to be a permutation group of degree n. For α ∈ Ω, the image of α under a permutation g of Ω is denoted by αg . A permutation group G on Ω induces an equivalence relation ∼G on Ω given by α ∼G β if and only if αg = β for some g ∈ G. The equivalence classes of ∼G are called G-orbits in Ω. So the G-orbit containing α is αG := {αg | g ∈ G}; G is said to be transitive on Ω if there is only one G-orbit, that is, if Ω = αG for some and hence every point α ∈ Ω. When studying automorphism groups of graphs we wish to consider their actions on several different sets: the vertices, edges, arcs etc. Thus it will be helpful to introduce the concept of a permutation representation. A permutation representation of a group G on a set Ω is a group homomorphism ϕ : G → Sym(Ω). The image (G)ϕ of G under ϕ is the permutation group on Ω induced by G under the representation ϕ and is often denoted by GΩ . The representation ϕ is said to be faithful if Ker(ϕ) = 1. When there is no likelihood of confusion we often suppress ϕ and write the permutation (g)ϕ of Ω simply as g, or sometimes as g Ω , for g ∈ G. In fact it is useful to consider more than one permutation representation when analysing any permutation group. For example, any transitive permutation group G on Ω has a faithful permutation representation on Ω×Ω given by (α, β)g := (αg , β g ) for g ∈ G and α, β ∈ Ω. The G-orbits in Ω×Ω are called the G-orbitals on Ω (or simply orbitals), and, for each point α ∈ Ω, 2

there is a 1-1 correspondence between the G-orbitals and the orbits in Ω of the stabilizer Gα of α (where Gα := {g ∈ Γ | αg = α}). If ∆ is a G-orbital then the corresponding Gα -orbit in Ω is ∆(α) := {β | (α, β) ∈ ∆}. Also we define a generalised orbital to be a non-empty G-invariant subset of Ω × Ω, that is a non-empty union of G-orbitals. For each (generalised) orbital ∆, there is a paired (generalised) orbital ∆∗ , namely ∆∗ := {(β, α) | (α, β) ∈ ∆}. If ∆ = ∆∗ we say that the (generalised) orbital ∆ is self-paired. Also, for each transitive group G on Ω, the set ∆0 := {(α, α) | α ∈ Ω} is a G-orbital and is called the trivial G-orbital. The number of G-orbitals, or equivalently the number of Gα -orbits in Ω, is called the rank of G. With each generalised G-orbital we will associate a di-graph, and with each self-paired generalised G-orbital not containing ∆0 we will associate a graph; each such di-graph and graph will have vertex set Ω and will admit the group G as a vertex-transitive subgroup of automorphisms. First we explain our notation. By a di-graph (or directed graph) we mean a pair Γ = (Ω, E) where Ω is a set whose elements are called the vertices of Γ, and E is a subset of Ω × Ω whose elements are called the edges or arcs of Γ. Similarly a graph is a pair Γ = (Ω, E) where Ω is the set of vertices of Γ, and E is a subset of unordered pairs of distinct vertices which are called the edges of Γ; the ordered pairs (α, β) of adjacent vertices (that is those for which {α, β} ∈ E) are called the arcs of Γ. Thus, for a di-graph, edges and arcs are the same, but this is not the case for a graph. Note that, according to our definition, a di-graph may or may not have some loops (edges of the form (α, α)), but a graph does not have any loops. An automorphism of a graph or di-graph Γ = (Ω, E) is an element g of Sym(Ω) such that eg ∈ E if and only if e ∈ E; and the set of all automorphisms forms a subgroup AutΓ of Sym(Ω) called the automorphism group of Γ. If a subgroup G of AutΓ is transitive on the vertex set Ω, the edge set E, or the set of arcs of Γ, we say that Γ is G-vertex-transitive, G-edge-transitive, or G-arc-transitive respectively. Often we omit the prefix “G’. Now let ∆ be a (generalised) G-orbital. Then the associated (generalised) orbital di-graph is the di-graph (Ω, ∆). As it will seldom cause confusion, we shall denote this di-graph also by ∆. It follows from this definition that G ≤ Aut∆, and G is transitive on the vertices of ∆. Also G is transitive on the arcs of ∆ if and only if ∆ is a G-orbital. Now a (generalised) orbital ∆ has the property (α, β) ∈ ∆ if and only if (β, α) ∈ ∆ if and only if ∆ is self-paired. Thus, if ∆ is self-paired and ∆0 6⊆ ∆, then the di-graph ∆ contains no loops and the edges occur in pairs of the form {(α, β), (β, α)}. If this is the case then we may replace each such pair by the unordered pair {α, β} and thereby obtain a graph, called the (generalised) orbital graph associated with ∆. Again if no confusion arises 3

we denote the (generalised) orbital graph by ∆. As for the (generalised) orbital di-graph, this (generalised) orbital graph also admits G as a subgroup of its automorphism group; G is transitive on vertices, and G is transitive on arcs if and only if ∆ is a self-paired nontrivial G-orbital. As examples of generalised orbital graphs ∆ for a transitive permutation group G ≤ Sym(Ω), we always have the empty graph (Ω, ∅) on Ω, by taking ∆ = ∆0 , and the complete graph on Ω, that is the graph in which every pair of distinct vertices is an edge, by taking ∆ = (Ω × Ω) \ ∆0 . If |Ω| = n then the complete graph on Ω is denoted by Kn . The most important (but elementary) observations to be made about generalised orbital graphs of transitive permutation groups on a set Ω are the following. Theorem 2.1

(a) G is a vertex-transitive subgroup of automorphisms of a graph Γ =

(Ω, E) if and only if Γ is a generalised orbital graph for G, namely for the self-paired generalised orbital ∆ := {(α, β) | {α, β} ∈ E}. (b) G is an arc-transitive group of automorphisms of a graph Γ = (Ω, E) if and only if Γ is an orbital graph for G, namely for the self-paired orbital ∆ := {(α, β) | {α, β} ∈ E}. We have already seen that generalised orbital graphs have the asserted properties. For the converse observe that, if G ≤ AutΓ and G is transitive on the vertex set Ω of Γ, then the set of arcs of Γ is a G-invariant subset of Ω × Ω and hence is a generalised orbital for the transitive permutation group G on Ω.

3

Partitions, blocks and primitivity

One of the traditional ways of studying a transitive permutation group G ≤ Sym(Ω) is to examine the G-invariant partitions of Ω. A partition P of Ω is said to be G-invariant if the elements of G permute the parts of P block-wise, that is, P g ∈ P for all P ∈ P and g ∈ G (where P g := {αg | α ∈ P }). The trivial partitions {Ω} and {{ω} | ω ∈ Ω} are G-invariant for all transitive groups G, and a transitive permutation group G on Ω is said to be primitive on Ω if these are the only G-invariant partitions of Ω. If G is transitive, but not primitive, on Ω, then G is said to be imprimitive on Ω. Let G be a transitive permutation group G on Ω. Then for every non-empty subset B of Ω, the collection of subsets B := {B g | g ∈ G} is G-invariant in the sense that Dg ∈ B for all g ∈ G and D ∈ B. Thus B is a G-invariant partition of Ω if and only if it is in fact a partition. It turns out that this is the case if and only if the subset B is a block of imprimitivity which is defined as follows. A non-empty subset B of Ω is a block of imprimitivity for G in Ω if, for all g ∈ G, either B g = B or B g ∩ B = ∅. It is not difficult to verify that a partition of Ω is G-invariant if and only if all of its parts are blocks of imprimitivity, and that the latter 4

is true if and only if at least one of the parts is a block of imprimitivity. In line with our definitions for G-invariant partitions, a block of imprimitivity is said to be trivial if either it contains only one element or it is equal to Ω. Thus G is primitive on Ω if and only if the only blocks of imprimitivity for G in Ω are the trivial ones. For any subset P of Ω the setwise stabilizer GP of P in G is defined as GP = {g ∈ G | P g = P }; clearly GP is a subgroup of G. For a G-invariant partition P of Ω, if P is the part of P containing a point α, then GP contains the point stabilizer Gα . Conversely, if Gα ≤ H ≤ G then the H-orbit P := αH containing α determines a G-invariant partition of Ω, namely P := {P g | g ∈ G}. In fact the lattice of all G-invariant partitions of Ω is isomorphic to the lattice of subgroups of G containing Gα , and hence G is primitive on Ω if and only if Gα is a maximal subgroup of G. For a G-invariant partition P of Ω, there correspond two (usually smaller) transitive permutation groups, namely the group GP of permutations of P induced by G, and the permutation group GPP induced on P by GP , where P ∈ P. We describe these groups below. The map g 7→ g P (g ∈ G) defines a permutation representation of GP on P with image the permutation group GPP . Of course, choosing different parts P, P 0 ∈ P gives us different 0

transitive permutation groups GPP and GPP 0 , but they are essentially the same in that they are permutationally equivalent. As in [71, p 32] by a permutational equivalence between two permutation groups H, K on ∆, Σ respectively we mean a pair (f, ϕ) such that f : H → K is a group isomorphism, ϕ : ∆ → Σ is a bijection, and for all h ∈ H, δ ∈ ∆, we have (δ h )ϕ = (δϕ)(h)f . (Also, if H = K so that the isomorphism f is an automorphism, then we call (f, ϕ) a permutational isomorphism.) It is easy to verify that each element x ∈ G such that P x = P 0 induces a bijection x ¯ from P to P 0 (by restriction) and an isomorphism x ˆ from 0

GPP to GPP 0 (by conjugation), and the pair (ˆ x, x ¯) is a permutational equivalence. For g ∈ G, we let g P denote the permutation of P induced by g. Then the map g 7→ g P defines a permutation representation of G on P. Thus, as an abstract group, the image GP of this permutation representation is a quotient group of G, and we view the permutation group GP on P as a quotient of the permutation group G on Ω. If we choose P such that GP is maximal in G for some, and hence all, P ∈ P, we see that GP must be primitive on P. Thus every transitive permutation group has at least one primitive quotient. We may view the transitive group G as being “built up” of the two permutation groups GP and GPP . If Ω is finite then, by repeating this procedure for the transitive group GPP (and so on), we will eventually obtain a finite sequence of primitive permutation groups of which, in some sense, G is “composed”. The traditional approach to the study of finite transitive permutation groups is to assert, at this point, that the primitive permutation groups are the building blocks for transitive 5

permutation groups, and that the central problem for finite permutation groups is to understand the structure of finite primitive permutation groups, and, incidentally, how they can be fitted back together to form the transitive groups. In our exploration of vertex-transitive graphs in this chapter we shall examine the extent to which this traditional approach has yielded an appropriate group theoretical framework for applications to problems about finite vertex-transitive graphs. For the problem of classifying distance-transitive graphs we shall see that this approach was ideal; for the problem of enumeration of vertex-transitive graphs of small order it yielded helpful but insufficient information; while for the problem of describing the structure of 2-arc transitive graphs we shall see that it was regrettably inappropriate, but that a generalisation of the structure theory of finite primitive groups to quasiprimitive groups was both achievable and useful.

4

Quotients of vertex-transitive graphs

Here we look briefly at the exact analogue of the discussion in Section 3 for finite vertextransitive graphs. First we define quotient graphs relative to arbitrary partitions of the vertex set. Let P be a partition of the vertex set Ω of a graph Γ = (Ω, E). Then the quotient graph ΓP of Γ relative to P is defined to be the graph ΓP = (P, EP ) where {P, P 0 } ∈ EP if and only if there exist α ∈ P and α0 ∈ P 0 such that {α, α0 } ∈ E. Quotient graphs inherit some of the properties of the original graph. For example quotient graphs of connected graphs are connected. Moreover, if P is invariant under some subgroup G of AutΓ, then ΓP admits GP as a subgroup of automorphisms. (A graph Γ = (Ω, E) is connected if, for all α, β ∈ Ω, there is a finite sequence γ0 , . . . , γn of vertices such that γ0 = α, γn = β, and {γi , γi+1 } ∈ E for all 0 ≤ i < n. If Γ is not connected then we say that Γ is disconnected.) Also, for P ⊆ Ω, let [P ] denote the subgraph of Γ induced on P , that is the graph with vertex set P such that a pair of vertices of P is an edge of [P ] if and only if it is an edge of Γ. Again, if P is invariant under some subgroup G of AutΓ (that is, if P is fixed setwise by G), then the permutation group GP induced on P by G is a subgroup of Aut[P ]. Now suppose that Γ is G-vertex-transitive. If G is imprimitive on Ω (that is, Γ is G-verteximprimitive) and P is a nontrivial G-invariant partition of Ω, then ΓP is GP -vertex-transitive, and [P ] is GPP -vertex-transitive. Furthermore the graphs [P ], for P ∈ P, are all isomorphic, and the transitive permutation groups GPP , for P ∈ P, are permutationally equivalent. Thus in some sense the vertex-transitive graph Γ is “composed of” the two smaller vertex-transitive graphs ΓP and [P ], and we could refine this decomposition (just as for transitive permutation groups in the previous section) to give a sequence of vertex-primitive graphs. However for several reasons this may not always be the most helpful path to an understanding of the

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structure of a finite vertex-transitive graph Γ. For example, it follows from the discussion in Section 3 that if G is arc-transitive on a connected graph Γ then, for a nontrivial G-invariant partition P, the group GP is arctransitive on ΓP . Thus if Γ, and hence also ΓP , are connected, then all edges of Γ join vertices in different parts of P; so in particular the induced subgraphs [P ], for P ∈ P, are all empty graphs (that is they have no edges). Thus the subgraphs [P ] may not give much information about Γ (although a more imaginative approach to the induced structure on P is possible and is the subject of Tony Gardiner’s lectures for this NATO Advanced Study Institute, see also [33]). From the point of view described here, most of the information will be contained in the quotient graph ΓP , and by choosing the nontrivial G-invariant partition P to have |P| as small as possible, we may ensure that the group GP is primitive. This suggests that one might study vertex-primitive graphs to gain insights into the nature of quotients of finite connected vertex-transitive graphs. However, if we are interested in a family of vertex-transitive graphs with some extra defining property, there are some additional difficulties. We will have in general no guarantee that this extra property will be inherited by ΓP and/or by [P ]. Sometimes just one of ΓP and [P ] will have this extra property (this is the case for example with the property of being connected and arc-transitive), and sometimes neither will inherit it. The examples discussed in this chapter will demonstrate a range of possibilities.

5

Enumeration of vertex-transitive graphs of small order

The order of a finite graph Γ = (Ω, E) is the number |Ω| of vertices. In this section we discuss briefly some of the ideas and techniques used to enumerate all vertex-transitive graphs of certain (small) orders. It turned out that a straight-forward application of the approach outlined in the previous section for analysing the structure of vertex-transitive graphs was not the most efficient way of proceeding. In fact, this example was chosen to demonstrate that, even though a theoretical framework may give a good description of the structure of certain finite vertex-transitive graphs, it may not provide a computationally efficient approach to their enumeration. First we make a few general elementary observations about finite vertextransitive graphs. 1. If a vertex-transitive graph Γ of order n is disconnected, then all of its connected components are isomorphic, and are themselves vertex-transitive. Thus, if all vertextransitive graphs of order a proper divisor of n have been enumerated, then we may read off from these lists, up to isomorphism, all the disconnected vertex-transitive graphs of order n. 7

2. If Γ = (Ω, E) is a vertex-transitive graph of order n, then also its complementary graph Γ := (Ω, E) is vertex-transitive of order n (where, for distinct points α and β of Ω, {α, β} ∈ E if and only if {α, β} 6∈ E). Also AutΓ = AutΓ. 3. Every finite vertex-transitive graph Γ is regular, that is, there is a constant k such that every vertex α is adjacent to exactly k other vertices. The constant k is called the valency of Γ, and is denoted by val(Γ). Thus to enumerate all vertex-transitive graphs Γ of a given order n, we may assume that both Γ and Γ are connected, and that val(Γ) < n/2. Moreover, we might expect (naively perhaps) that those of valency close to n/2 might be more difficult to enumerate than those of small valency. Up to 1985 the only catalogues of vertex-transitive graphs available in the literature were the list of vertex-transitive graphs of order at most 13 by H. P. Yap [94] (excluding those of valency 5 on 12 vertices), and the list of all vertex-transitive graphs of order at most 19 by B. D. McKay [63, 64]. In the latter case, McKay had used complicated matrix methods to construct his list. The vertex transitive graphs of orders n = 20, 21, 22 and 23 were constructed by McKay and Royle [67]. These constructions were based on an elementary group-theoretic lemma exploiting the fact that in all cases the prime decomposition of n contained a relatively large prime factor p. The lemma showed that every vertex-transitive subgroup of the automorphism group of such a graph contained a permutation of order p which had relatively few cycles of length p and no fixed vertices. However, this method did not easily carry over to the case n = 24 as the existence of such automorphisms could not be guaranteed. The enumeration of the vertex-transitive graphs of order 24 was completed by G. F. Royle and the author [81] making use of the theory outlined in the previous section. Further, it required the classification of the transitive permutation groups of degree 12 (see [80]), it involved heavy use of the computer sytem CAYLEY (which has now evolved into the new system MAGMA [17]) and a suite of programs to compute second cohomology groups developed by D. F. Holt [39]. We outline some of the main ideas of this enumeration. All primitive groups of degree 24 are 2-transitive. (A permutation group G on Ω is said to be 2-transitive on Ω if G is transitive on the ordered pairs of distinct points of Ω.) Hence the only vertex-primitive graphs of order 24 are 24K1 and K24 . So we may assume that the automorphism group of a vertex-transitive graph Γ = (Ω, E) of order 24 is imprimitive, and by our remarks above we may also assume that Γ is connected and has valency less than 12. Let G be a vertex-transitive subgroup of AutΓ. Then certainly G is imprimitive on Ω, and for each nontrival G-invariant partition P of Ω, the two graphs ΓP and [P ] (where P ∈ P) are 8

vertex-transitive with orders being proper divisors of 24. Since all vertex-transitive graphs of order a proper divisor of 24 are known, the search procedure might be subdivided as follows. For each divisor d of 24 with 1 < d < 24, find all pairs (Γ, G) with Γ a graph of order 24, G a vertex-transitive subgroup of AutΓ, and such that there is a G-invariant partition P of vertices for which |P| = d and |P| is maximal over all nontrivial G-invariant partitions P. Although this was the basic underlying strategy used for the enumeration, the large number of graphs to be identified presented serious problems of storage and isomorphism testing, besides the other group theoretic problems associated with the search. (It turned out that there are, up to isomorphism, 7753 vertex-transitive graphs of order 24.) From their experience with the classification of the smaller vertex-transitive graphs, McKay and Royle guessed that the majority of the examples would be Cayley graphs, and suggested that these could be found separately. Then, when following the search strategy above, whenever a graph was found we could first test to see if it was a Cayley graph and, if so, discard it. We now explain what Cayley graphs are. Definition 5.1 Let G be a finite group, and let S ⊆ G such that 1G 6∈ S. The Cayley digraph Cay(G, S) of G relative to S is the di-graph with vertex set G such that, for x, y ∈ G, the pair (x, y) is an arc if and only if xy −1 ∈ S. Further, if S = S −1 := {s−1 | s ∈ S}, then (x, y) is an arc if and only if (y, x) is an arc, and we define the Cayley graph of G relative to S as the graph with vertex set G such that {x, y} is an edge if and only if xy −1 ∈ S. We abuse notation somewhat and denote this graph also by Cay(G, S), as its meaning will be clear from the context. A subset S of G such that 1 6∈ S and S = S −1 will be called a Cayley subset of G. The group G acting on itself in its right regular representation (that is, g : x 7→ xg) is a subgroup of the automorphism group of Cay(G, S) and acts transitively on G. Thus all Cayley graphs are vertex-transitive. Further, this action of G on itself is regular, where we say that a permutation group H on a set ∆ is regular on ∆ if H is transitive on ∆ and the only element of H which fixes a point of ∆ is the identity 1H . The existence of a subgroup of AutΓ acting regularly on vertices characterises Cayley graphs. Lemma 5.2 ([13, Lemma 16.3]) A graph Γ = (Ω, E) is isomorphic to a Calyey graph for a group G if and only if AutΓ has a subgroup G isomorphic to G such that G is regular on Ω. Thus the search for incomplete vertex-transitive graphs of order 24 had two phases, finding the Cayley graphs of order 24, and finding the remaining examples, the so-called non-Cayley

9

graphs of order 24. We require a list containing exactly one copy of a graph from each isomorphism class of vertex-transitive graphs of order 24. Note that, if Γ = Cay(G, S), and if σ ∈ Aut(G), then σ induces an isomorphism from Γ to Cay(G, S σ ). There may be other isomorphisms between Cayley graphs for G, and it is possible for a graph to be a Cayley graph for more than one group. Cayley graphs of order 24:

All groups of order 24 are known, up to isomorphism. For

each such group G, the following procedure, written schematically below, was used to find, up to isomorphism, all Cayley graphs for G. Then graphs in the resulting collection were tested for isomorphism. for each group G of order 24 for each orbit of Aut(G) on Cayley subsets choose one representative S and test isomorphism of the resulting list of Cayley graphs Cay(G, S).

The isomorphism test was performed using McKay’s program nauty for computing graph automorphisms and isomorphisms (see [65]). In this way 7697 pairwise non-isomorphic Cayley graphs of order 24 were found. Non-Cayley graphs of order 24:

For each vertex- transitive non-Cayley graph Γ = (Ω, E)

of order 24, the automorphism group AutΓ has at least one minimal transitive subgroup G, that is, a subgroup G which is transitive on Ω but has the property that every proper subgroup of G is intransitive. Since Γ is a non-Cayley graph, G is not regular on Ω. Moreover G is imprimitive on Ω. Let d be maximal such that d < 24 and there exists a G-invariant partition of Ω having d parts, and let P = {P1 | . . . | Pd } be such a partition. This phase of the search found, for all divisors d of 24 with 1 < d < 24, all pairs (Γ, G) with Γ, G as above. Since the choice of the minimal transitive subgroup G was not necessarily unique, isomorphism testing was necessary. We sketch some of the details. It should be emphasised that, even if Γ admits a minimal transitive subgroup G which is not regular on vertices, Γ may still be a Cayley graph for some other subgroup of AutΓ. Thus, our analysis may produce Cayley graphs, and this must be checked for. Let Γ, G, d be as above. Let K be the kernel G(P) of the action of G on P, so that GP = G/K. By the maximality of d, the group GP1 is primitive. If K = 1, then GP ∼ =G P1

is a transitive, but not necessarily minimal transitive permutation group of degree d, and 24 divides |GP |. If K 6= 1, then GP is a minimal transitive permutation group of degree d. A careful theoretical analysis showed, using the fact that Γ is a non-Cayley graph, that 10

d ≥ 6 and d 6= 8. Also, in the case d = 6 it turned out that all the possibilities for Γ were Cayley graphs, but discovering this required a combination of theoretical analysis and some computation. The heart of the problem was the case d = 12. Here, if α ∈ P1 then Gα fixes some union of parts of P pointwise. A combination of computation and theory showed that the number f of fixed points of Gα was not 2, 4, or 6. So f = 8, 12 or 24. Thus |K| ≤ 23 . Suppose first that K 6= 1. Then GP = G/K is one of the 17 minimal transitive groups of degree 12. (Classifying these formed part of the project!) Thus G is an extension of the elementary abelian 2-group K by a known minimal transitive group GP of degree 12. The programs of Derek Holt (see [39]) were used to construct the second cohomology group for each pair (K, GP ) for each possible action of GP on K. This yielded all possibilities for the group G together with the setwise stabilizer GP1 . Next all possibilities for the point stabilizer Gα (an index 2 subgroup of GP1 ) had to be identified, and then all generalised orbital graphs for these permutation groups had to be examined to determine which of them were non-Cayley graphs. In this way a list of 56 pairwise non-isomorphic non-Cayley graphs of order 24 were constructed. If K = 1 the group GP ∼ = G is transitive of degree 12, but is not necessarily a minimal transitive group. Each group on Royle’s list [80] of transitive groups of degree 12 was therefore a candidate for GP . Each of these was examined as a possibility for G, with GP1 a point stabilizer in the action of degree 12. The candidates for Gα were the subgroups of GP1 of index 2. These were all located and the corresponding transitive permutation groups of degree 24 were computed and their generalised orbital graphs constructed. The only non-Cayley graphs found were among the list of 56 graphs already constructed . The fact that most of the vertex-transitive graphs of order at most 24 are Cayley graphs led McKay and Praeger [48] to make the following conjecture. Conjecture 5.3 For n ≥ 1, let vtr(n) and cay(n) denote the numbers of isomorphism types of vertex-transitive graphs and Cayley graphs, respectively, with order at most n. Then lim

n→∞

6

cay(n) = 1. vtr(n)

Primitive groups and the O’Nan-Scott Theorem

For a proper discussion of our next problem, that of classifying finite distance transitive graphs, we need to understand the possible structures of finite primitive permutation groups. The type of information needed is contained in the so-called O’Nan-Scott Theorem (see [61] or [84]) which was proved independently by Michael O’Nan and Leonard Scott in 1980. It 11

has become one of the most important and useful theorems for studying finite primitive permutation groups and their applications. The O’Nan-Scott Theorem provides essentially an identification of several types of finite primitive permutation groups such that for each type we have additional information about either the abstract group theoretical structure, or the nature of the action, or both. The ordering and the presentation of the various types, and the amount of subdivision of the types preferred by those who wish to use this theorem varies according to the requirements of the different applications. For some requirements the nature of the socle or the minimal normal subgroups is most important, while (for example) for others it is the nature of the suborbits, or the existence of regular subgroups that is needed. (The socle soc(G) of a group G is the product of the minimal normal subgroups of G.) For this exposition I have chosen a subdivision into types which was suggested by Laci Kovacs in 1985. It is a little finer than the subdivision used in [61] or [84] (but it is the subdivision used in [74]). We shall define eight types of finite primitive permutation groups. The O’Nan-Scott Theorem states that every finite primitive permutation group belongs to exactly one of these types. Let G be a finite primitive permutation group on Ω and let α ∈ Ω. Then G has at most two minimal normal subgroups (see the first part of the proof in [61, Section 2]), and if there are two minimal normal subgroups then they are isomorphic and each is equal to the centraliser of the other. In particular if G has an abelian minimal normal subgroup N then N is the unique minimal normal subgroup of G, that is N is the socle of G. Further N is elementary abelian and is regular on Ω, and G is the semidirect product G = N Gα , for α ∈ Ω. Thus we may identify Ω with N = Zdp , which may be viewed as a d-dimensional vector space over a field of prime order p, and choosing α as the zero vector we then have that Gα is an irreducible subgroup of nonsingular linear transformations of N . This is our first type, and is named HA since G is contained in the Holomorph of the Abelian group N . The holomorph Hol(N ) of a group N is the semidirect product N.Aut(N ), where Aut(N ) acts naturally on the normal subgroup N . In the case where N is an elementary abelian group Zdp , Hol(N ) is the affine group AGL(d, p) of N regarded as a finite affine space. HA (holomorph of an abelian group):

Ω = Zdp for a prime p and positive integer d and G

is the semidirect product G = N.Go , a subgroup of the affine group AGL(d, p) on Ω, where N is the group of translations and Go is an irreducible subgroup of GL(d, p). For all other types each minimal normal subgroup N of G is nonabelian, and hence N = T1 × . . . × Tk for some positive integer k where each Ti ∼ = T , a nonabelian simple group. If there is a second minimal normal subgroup M 6= N , then M ∼ = N , both M and N are regular on Ω, and soc(G) = N ×M . In this case G is contained in the holomorph of N , and the 12

second minimal normal subgroup M = {xϕ−1 x | x ∈ N }, where ϕx is the inner automorphism of N induced by x, that is, ϕx : g 7→ g x . This case is subdivided into two primitive types, namely the cases where k = 1 and k > 1. In the former case the type is named HS since G is contained in the Holomorph of the Simple group T . The case where k > 1 is named HC since G is contained in the Holomorph of a “Compound” group N . The word compound is used here because G is also a subgroup of a group of compound diagonal type, a type which will be described below. For a group N , and for x ∈ N and σ ∈ Aut(N ), we denote the image of x under σ by xσ . Also the set {ϕx | x ∈ N } of all inner automorphisms of N forms the inner automorphism group Inn(N ) of N . HS (holomorph of a simple group)

Ω = T , a finite nonabelian simple group, and we have

T.Inn(T ) ≤ G ≤ Hol(T ) = T.Aut(T ), where for x ∈ Ω, t ∈ T and σ ∈ Aut(T ), tσ : x 7→ xσ tσ . If α = 1T ∈ Ω, then Inn(T ) ≤ Gα ≤ Aut(T ). HC (holomorph of a compound group)

Ω = T k , where k > 1 and T is a finite nonabelian

simple group, and N.Inn(N ) ≤ G ≤ Hol(N ) = N.Aut(N ). As for type HS, for x, n ∈ N and σ ∈ Aut(N ), nσ : x 7→ xσ nσ . If α = 1N ∈ Ω, then Inn(N ) ≤ Gα ≤ Aut(N ) and Gα acts transitively by conjugation on the simple direct factors of N . There are five further primitive types and for each of these types, N = soc(G) is the unique minimal normal subgroup of G. These five types are the type AS where G is Almost Simple, type SD where G has a Simple Diagonal action on Ω, type CD where G has a Compound Diagonal action on Ω, type TW where G is a Twisted Wreath product with regular socle, and type PA where T induces a Product Action on a cartesian product structure on Ω. We begin with the description of the type AS. A finite group G is said to be almost simple if T ≤ G ≤ Aut(T ) for some finite nonabelian simple group T . AS (almost simple) T ≤ G ≤ Aut(T ), where T is a finite nonabelian simple group, and G = T Gα with Gα 6≥ T ; the stabilizer Gα must be maximal in G, and in particular Gα 6= {1G }. The next type is the simple diagonal type SD. A primitive permutation group of type SD is a subgroup of the group

W = {(a1 , . . . , ak ) · π | ai ∈ Aut(T ), π ∈ Sk , ai ≡ aj

(mod Inn(T )) for all i, j}

(1)

where π −1 (a1 , . . . , ak )π = (a1π−1 , . . . , akπ−1 ). Further, soc(W ) is {(t1 , . . . , tk ) | ti ∈ Inn(T )}, and a primitive action of W on T k−1 (which we identify with Inn(T )k−1 ) is defined by −1 (a1 , . . . , ak ) : (t1 , . . . , tk−1 ) 7→ (a−1 k t1 a1 , . . . , ak tk−1 ak−1 ), and

13

−1 π : (t1 , . . . , tk−1 ) 7→ (t−1 kπ −1 t1π −1 , . . . , tkπ −1 t(k−1)π −1 ),

for (a1 , . . . , ak )π ∈ W , where tk = 1T . Thus for α = (1T , . . . , 1T ) ∈ T k−1 , Wα = A×Sk where A = {(a, . . . , a) | a ∈ Aut(T )}. The condition for a subgroup G of W which contains soc(W ) to be primitive in this action is that G permutes the simple direct factors of N primitively. SD (simple diagonal) N = soc(W ) ≤ G ≤ W , Ω = T k−1 with the action defined above, and G acts primitively by conjugation on the k simple direct factors of N , (k > 1). The name simple diagonal comes from the fact that Nα is the full diagonal subgroup {(t, . . . , t) | t ∈ Inn(T )} of N . For the compound diagonal type CD the group G preserves a product structure on Ω, that is Ω = ∆l for some l ≥ 2 and G ≤ Sym(∆) o Sl where, for δ = (δ1 , . . . , δl ) ∈ Ω and (a1 , . . . , al )π ∈ Sym(∆) o Sl , (a1 , . . . , al ) : δ 7→ (δ1π−1 , . . . , δlπ−1 ) and π : δ 7→ (δ1a1 , . . . , δlal ). Moreover the subgroup of Sym(∆) involved is a primitive group of type SD, hence the name Compound Diagonal. CD (compound diagonal)

Ω = ∆l , and N = T k ≤ G ≤ H o Sl ≤ Sym(∆) o Sl , for some

divisor l of k, where l ≥ 2 and k/l ≥ 2, and H ≤ Sym(∆), soc(H) = T k/l and H is primitive of type SD; G acts transitively by conjugation on the simple direct factors of N. The next type is the twisted wreath type TW. The original definition of a twisted wreath product was given by B. H. Neumann in [69]. The exposition here follows that in Suzuki’s book [86, p. 269] and is the same as in [61]. Its use was suggested by Laci Kovacs. The core of a subgroup H of a group G is coreG (H) := ∩g∈G H g . The twisted wreath product T twrϕ P of groups T and P relative to ϕ may be defined as follows. Let P have a transitive action on {1, . . . , k} and let Q be the stabilizer of the point 1 in this action. Suppose that there is a homomorphism ϕ : Q → Aut(T ) such that coreP (ϕ−1 (Inn(T )) = {1P }. Define N := {f : P → T | f (pq) = f (p)ϕ(q) for all p ∈ P, q ∈ Q}. Then N is a group under pointwise multiplication and N ∼ = T k . Let P act on N by f p (x) := f (px) for all p, x ∈ P. We define T twrϕ P to be the semidirect product of N by P relative to this conjugation action of P . Such a twisted wreath product T twrϕ P has a transitive action on N such that N acts regularly by right multiplication and for f ∈ N and p ∈ P , p : f 7→ f p . Thus the stabilizer of 14

the point 1N in T twrϕ P is the subgroup P . The conditions on N and P , for P to be maximal in T twrϕ P , are quite subtle (see [2, 50]) and imply that the image of ϕ contains Inn(T ), and that coreP (Q) = 1. TW (twisted wreath product)

G is the twisted wreath product T twrϕ P and Ω = N with

the action as defined above; moreover, coreP (Q) = 1, ϕ(Q) contains Inn(T ), and ϕ cannot be extended to any larger subgroup of P . For the final primitive type PA, G preserves a product structure ∆k on Ω and the subgroup of Sym(∆) involved is primitive of type AS with socle T . Note that, for α = (δ, . . . , δ) ∈ Ω where δ ∈ ∆, we have Nα = Tδk 6= 1. PA (product action)

Ω = ∆k , N = T k ≤ G ≤ H o Sk ≤ Sym(∆) o Sk , where H is a

primitive permutation group on ∆ of type AS with socle T , and G acts transitively by conjugation on the k simple direct factors of N , (k > 1). It is clear from the descriptions of these eight types that a finite primitive permutation group belongs to at most one of these types. The O’Nan-Scott Theorem asserts that the converse is also true. Theorem 6.1 ([61, 84]) Each finite primitive permutation group is permutationally equivalent to a primitive group in exactly one of the primitive types HA, HS, HC, AS, SD, CD, TW, PA.

7

Distance transitive graphs

The problem of classifying the finite distance-transitive graphs is one for which the theoretical developments in group theory in the 1980’s were almost perfectly suited to the graph theoretic application.

7.1

Definitions and examples

First we explain what these graphs are and give some examples. For vertices α, β of a graph Γ = (Ω, E), a path from α to β is a finite sequence of vertices γ0 , . . . , γn such that γ0 = α, γn = β, and {γi , γi+1 } ∈ E for all i with 0 ≤ i < n; and n is called the length of the path. The smallest value for n such that there is a path of length n from α to β is called the distance from α to β and is denoted d(α, β). The diameter d(Γ) of a connected graph Γ is the maximum of d(α, β) over all α, β ∈ Ω. If Γ is connected then, for each i ∈ {0, 1, . . . , d(Γ)}, we set Γi := {(α, β) | d(α, β) = i}, the set of all pairs of vertices at distance i. Then, for 15

G ≤ AutΓ, we say that Γ is a G-distance transitive graph if G is transitive on each of the sets Γ0 , . . . , Γd where d = d(Γ); Γ is distance transitive if it is (AutΓ)-distance transitive. Notice that, for a G-distance transitive graph Γ = (Ω, E), the requirements that G be transitive on Γ0 , and Γ1 are equivalent to the conditions that G be transitive on vertices and arcs, respectively. Also, the sets Γi are the G-orbitals in Ω. It is easy to see that, for each n, the complete graph Kn and the cycle Cn on n vertices are distance transitive, as is the complete bipartite graph Kn,n on 2n vertices. (For n ≥ 3, the cycle Cn is the graph with vertex set Zn and edges {i, i + 1} for each i ∈ Zn ; and for n ≥ 2, the complete bipartite graph Kn,n is the graph with vertex set Zn × Z2 such that {(i, j), (i0 , j 0 )} is an edge if and only if j 6= j 0 .) There are other interesting infinite familes of distance transitive graphs, and we describe a few of them below. Example 7.1

(a) Two of the families of finite distance transitive graphs have, as vertex

set, the set Ω of k-element subsets of a set of size n, for some k, 1 ≤ k < n. For the first family, the Johnson graphs J(n, k), two k-element subsets α, β are adjacent if and only if the intersection α ∩ β has size k − 1. The valency of J(n, k) is k(n − k), and Aut J(n, k) = Sn . For the second family, the Odd graphs Ok+1 , we have n = 2k + 1, and two k-element subsets α, β are adjacent if and only if they are disjoint. The valency of Ok+1 is k + 1 and Aut Ok+1 = Sn . (See [18, Section 9.1].) (b) The Hamming graph H(d, n) (where d ≥ 2, n ≥ 2) has vertex set Zdn and the vertices α and β are joined by an edge if and only if α − β has exactly one non-zero entry. The valency of H(d, n) is d(n − 1), and the diameter is d; its automorphism group is Sn o Sd , and it is vertex-primitive if and only if n ≥ 3. When n = 2, the Hamming graph H(d, 2) = H(d), is often called the d-cube graph. (See [18, Section 9.2].) For a vertex α, and i ≤ d(Γ), define Γi (α) := {β | d(α, β) = i}, the set of vertices at distance i from α. The following elementary lemma follows immediately from our assertions in Section 2, since a graph Γ is G-distance transitive if and only if the sets Γi are the Gorbitals. It shows that, for a vertex-transitive graph, distance transitivity may be determined by examining the actions of the point stabilizer Gα on the sets Γi (α). Lemma 7.2 Let Γ = (Ω, E) be a connected graph, let α ∈ Ω, and let G ≤ AutΓ. Then Γ is G-distance transitive if and only if G is transitive on Ω and Gα is transitive on Γi (α) for all i ∈ {0, . . . , d(Γ)}. Thus the sets Γi (α) are the Gα -orbits in Ω.

16

7.2

Parameters of finite distance transitive graphs

Let Γ = (Ω, E) be a finite distance-transitive graph, and define the parameters ki , ai , bi and ci of Γ as follows. Let α ∈ Ω. For each i = 0, . . . , d(Γ), define ki := |Γi (α)|. For each i, choose a vertex β ∈ Γi (α), and let ai , bi , ci be the number of vertices joined to β and lying in Γi (α), Γi+1 (α) (if i < d), and Γi−1 (α) (if i > 0), respectively. By distance transitivity these numbers are independent of the choices of the points α and β. The parameters are subject to many restrictions (see for example [18, Section 4.1]). We mention here only a few of them. If both the valency and the diameter of Γ are at least 3, then the parameters ki , i = 0, . . . , d(Γ), have a unimodal property [78, Lemma 1.1], namely, there exist i and j such that i + j ≤ d(Γ), k1 < k2 < · · · < ki = · · · = ki+j

and if i + j < d(Γ) then

ki+j > · · · > kd .

Thus each positive integer can occur as a value of ki for at most two subscripts i, unless it is equal to the maximum of the ki . It is also true that the valency of Γ, which by definition is equal to k1 , must be the smallest or the second smallest of the ki .

7.3

Imprimitive distance transitive graphs

Now suppose that G acts imprimitively on the vertex set Ω of Γ, that is, there is a non-trivial G-invariant partition P = {P1 | . . . | Pr } of Ω. Let α, β ∈ P1 and set i := d(α, β). As each g ∈ Gα fixes α, it must fix P1 setwise since P1 is a block of imprimitivity for G in Ω. Hence Γi (α) ⊆ P1 . This observation was strengthened by D. H. Smith [85] to show that there are just two different kinds of G-invariant partitions possible, corresponding to bipartite and antipodal graphs. A graph Γ is called bipartite if its vertex set may be partitioned into two sets, say P1 , P2 , such that, for i = 1, 2, no two vertices of Pi are joined by an edge of Γ. A graph Γ is said to be antipodal if its diameter d(Γ) = d is greater than 1, and (the generalised AutΓ-orbital graph) Γd is a disjoint union of complete graphs. Theorem 7.3 ([85]) Let Γ = (Ω, E) be a finite G-distance transitive graph of valency greater than 2. Then G is imprimitive on Ω if and only if Γ is either bipartite or antipodal. Thus if P1 is a nontrivial block of imprimitivity containing α, then either Γ is bipartite and P1 = {α} ∪ Γ2 (α) ∪ Γ4 (α) ∪ . . . , or Γ is antipodal of diameter d and P1 = {α} ∪ Γd (α). Note that an imprimitive distance-transitive graph may be both bipartite and antipodal (for example the d-cube graph for all d ≥ 2). In each of these two imprimitive cases it is possible to produce smaller distance transitive graphs. If the G-distance transitive graph Γ is bipartite, then we may represent Γ by the following diagram. 17

The quotient graph ΓP of Γ modulo the bipartition P = {P1 , P2 } is the complete graph K2 , and the induced subgraph [P1 ] is an empty graph. It turns out that it is helpful to consider the distance-2 graph, that is the orbital graph corresponding to the G-orbital Γ2 . If Γ is connected and of diameter at least 2 then Γ2 has two connected components, namely P1 and P2 and the graphs induced on these connected components are denoted by Γ+ and Γ− and are called the bipartite halves of Γ. Smith [85] proved that Γ+ and Γ− are isomorphic distance-transitive graphs. If Γ is an antipodal G-distance transitive graph, then we can represent Γ by the following diagram.

In this case Smith [85] proved that the antipodal quotient ΓP = (P, EP ) of Γ relative to the G-invariant partition P generated by {α} ∪ Γd(Γ) (called the antipodal partition) is a distance transitive graph. Moreover Γ is a cover of ΓP . (For a graph Γ = (Ω, E) and a partition P of Ω, we say that Γ covers ΓP = (P, EP ), or Γ is a cover of ΓP , if, whenever {P, P 0 } ∈ EP , each point of P, P 0 is joined to exactly one point of P 0 , P respectively. In other words the induced subgraph [P ∪ P 0 ] is a perfect matching between P and P 0 .) The distance transitive graph Γ is called an antipodal cover of ΓP . Following Cameron’s discussion in [21, p. 11], if Γ is a finite bipartite distance transitive graph of valency greater than 2, then its bipartite half Γ+ is not bipartite, and either Γ+ is primitive, or Γ+ is antipodal and the antipodal quotient of Γ+ is primitive; on the other hand if Γ is a finite non-bipartite distance transitive graph, then its antipodal quotient ΓP is neither antipodal nor bipartite, and hence is primitive. Thus by forming a sequence of antipodal quotients and/or bipartite halves of a given finite distance transitive graph Γ, we obtain a primitive distance transitive graph in at most three steps, and in at most two steps if we do the process carefully. Thus the classification problem for finite distance transitive graphs may be divided into the problem of classifying the finite, vertex-primitive, distance transitive graphs, and the problem of reconstructing general finite distance transitive graphs from the primitive examples.

7.4

Primitive distance transitive graphs

The analysis of finite primitive distance transitive graphs using the O’Nan-Scott Theorem was begun by the author and Jan Saxl in 1983. It was this project which led to the statement 18

and proof of the O’Nan-Scott Theorem given in [61]. The O’Nan-Scott Theorem identifies eight types of finite primitive permutation groups, and it turns out that only three of these types can occur as vertex-primitive groups G acting distance transitively, namely the types HA, AS, and PA. Moreover all the primitive distance transitive graphs of type PA could be identified. In proving this result, the most unpleasant types to deal with were the types HS and SD, and the original rather ugly proof was superseded by a more pleasant combinatorial approach due to Kazuhiro Yokoyama. It is possible to get a good understanding of a graph Γ = (Ω, E) admitting a vertexprimitive group G of type HS or SD by considering the case where G = T × T, for some nonabelian simple group T . We may identify the set Ω with T and then the action of G on Ω is given by (x, y) : g 7→ x−1 gy. The stabilizer Gα of the point α = 1T is Gα = {(x, x) | x ∈ T }, and the Gα -orbits in Ω are the conjugacy classes of T. Moreover each of the simple direct factors T of G acts regularly on Ω, and hence by Lemma 5.2, Γ is a Cayley graph Cay(T, S) for T , for some Cayley subset S of T . Since S is Gα -invariant, it is a union of conjugacy classes of T . If now Γ is G-distance transitive, the set S must be a single conjugacy class of non-identity elements of T by Lemma 7.2, and must be the smallest or second smallest of these (see Subsection 7.2). So the question was: Can such a Cayley subset give rise to a distance transitive Cayley graph? The negative answer involved a delicate interplay of group theory and combinatorics. The result of this first analysis of finite primitive distance transitive graphs is the following theorem. Theorem 7.4 ([78]) Let Γ = (Ω, E) be a graph and G ≤ AutΓ. If Γ is G-distance transitive, and G is primitive on Ω, then one of the following holds: (i) Γ is a Hamming graph H(d, n) or its complement H(2, n) (if d = 2), for some n ≥ 3, and G is of type PA. (ii) G is of type AS. (iii) G is of type HA. Thus to complete the classification of finite primitive distance transitive graphs, it is necessary to find all those admitting primitive groups of types HA and AS. Suppose that Γ = (Ω, E) is G-distance transitive with G primitive on Ω. If G is almost simple, then T ≤ G ≤ Aut(T ), for some nonabelian simple group T . Consider the permutation character of G on Ω, that is the map π : G → C defined by π(g) = |fixΩ g|, the number of fixed points of g in Ω. It is a basic result from group representation theory (see, for example [42, Theorem 2.8]) that π is the sum of so-called irreducible 19

characters. Moreover, because Γ is G-distance transitive, all of the G-orbitals are self-paired, and as a consequence (see [70, Theorem 8]) the irreducible characters occurring in π are distinct. Since π(1G ) = |Ω|, it follows that |Ω| is at most the sum of the degrees of all irreducible characters of G. (The degree of a character χ : G → C is the χ(1G ).) This places a strong restriction on the order of Γ. More importantly it means that |Gα | = |G|/|Ω| has to be quite large relative to |G|, and since G is primitive on Ω, Gα is a maximal subgroup of G. In most cases, all maximal subgroups of G of such a large size have been classified. Thus the possibilities for the permutation representation of G on Ω are known, and Γ must be an orbital graph for one of these permutation representations. In fact Γ must correspond to one of the two smallest Gα -orbits. Thus the strategy is, for each of these permutation representations, to look for the two smallest Gα -orbits, which are the possibilities for Γ1 (α) and to examine these orbital graphs to decide whether they are distance transitive. By its very nature, this approach to a classification depends on the finite simple group classification. An excellent reference for a more detailed account of the current state of the finite distance transitive graph classification is the survey article of A. A. Ivanov [44]. We give a brief overview of the situation. The strategy suggested above for classifying the primitive distance transitive graphs of type AS has been followed for some classes of almost simple groups G, and has led to a classification of G-vertex-primitive, G-distance transitive graphs for these groups. In the case where soc(G) = An with n > 18, the possibilities for the maximal subgroup Gα for which the associated permutation character is multiplicity free were determined by Saxl [83], and the corresponding distance transitive graphs, together with the examples in the small cases 5 ≤ n ≤ 18 were classified by Liebeck, Saxl and the author in [60]. Independently the classification of distance transitive graphs with soc(G) = An (not just the primitive examples) was done by A. A. Ivanov [43]. Next, the primitive distance transitive graphs with soc(G) = PSL(n, q) have been classified as a result of work by several groups, see [16, 15, 29, 41]. Those with soc(G) one of the other classical groups were studied by Inglis in his thesis [40] and a precise strategy for a classification for classical groups of dimension at least 13 was formulated (including a classification of the multiplicity free permutation representations for these groups). However the full details of such a classification for the classical groups has not yet been published, though a group including van Bon, Inglis and Saxl have made good progress on it. Similarly the classification for soc(G) an exceptional Lie type simple group is still an open problem. The examples for which soc(G) is one of the 26 sporadic simple groups have very recently been classified in a joint research effort involving heavy and creative use of computation by A. A. Ivanov, S. A. Linton, K. Lux, J. Saxl and L. Soicher [46].

20

Problem 7.5 Complete the classification of the finite primitive distance transitive graphs of type AS. The situation for the classification of the primitive distance transitive graphs of type HA is not so clear. We outline what seems to be the most helpful strategy, and refer the reader to [44] for more details. If Γ = (Ω, E) is G-distance transitive, with G primitive on Ω of type HA, then G = N Go with N = Znp and Go an irreducible subgroup of GL(n, p), for some prime p and integer n ≥ 1. Further, N is regular on Ω and so by Lemma 5.2, Γ is isomorphic to the Cayley graph Cay(N, Γ1 (α)), for α the zero vector of the n-dimensional vector space N over GF(p). Without loss of generality we may assume that Γ is not a complete graph or a cycle, so d(Γ) ≥ 2 and k1 = |Γ1 (α)| ≥ 3. Moreover the classification of the examples having diameter 2 follows from Martin Liebeck’s classification of the finite affine rank 3 permutation groups [59]. So we may assume that d(Γ) ≥ 3. Choose x ∈ Γ1 (α), and suppose that the 1-dimensional space hxi over GF(p) generated by x is not contained in Γ1 (α) ∪ {α}. Then there exist λ1 , λ2 ∈ GF(p)# such that λ1 x, λ2 x ∈ Γ1 (α), but y := (λ1 + λ2 )x 6∈ Γ1 (α). It follows from the definition of a Cayley graph that y ∈ Γ2 (α), and so we find that Γ2 (α) = (λ1 + λ2 )Γ1 (α) and hence that k1 = |Γ1 (α)| = |Γ2 (α)| = k2 . By an extension of the unimodality property discussed in Subsection 7.2, see [44, Lemma 2.3.2 (vi)], it follows that Γ is an antipodal 2-fold covering of a complete graph and in particular is not vertex-primitive. This is a contradiction, and hence hxi ⊆ Γ1 (α) ∪ {α}. Since Γ is connected, Γ1 (α) is a spanning set for N , and it follows that every element of N can be expressed as a sum of at most n elements of Γ1 (α). In other words, Γ has diameter d(Γ) ≤ n. Hence Go has at most n orbits on the non-zero vectors of N , and it turns out that this is a strong restriction on Go . In organising the possibilities for Go , it is helpful also to use the Aschbacher classification of the subgroups of GL(n, p), see [1, 49]. Jon van Bon has been pursuing this approach to analysing the possibilities for Γ. However it is not clear how far he is towards a classification. Problem 7.6 Complete the classification of the finite primitive distance transitive graphs of type HA.

8

Classifying finite imprimitive distance transitive graphs

Several aspects of the problem of constructing all the finite distance transitive graphs from the primitive examples have received attention. Again we refer to [44] for an account of the status of this up to 1994, and make only a few remarks. J. Hemmeter, and A. Brouwer [19, 36, 37, 38] have classified the bipartite doublings of some of the infinite families of finite distance transitive graphs of large dimension, and Ivanov [44, Section 6.8] discusses this problem for the 21

small diameter case. A general study of distance transitive antipodal covers Γ of finite distance transitive graphs of valency k was made by Tony Gardiner [30]: he showed that the antipodal blocks must have size at most k, and in the case of equality, either the graph Γ is one of two well known examples, or its antipodal quotient is the complete bipartite graph Kk,k . In the latter case, he showed further in [31] that the examples were in 1–1 correspondence with a certain family of finite projective planes. This family of projective planes has been determined explicitly (see [22, 62]) and consists of the Desarguesian projective planes and a family of projective planes defined over the twisted fields of Albert. Van Bon and Brouwer [14] have classified the distance transitive antipodal covers of classical distance transitive graphs of sufficently large diameter (finding no previously unknown examples). On the other hand the recent classification [45] of the distance transitive antipodal covers of the complete bipartite graphs Kk,k produced several new examples. The distance transitive double covers of complete graphs are in 1–1 correspondence with the so-called doubly-transitive regular twographs, and these have recently been classified by Don Taylor [87]. A general investigation of distance transitive antipodal covers of complete graphs [34] has also produced some new families of examples.

9

s-arc transitive graphs

For a positive integer s, an s-arc in a graph Γ = (Ω, E) is an (s + 1)-tuple (α0 , . . . , αs ) of vertices with the property that (αi−1 , αi ) ∈ E for 1 ≤ i ≤ s and αi−1 6= αi+1 for 1 ≤ i ≤ s−1. Further, Γ is said to be (G, s)-arc transitive if G ≤ AutΓ and G is transitive on the s-arcs of Γ; also an (AutΓ, s)-arc transitive graph will be called simply s-arc transitive. A 1-arc is just an arc, and hence Γ is (G, 1)-arc transitive if and only if it is an orbital graph for G. Thus the property of being s-arc transitive is mainly of interest if s ≥ 2. In this section we shall study the property of s-arc transitivity and the structure of finite s-arc transitive graphs, for s ≥ 2.

9.1

Examples, and Tutte’s Theorem

Although s-arc transitivity (s ≥ 2) seems a strong restriction on a finite graph, there are many families of examples known. We list a few of them. Example 9.1

(a) For n ≥ 3, the cycle Cn is s-arc transitive for all s ≥ 1.

(b) For n ≥ 4, the complete graph Γ = Kn is 2-arc transitive, but not 3-arc transitive. (c) For n ≥ 3, the complete bipartite graph Γ = Kn,n is 3-arc transitive, but not 4-arc transitive. 22

(d) Also, for n ≥ 3, the Odd graph Γ = On (defined in Example 7.1 (a)) is 3-arc transitive but not 4-arc transitive. There are also examples known of 3-valent graphs which are 4-arc transitive but not 5-arc transitive, and 5-arc transitive but not 6-arc transitive, for example the Coxeter graph on 28 vertices (see [18, Section 12.3]) and the (3, 8)-cage on 30 vertices (see [18, Section 6.9]). A remarkable theorem of Tutte (proved in 1947 and 1959) shows that for 3-valent graphs the parameter s can be no larger than 5. Theorem 9.2 ([89, 90]) If Γ = (Ω, E) is a (G, s)-arc transitive graph of valency 3, with G transitive on Ω, then s ≤ 5. This surprising result raised the question of whether there might be an absolute bound on s for finite vertex-transitive, s-arc transitive graphs of valencies greater than 3. This question received much attention in the literature and was ultimately answered in the affirmative by Richard Weiss in 1983. However, whereas Tutte’s Theorem had a beautiful and elementary proof, the proof of Weiss’s Theorem relies on the finite simple group classification. We continue our discussion of this below. An excellent survey of the status of the work in 1978, including a proof of Tutte’s theorem, can be found in [91].

9.2

Vertex-transitivity, local 2-transitivity, and Weiss’s Theorem

Although every 1-arc transitive graph is vertex-transitive this is not necessarily the case when s ≥ 2. However every connected s-arc transitive graph which is not vertex-transitive, and which contains at least one s-arc, is a tree. (A tree is a graph which has no cycles of length greater than 2.) Theorem 9.3 If Γ = (Ω, E) is a connected (G, s)-arc transitive graph containing at least one s-arc, for some positive integer s, then one of the following holds. (a) G is transitive on Ω and Γ is (G, i)-arc transitive for each i satisfying 1 ≤ i ≤ s; (b) Γ is a tree of diameter s, and if α = (α0 , α1 , . . . , αs ) and β = (β0 , β1 , . . . , βs ) are s-arcs of Γ, then for some ω ∈ {β0 , βs } and ω 0 ∈ {α0 , αs }, d(ω, ω 0 ) = s. Proof. Set v := |Γ1 (α0 )|. Suppose first that v ≥ 2. We shall show that (a) holds. For each γ ∈ Γ1 (α0 ) \ {α1 }, (γ, α0 , α1 , . . . , αs−1 ) is an s-arc, and so there exists an element g ∈ G which maps α to this s-arc, and hence α0g = γ. Thus |Γ1 (γ)| = |Γ1 (α0 )| = v ≥ 2 for each such vertex γ. Repeating this argument, with γ and (γ, α0 , α1 , . . . , αs−1 ) replacing α0 and α respectively, we see that |Γ1 (ω)| = v for each vertex ω at distance at most 2 from α0 . 23

Continuing in this way, we deduce, since Γ is connected, that Γ is regular of valency v. Now choose γ0 ∈ Ω, γ0 6= α0 . Since Γ is connected there exists a path γ = (γ0 , γ1 , . . . , γt ) from γ0 to α0 (that is, γt = α0 ), and we choose such a path with t as small as possible. Then γ is a t-arc. If t ≥ s then (γ0 , γ1 , . . . , γs ) is an s-arc, while if t < s then, since v ≥ 2, we may extend γ to an s-arc. Since Γ is (G, s)-arc transitive, some element of G maps α to this s-arc, and hence maps α0 to γ0 . Thus G is transitive on Ω. Moreover, for each i such that 1 ≤ i < s, each i-arc may be extended to an s-arc. Then a simlar argument shows that Γ is (G, i)-arc transitive also. Thus we may suppose that v = 1. Since Γ is (G, s)-arc transitive, the first vertex β0 of every s-arc satisfies |Γ1 (β0 )| = 1 also. We claim that Γ is a tree. For if not, then some vertex ω lies in a cycle of Γ of length at least 3, and hence occurs as the first vertex of an s-arc (winding around this cycle) while |Γ1 (ω)| ≥ 2. Thus Γ is a tree. Since Γ contains s-arcs, the diameter d(Γ) ≥ s. However, if d(Γ) were greater than s then Γ would contain a d(Γ)arc (δ0 , δ1 , . . . , δd(Γ) ) with |Γ1 (δ0 )| = 1. Since Γ is (G, s)-arc transitive, there exists g ∈ G such that (δ0 , δ1 , . . . , δs )g = (δ1 , δ2 , . . . , δs+1 ), which is a contradiction since |Γ1 (δ1 )| ≥ 2. Therefore, d(Γ) = s. Note that, since Γ is a tree, for vertices ω, ω 0 , there is a unique path p(ω, ω 0 ) with no repeated edges from ω to ω 0 (having length zero if ω = ω 0 ). Let β = (β0 , β1 , . . . , βs ) be an arbitrary s-arc. If, for each ω ∈ {β0 , βs } and ω 0 ∈ {α0 , αs }, p(ω, ω 0 ) had length strictly less than s/2, then we would have d(α0 , αs ) ≤ d(α0 , β0 ) + d(β0 , αs ) < s, which is not the case. Thus we may assume without loss of generality that p(β0 , α0 ) has length d0 = d(β0 , α0 ) ≥ s/2. Of course d0 ≤ d(Γ) = s. Let i be the greatest integer such that 0 ≤ i ≤ s and αi belongs to p(β0 , α0 ). Then d(β0 , αi ) = d0 − i. Also since s = d(β0 , βs ) ≤ d(β0 , αi ) + d(αi , βs ), it follows that d(αi , βs ) ≥ s − d0 + i, with equality if and only if αi belongs to the s-arc β. Suppose that αi does not belong to β. Let j be the greatest integer such that βj belongs to p(β0 , α0 ). Then βj lies on the path p(β0 , αi ) and j < d0 − i, see the diagram below.

By the maximality of i and j, the shortest path from βs to αs runs first along part of the (s − j)-arc β − := (βs , . . . , β0 ) from βs to βj , then along part of p(β0 , α0 ) from βj to αi , and finally along part of α from αi to αs . Thus s ≥ d(βs , αs ) = (s − j) + (d0 − i − j) + (s − i) = 2s + d0 − 2(i + j),

24

whence i + j ≥ (s + d0 )/2. However we have also shown that i + j < d0 , and it follows that d0 > s which is a contradiction. Hence αi belongs to β and d(βs , αi ) = s − d0 + i. With the same meaning for j as above, we see that the point βj must lie on the path p(αi , α0 ), so βj = αt for some t such that 0 ≤ t ≤ i. So the shortest path from βs to αs runs first along β − from βs to βj = αt , then along α from αt through αi to αs . Thus s ≥ d(βs , αs ) = (s − j) + (s − t), whence j + t ≥ s. However d0 = d(β0 , α0 ) = d(β0 , βj ) + d(αt , α0 ) = j + t (since βj = αt ), and d0 ≤ s. It follows that d0 = s, as required.

We shall be concerned with finite G-vertex-transitive, (G, s)-arc transitive graphs of valency at least 3, with s ≥ 2. By Theorem 9.3, such graphs are (G, 2)-arc transitive, and this property for vertex-transitive graphs is equivalent to a condition on the vertex stabiliser Gα , a so-called “local condition” on Γ. For a group-theoretic property Q, we shall say that Γ is Γ (α)

G-locally Q if the permutation group Gα1

has property Q.

Lemma 9.4 Suppose that the graph Γ = (Ω, E) is G-vertex-transitive and let α ∈ Ω. Then Γ (α)

Γ is (G, 2)-arc transitive if and only if Gα1

is 2-transitive on Γ1 (α).

Proof. Suppose that Γ is (G, 2)-arc transitive. Let (β1 , β2 ), (γ1 , γ2 ) be pairs of distinct points of Γ1 (α). Then (β1 , α, β2 ) and (γ1 , α, γ2 ) are 2-arcs of Γ. Thus there exists g ∈ G such Γ (α)

that (β1 , α, β2 )g = (γ1 , α, γ2 ). In particular, g ∈ Gα and (β1 , β2 )g = (γ1 , γ2 ). So Gα1

is

2-transitive. Conversely, suppose that for some, and hence for every, α ∈ Ω, the permutation group Γ (α) Gα1

is 2-transitive. Let (α1 , α2 , α3 ) and (β1 , β2 , β3 ) be 2-arcs of Γ. Since G is transitive

on Ω, there exists g ∈ G such that α2g = β2 . Then α1g , α3g are distinct points of Γ1 (β2 ), as are β1 , β3 . Since Gβ2 is 2-transitive on Γ1 (β2 ), there exists h ∈ Gβ2 such that (α1g , α3g )h = (β1 , β3 ). Then gh ∈ G and gh maps (α1 , α2 , α3 ) to (β1 , β2 , β3 ), whence Γ is (G, 2)-arc transitive.

It follows from the finite simple group classification that all finite 2-transitive permutation Γ (α)

groups are known, see [20]. Thus all possibilities for the group Gα1

with |Γ1 (α)| ≥ 3 are Γ (α)

known. It was proved in a series of papers that, for each of the possibilities for Gα1

, the

graph Γ could not be as much as (G, 8)-arc transitive. The final steps of the proof were completed by Richard Weiss [92] in 1983 (and references to the other papers contributing to the proof can be found in [92]). Theorem 9.5 (Weiss) Suppose that Γ = (Ω, E) is a finite G-vertex-transitive, (G, s)-arc transitive graph of valency at least 3. Then s ≤ 7.

25

9.3

Quotients of finite 2-arc transitive graphs

In Section 4 we defined the quotient ΓP = (P, EP ) of a graph Γ = (Ω, E) relative to a partition P of its vertex set Ω. In the special case where Γ is a cover of ΓP (as defined in Section 7.3), Γ and ΓP have very similar local properties. In particular, if Γ is regular, then also ΓP is regular, and val(Γ) = val(ΓP ). In this subsection we begin a discussion of the role of covers in describing the structure of finite 2-arc transitive graphs. Although 2-arc transitivity is a very strong group theoretic condition for the automorphism group of a graph to satisfy, the family of finite, connected, vertex-transitive, 2-arc transitive graphs is very large. Laci Babai showed in [6] that every finite regular graph has a finite 2-arc transitive cover. Thus the family of all finite regular graphs is contained in the class of all quotient graphs of finite 2-arc transitive graphs; and we note that for most regular graphs, their only automorphism is the identity map. This observation of Babai suggests that the family of finite 2-arc transitive graphs might be a rather uncontrollable class for which no sensible organisational principle could be expected. However, in 1985, a preliminary study was conducted in [73] of certain families of arctransitive graphs with extra group theoretic defining properties, one of these properties being 2-arc transitivity. The philosophy underlying [73] was that, in order to understand the structure of typical graphs in such a family F, the only quotients of graphs from F which should be studied were those relative to a vertex-partition invariant under a group which witnessed membership of the graph in F. For a graph Γ = (Ω, E) and a given subgroup G of AutΓ, most partitions P of Ω will neither be invariant under G, nor have the property that Γ is a cover of ΓP . Babai’s approach was to consider partitions P with the latter property, while the approach in [73] was to consider partitions with the former property. However, although the quotient ΓP of a (G, 2)-arc transitive, G-vertex-transitive graph Γ relative to a G-invariant partition P is (GP , 1)-arc transitive, it is in general far from being (GP , 2)-arc transitive. It turned out that, by choosing a rather natural sub-class of G-invariant partitions, it could be guaranteed that for such partitions P the quotient ΓP was (GP , 2)-arc transitive, and also that Γ was a cover of ΓP . These partitions are called normal partitions and will be discussed in the next section. The assertion that a quotient of a (G, 2)-arc transitive graph relative to a G-invariant partition might (by chance) be, but certainly need not be 2-arc transitive, can be illustrated effectively by the case where G = Sz(q), a Suzuki simple group. The (Sz(q), 2)-arc transitive graphs were classified by Fang and the author in [26], and it turns out that there are essentially two kinds of such graphs. Theorem 9.6 ([26]) Let G = Sz(q) where q = 2c for some odd integer c ≥ 3, and suppose 26

that Γ = (Ω, E) is a (G, 2)-arc transitive graph. Then, for α ∈ Ω, one of the following holds, and in each case examples of such graphs exist. (a) Γ has valency 2a , for some divisor a of c with a > 1, and Gα ∼ = Za2 o Z2a −1 ; (b) Γ has valency 5, and Gα ∼ = Z5 o Z4 . Each of the graphs Γ(2a ) = (Ω, E) occuring in part (a) has a quotient Γ(2a )P (relative to a certain Sz(q)-invariant partition P of Ω) which is a complete graph Kq2 +1 and on which Sz(q) acts 2-transitively on vertices: hence Sz(q) is primitive on vertices and transitive on 0

2-arcs of this quotient Γ(2a )P . Moreover if a divides a0 which divides c, then Γ(2a ) is a 0

quotient of Γ(2a ), and Γ(2c ) is a quotient of both Γ(2a ) and Γ(2a ), all relative to Sz(q)invariant partitions. The situation for the graph Γ(5) in part (b) is quite different. The graph Γ(5) = (Ω, E) has a unique vertex-primitive quotient Γ(5)P relative to a Sz(q)-invariant partition P of Ω, and Sz(q) is far from transitive on its 2-arcs. In fact this quotient is not even Sz(q)-locally primitive if q > 8. Also in none of these examples is the original graph a cover of the quotient.

10

Normal quotients of finite locally primitive graphs

For a transitive permutation group G on a set Ω, a partition P of Ω will be called G-normal if P is the set of N -orbits in Ω for some normal subgroup N of G. It is not difficult to see that all G-normal partitions of Ω are G-invariant. Lemma 10.1 Let G be a permutation group on a set Ω, and let N be a normal subgroup of G. Then the set PN of N -orbits in Ω is a G-invariant partition of Ω. Proof.

By definition PN is a partition on Ω. We need to show that it is G-invariant. Let

P ∈ PN and g ∈ G. Suppose that P = αN , the N -orbit containing α. Then P = {αn | n ∈ N }, and so P g = {αng | n ∈ N } = {(αg )g

−1 ng

| n ∈ N }.

Since N is normal in G, each element g −1 ng lies in N , and moreover g −1 ng runs over all elements of N as n does. Thus P g = {(αg )n | n ∈ N }, that is P g is the N -orbit containing αg .

The set {{α}|α ∈ Ω} of orbits of the identity subgroup, and the set {Ω} consisting of the single G-orbit Ω, are both G-normal partitions (corresponding to the identity subgroup and to G), and they are called trivial G-normal partitions. All other G-normal partitions of Ω (if any) are said to be nontrivial. It may be the case that the only G-normal partitions are 27

the trivial ones, and in this case G is said to be quasiprimitive on Ω. From this definition it is clear that every primitive permutation group is quasiprimitive, but there are many quasiprimitive permutation groups which are not primitive. For example, any transitive permutation representation of a nonabelian simple group for which a point stabiliser is a non-maximal subgroup determines a quasiprimitive permutation group which is not primitive. Suppose now that Γ = (Ω, E) is a finite G-vertex-transitive graph, for some subgroup G of AutΓ. Let N be a normal subgroup of G which is not transitive on Ω, and let PN denote the set of N -orbits in Ω and ΓN the corresponding quotient graph ΓPN relative to the G-normal partition PN . Then ΓN is called a G-normal quotient graph of Γ. We shall consider normal quotients of graphs belonging to a larger family of finite vertextransitive graphs than the family of finite vertex-transitive, 2-arc transitive graphs, namely we shall consider the family of finite vertex-transitive, locally primitive graphs. Recall from Section 9.2 that a G-vertex-transitive graph Γ = (Ω, E) is said to be G-locally primitive if, for α ∈ Ω, Gα is primitive on Γ1 (α). By Lemma 9.4, if Γ is (G, 2)-arc transitive and Gvertex-transitive, then Γ is G-locally 2-transitive. Since every 2-transitive permutation group is primitive, it follows that such a graph Γ is G-locally primitive. Now let Γ = (Ω, E) be a connected G-locally primitive graph, for some vertex-transitive subgroup G of AutΓ, and let N be a normal subgroup of G which is intransitive on Ω. It was shown in [73] that, provided |PN | > 2, ΓN is GPN -locally primitive and GPN -vertextransitive, Γ is a cover of ΓN , and GPN = G/N with N semiregular on Ω. (A permutation group N on Ω is semiregular if the only element of N which fixes a point of Ω is the identity element.) Theorem 10.2 Let Γ = (Ω, E) be a connected G-vertex-transitive and G-locally primitive graph of valency v, and let N be a normal subgroup of G. Then one of the following holds. (a) N is transitive on Ω; or (b) Γ is bipartite, the N -orbits in Ω are the two parts of the bipartition, and the graph ΓN is the complete graph K2 ; or (c) ΓN = (PN , EN ) is a connected GPN - vertex-transitive and GPN -locally primitive graph of valency v, and Γ is a cover of ΓN . The subgroup N is semiregular on Ω, and GPN ∼ = G/N . Moreover if Γ is (G, 2)-arc transitive then ΓN is (GPN , 2)-arc transitive. Proof.

Suppose that |PN | ≥ 2. We have already observed that ΓN is connected and GPN -

vertex-transitive. Let α be a vertex in a part P ∈ PN such that some vertex β ∈ Γ1 (α) belongs to a part P 0 of PN different from P . It is easy to check that Γ1 (α) ∩ P 0 is a block of 28

Γ (α)

imprimitivity for the permutation group Gα1

. Since this group is primitive, Γ1 (α) ∩ P 0 is

a trivial block of imprimitivity, and hence is equal either to {β} or to the whole set Γ1 (α). Suppose the latter holds so that Γ1 (α) ⊆ P 0 . Since G is transitive on arcs, there is an element g ∈ G which maps the arc (α, β) to (β, α), and hence interchanges α and β. This element g must interchange the parts P and P 0 containing α and β, and also g must interchange Γ1 (α) and Γ1 (β). It follows that Γ1 (β) ⊆ P , and hence that P contains all vertices which are at distance 2 from α. Continuing this argument, and using the fact that Γ is connected, we see that all vertices of Γ must lie in either P or P 0 , and hence that part (b) holds. Thus we may assume that Γ1 (α) ∩ P 0 = {β}. This means that the v vertices adjacent to α lie in exactly v distinct parts of PN , say P1 = P 0 , P2 , . . . , Pv . Since P and each of the parts Pi are N -orbits, it follows that each vertex of P is adjacent to exactly one vertex of each of the Pi . Similarly, for each i, each vertex of Pi is adjacent to exactly one vertex of P . Thus the subgraph induced on P ∪ Pi is a complete matching, and since G is transitive on the arcs of Γ it follows that this assertion holds for every pair of parts adjacent in ΓN . Thus Γ is a cover of ΓN . Now the setwise stabiliser GP of P is equal to N Gα (since N Gα clearly fixes P setwise, contains Gα and is transitive on P ), and it is not difficult to check Γ (P )

that the permutation group GPN

induced by GP on the v parts of ΓN adjacent to P is

permutationally equivalent to the permutation group induced by Gα on Γ1 (α). In particular Γ (P )

GPN

Γ (α)

is primitive, and is even 2-transitive if Gα1

is 2-transitive. Thus ΓN is GPN -locally

primitive and, if Γ is (G, 2)-arc transitive, then ΓN is (GPN , 2)-arc transitive. Finally we show that N is semiregular and GPN = G/N . Let K be the kernel of the action of G on PN . Then N ≤ K. Suppose that Kγ 6= 1 for some vertex γ. Conjugating Γ (α)

Kγ by an element of G which maps γ to α we have that Kα 6= 1. Consider Kα 1

. Since

Kα fixes each part of PN setwise, and since distinct vertices of Γ1 (α) lie in distinct parts of PN , it follows that Kα fixes Γ1 (α) pointwise. Similarly, for each β ∈ Γ1 (α), we have that Kβ fixes Γ1 (β) pointwise, and it follows that Kα fixes all points at distance 2 from α. Continuing this argument, we see, since Γ is connected, that Kα fixes each vertex of Γ, contradicting the assumption that Kα 6= 1. Thus K is semiregular, and hence its subgroup N is also semiregular. Moreover, for each g ∈ K, αg ∈ αN (since K fixes αN setwise). Hence αg = αn for some n ∈ N , and so gn−1 ∈ K ∩ Gα = Kα = 1. Therefore K = N and GPN = G/N .

Remark 10.3 It was shown in the proof that, in Theorem 10.2 (c), the permutation groups Γ(α)

Gα

Γ (P )

and GPN

are permutationally equivalent, where P = αN . Further we did not assume,

and the proof did not use, finiteness of Γ. Suppose now that Γ = (Ω, E) is a finite graph as in Theorem 10.2, and that Γ is not bipartite. Thus, choosing N to be maximal by inclusion subject to being intransitive on Ω, we 29

find that GPN = G/N is quasiprimitive on PN in addition to ΓN being GPN -locally primitive. We say that a graph Γ is a quasiprimitive, locally primitive graph if there is a subgroup G ≤ AutΓ such that Γ is G-locally primitive, and also G is quasiprimitive on vertices. Similarly, Γ is a quasiprimitive, 2-arc transitive graph if there is a subgroup G ≤ AutΓ such that Γ is (G, 2)-arc transitive, and also G is quasiprimitive on vertices. Thus we have proved the following result. Theorem 10.4 Let Γ = (Ω, E) be a finite connected G-vertex-transitive and G-locally primitive graph which is not bipartite. Then there is a semiregular normal subgroup N of G which is intransitive on Ω (possibly N = 1), such that Γ is a cover of ΓN = (PN , EN ), GPN ∼ = G/N , GPN is quasiprimitive on PN , and ΓN is GPN -locally primitive. Thus ΓN is a quasiprimitive, locally primitive normal quotient of Γ which is covered by Γ. Moreover if Γ is (G, 2)-arc transitive then ΓN a quasiprimitive, 2-arc transitive normal quotient of Γ. Remark 10.5 (a) Similar assertions may be made about the existence of special kinds of vertex-transitive, locally primitive quotients covered by the finite bipartite graphs satisfying the conditions of Theorem 10.2 (by choosing normal subgroups of the given group which are maximal with respect to having more than two orbits on vertices, see [76]). However the structure for such graphs is not as easily described as that for the non-bipartite examples. (b) Theorem 10.4 shows that each finite, non-bipartite, vertex-transitive, locally primitive graph Γ is a cover of at least one of its quasiprimitive, locally primitive, normal quotients. If in addition Γ is 2-arc transitive, then Γ is a cover of at least one of its quasiprimitive, 2-arc transitive, normal quotients. Thus there is, after all, a strong organisational principle underlying the structure of finite, vertex-transitive, 2-arc transitive graphs. To understand the effectiveness of these observations, we need information about the structure of finite quasiprimitive permutation groups. At the time when Theorems 10.2 and 10.4 were proved no information of this kind was available in the literature. However, the wish to explore the structure of finite 2-arc transitive graphs stimulated an investigation of finite quasiprimitive groups leading to a theorem akin to the O’Nan-Scott Theorem. We discuss this theorem in Section 12, and its application to 2-arc transitive graphs in Section 13. (c) The quasiprimitive, locally primitive normal quotient ΓN obtained in Theorem 10.4 is not in general determined uniquely (even up to isomorphism) by Γ and G. In fact it is possible for there to be an arbitrarily large number of pairwise non-isomorphic

30

quasiprimitive 2-arc transitive normal quotients of a finite 2-arc transitive graph, as Example 11.2 in the next section demonstrates. Before moving to the next section, we mention the following conjecture of Richard Weiss. Conjecture 10.6 ([91]) (Weiss) There exists a function f on the natural numbers N such that if Γ = (Ω, E) is a finite, G-vertex-transitive, G-locally primitive graph of valency v then, for α ∈ Ω, |Gα | ≤ f (v). For non-bipartite graphs Γ, it follows from Theorem 10.4 (see [73]) that Conjecture 10.6 is true if it is true for finite, quasiprimitive, locally primitive graphs.

11

Sabidussi’s construction of arc-transitive graphs

Let Γ = (Ω, E) be a G-arc transitive graph. Then by Theorem 2.1, Γ is an orbital graph for the transitive permutation group G on Ω. In [82] Gert Sabidussi introduced a way of identifying the self-paired orbital involved, and a group theoretic method of constructing an isomorphic copy of Γ which has proved useful in classification problems. Let α and β be adjacent vertices of Γ, and let H := Gα , the stabilizer of α. Then the subgroup H is core-free in G, that is ∩x∈G H x = 1, since G acts faithfully on the vertices of Γ. Moreover since G is transitive on the arcs of Γ, there is an element g ∈ G which maps the arc (α, β) to (β, α), and hence interchanges α and β. Clearly we may require g to be a 2-element (that is, to have order a power of 2). It is not difficult to see that g 6∈ NG (H) (since H g = Gβ 6= Gα ), and g 2 ∈ H ∩ H g = Gαβ , and a little more tricky to see that hH, gi = G if and only if Γ is connected (since the elements of hH, gi map α to vertices in the connected component of Γ containing α). It was shown by Sabidussi [82] that the given G-arc transitive graph Γ is isomorphic to the graph Γ(G, H, HgH) = (Ω∗ , E ∗ ) defined by Ω∗ := [G : H], and E ∗ := {{Hx, Hy}|xy −1 ∈ HgH}, where we write [G : H] = {Hx|x ∈ G}. Conversely, if a group G is given with a core-free subgroup H and a 2-element g such that g 6∈ NG (H), g 2 ∈ H ∩ H g , and hH, gi = G, then the graph Γ(G, H, HgH) is connected and G-arc transitive, with G acting on Ω∗ by right multiplication. Sabidussi’s results were refined in [26, Theorem 2.1] yielding precise conditions under which Γ(G, H, HgH) is (G, 2)-arc transitive.

31

Theorem 11.1 Let G be a finite group with a core-free subgroup H and a 2-element g. Then the graph Γ(G, H, HgH) is a finite, connected, (G, 2)-arc transitive graph with G transitive on vertices (acting by right multiplication) if and only if g 6∈ NG (H), g 2 ∈ H, hH, gi = G, and the action of H on [H : H ∩ H g ] by right multiplication is 2-transitive. Thus a classification of all connected (G, 2)-arc transitive graphs is equivalent to a classification of all core-free subgroups H and 2-elements g of G satisfying the conditions of Theorem 11.1. We illustrate this construction with a family of finite non-bipartite, vertextransitive, 2-arc transitive graphs of valency 4. These examples also demonstrate that such a graph may have an arbitrarily large number of pairwise non-isomorphic quasiprimitive, 2-arc transitive quotients. Example 11.2 ([32, Example 2.1]) Let pi , where i = 1, . . . , n, be distinct primes such that pi ≡ ±3 (mod 8) for all i. For each i, let Gi = PSL(2, pi ) and let Hi be a subgroup of Gi g with Hi ∼ = = A4 . Then there exists a 2-element gi ∈ Gi such that Gi = hHi , gi i and Hi ∩ H i ∼ i

Z3 . Note that, for each i, the triple Gi , Hi , gi satisfies the conditions of Theorem 11.1, and hence the graph Γ(i) := Γ(Gi , Hi , Hi gi Hi ) is connected, Gi -vertex-transitive, and (Gi , 2)-arc transitive of valency 4. Moreover, Gi is quasiprimitive on the vertices of Γ(i). However we wish to push the construction a little further. Set G := G1 × . . . × Gn , and let H be a diagonal subgroup of H1 × . . . × Hn , that is, for each i choose an isomorphism ϕi : A4 → Hi and set H := {(g ϕ1 , . . . , g ϕn ) | g ∈ A4 }. Then H ∼ = A4 . Further let g = g1 . . . gn . Then H is core-free in G, G = hH, gi and H ∩ Hg ∼ = Z3 . It follows from Theorem 11.1 that the graph Γ := Γ(G, H, HgH) is connected, G-vertex transitive, and (G, 2)-arc transitive of valency 4. Since G has no subgroups of index 2, Γ is not bipartite. For any i ≤ n, set Ni := j6=i Gj . Then Ni is normal in G and ∼ Gi , and hence Ni is a maximal normal subgroup of G. Moreover Ni ∩ H = 1, and G/Ni = Q

hence Ni is semiregular on the vertex set of Γ. It follows from Theorem 10.4 that the quotient graph ΓNi is a (Gi , 2)-arc transitive graph with Gi quasiprimitive on vertices. In fact it is not difficult to see that ΓN ∼ = Γ(i). Thus Γ is a cover of each of its (pairwise non-isomorphic) i

quasiprimitive quotients Γ(1), . . . , Γ(n).

32

12

Finite quasiprimitive permutation groups

The class of finite quasiprimitive permutation groups may be described (see [75]) in a fashion very similar to the description given by the O’Nan-Scott Theorem 6.1 for finite primitive permutation groups. Eight types of finite quasiprimitive permutation groups can be defined, which in most cases parallel the primitive types from the O’Nan-Scott Theorem, and it is proved in [75] that every finite quasiprimitive permutation group belongs to exactly one of these types. The subdivision into types given here was used in [74] for primitive groups and in [11] for quasiprimitive groups; and moreover in the latter paper a refinement of it was needed. Let G be a finite quasiprimitive permutation group on Ω and let α ∈ Ω. Then, as in the case of finite primitive groups, G has at most two minimal normal subgroups (see the first portion of [75, Section 3]), and if there are two minimal normal subgroups then they are isomorphic and each is equal to the centraliser of the other. The first three quasiprimitive types, namely types HA, HS, and HC, are exactly the same as the corresponding primitive types, that is, all quasiprimitive permutation groups of these types are primitive. We therefore suppose that G is not of these types, and hence (see the discussion in Section 6) G has a unique minimal normal subgroup N = soc(G) = T1 × . . . × Tk , for some positive integer k, where each Ti ∼ = T , a nonabelian simple group. Each of the five remaining quasiprimitive types corresponds to a primitive type. However, unlike the types HA, HS, and HC, each of these five types contains some imprimitive quasiprimitive permutation groups. For the first four of them the parallel with the corresponding primitive type is very close. These four types are the types AS, SD, CD, and TW. We describe these four types below, highlighting the differences from the corresponding primitive types. AS (almost simple) T ≤ G ≤ Aut(T ), where T is a finite nonabelian simple group, and G = T Gα . In this type we may have Gα = {1G }, that is N = T may be regular on Ω. Recall that for G to be primitive of type AS the stabilizer Gα must be maximal in G, and in particular Gα 6= {1G }. The next type is the simple diagonal type SD. A quasiprimitive permutation group of type SD is a subgroup of the group W defined in Section 6. The difference here from the primitive type SD is that G is only required to act transitively, not primitively, on the simple direct factors of N . SD (simple diagonal) N = soc(W ) ≤ G ≤ W , Ω = T k−1 , where k > 1, with the action defined in Section 6, and G acts transitively by conjugation on the k simple direct factors of N . 33

Similarly for the compound diagonal type CD the group G preserves a product structure Ω = ∆l , for some l ≥ 2, and the subgroup of Sym(∆) involved is a quasiprimitive group, rather than a primitive group, of type SD. CD (compound diagonal)

Ω = ∆l , and N = T k ≤ G ≤ H o Sl ≤ Sym(∆) o Sl , for some

divisor l of k, where l ≥ 2 and k/l ≥ 2, and H ≤ Sym(∆), soc(H) = T k/l and H is quasiprimitive of type SD; G acts transitively by conjugation on the simple direct factors of N . For the quasiprimitive type TW, again G = T twrϕ P = N.P , the subgroup Q := NP (T1 ) has index k in P (k > 1), ϕ : Q → Aut(T ), and coreP (ϕ−1 (Inn(T )) = {1P }. TW (twisted wreath product) G is the twisted wreath product T twrϕ P , and Ω = N with the action as defined in Section 6. The differences between the conditions on P, Q for primitivity and quasiprimitivity of T twrϕ P are rather subtle. A discussion can be found in [75, Remark 2.1]. We note in particular that for the quasiprimitive type TW we do not require that the image of ϕ contains Inn(T ). For the final quasiprimitive type PA, G preserves a product structure ∆k on a G-invariant partition of Ω and the subgroup of Sym(∆) involved is quasiprimitive of type AS with socle T . Thus a quasiprimitive group G of type PA induces a faithful product action on this partition of Ω. For a group R and positive integer k ≥ 2, a subdirect product of Rk is a subgroup of Rk which projects onto each of the k direct factors R. PA (product action)

N = T k ≤ G ≤ H o Sk ≤ Sym(∆) o Sk , where H is a quasiprimitive

permutation group on ∆ of type AS with non-regular socle T , and G acts transitively by conjugation on the simple direct factors of N . Choose δ ∈ ∆ and set R := Tδ . There is a (possibly trivial) G-invariant partition Ω0 of Ω such that for some ω ∈ Ω0 , Nω = Rk , and for α ∈ ω, Nα is a subdirect product of Rk . This last quasiprimitive type is the furthest from the corresponding primitive type. It is clear from the descriptions of these eight types that a finite quasiprimitive permutation group belongs to at most one of these types. The main theorem in [75] shows that the converse is also true. Theorem 12.1 ([75, Theorem 1]) Each finite quasiprimitive permutation group is permutationally equivalent to a quasiprimitive group in exactly one of the quasiprimitive types HA, HS, HC, AS, SD, CD, TW, PA. 34

In the next section we describe how this structure theorem for finite quasiprimitive permutation groups can be used to study the structure of finite quasiprimitive 2-arc transitive graphs. We have already noted, in Remark 10.5 that the structure of the finite bipartite, vertex-transitive, 2-arc transitive graphs is more complicated than that of the non-bipartite examples. However, quasiprimitive groups certainly arise when describing their normal 2-arc transitive quotients (see [76]). Finally we remark that this structure theorem has also been used very recently, by Conder, Li and the author, to study Weiss’s Conjecture 10.6 and it seems that we can now show that Conjecture 10.6 is true provided it is true for almost simple, locally primitive graphs, that is, for graphs Γ which admit a quasiprimitive subgroup G of AutΓ of type AS such that Γ is G-locally primitive.

13

Finite quasiprimitive 2-arc transitive graphs

It is important to test the effectiveness or power of our observations about the role of quasiprimitive graphs in understanding typical finite, vertex-transitive, 2-arc transitive graphs. To this end, Theorem 12.1 was used to study the possible structures of the finite quasiprimitive 2-arc transitive graphs. It turned out that only half of the quasiprimitive types could occur as quasiprimitive 2-arc transitive groups of automorphisms. Theorem 13.1 ([75, Lemmas 5.2 and 5.3]) If Γ = (Ω, E) is a finite (G, 2)-arc transitive graph such that G is quasiprimitive on Ω, then G has quasiprimitive type HA, AS, TW, or PA. Further, it was observed in [75, Section 6] and [77] that, for each of the types HA, AS, TW, and PA, there are examples of (G, 2)-arc transitive graphs with G quasiprimitive of the given type. The next step in the study of quasiprimitive 2-arc transitive graphs was to investigate further the nature of the graphs corresponding to each of these quasiprimitive types. It was in fact possible to complete the classification of those of type HA. This was done in [47] by Ivanov and the author. Recall that all quasiprimitive permutation groups of type HA are primitive. Theorem 13.2 ([47]) If Γ = (Ω, E) is a finite (G, 2)-arc transitive graph of valency k such that G is (quasi)primitive on Ω of type HA, then |Ω| = 2n , G = Zn2 ·Go ≤ Zn2 ·(AutΓ∩GL(n, 2)) with Go irreducible, and Γ, n, k, AutΓ ∩ GL(n, 2) are as in one of the lines of Table 1. If Γ = (Ω, E) is a finite primitive (G, 2)-arc transitive graph of type HA having valency k ≥ 3, then it was shown in [47] that G must be of the form G = N · Go , where the normal 35

Γ Kk+1 2k Pm (a)

n log2 (k + 1) k−1 ma

U (q) Γ(C23 ) Γ(C22 )

≥ q2 − q + 1 11 10

k k = 2n − 1 k odd 2am −1 2a −1 3

AutΓ ∩ GL(n, 2) GL(n, 2) Sk PΓL(m, 2a )

q +1 23 22

Comments – – m≥3 q≡3

PΓU(3, q) M23 M22 · 2

(mod 4); not confirmed – –

Table 1: Quasiprimitive 2-arc transitive graphs of type HA. subgroup N = Zn2 acts regularly on vertices, and the subgroup Go (which is the stabilizer of the identity element of N ) has a faithful 2-transitive action of degree k. Furthermore it was shown that N may be identified with a quotient of the natural GF(2)Go -permutation module Zk2 for the 2-transitive action of Go relative to some maximal submodule W such that Go is

faithful on Zk2 /W , and that N and hence W determine Γ. In fact we have Ω = N = Zk2 /W and {x + W, y + W } is an edge if and only if x + y ≡ e

(mod W ) for some weight 1 vector

e ∈ Zk2 . Thus, for a 2-transitive permutation group Go of degree k, the finite primitive (G, 2)arc transitive graphs of type HA, with point stabilizer Go , are in one-to-one correspondence with the maximal GF(2)Go -submodules W of the GF(2)Go -permutation module Zk2 such that Go is faithful on Zk2 /W ; alternatively they are in one-to-one correspondence with the minimal faithful GF(2)Go -submodules of Zk2 . Most of the examples in Theorem 13.2 are therefore defined in terms of a maximal submodule of the GF(2)Go -permutation module for the 2-transitive group Go . Occasionally the examples turn out to be isomorphic to some well-known graphs, and in such cases we use their more familiar descriptions. This is the case for the complete graphs in line 1 of the table, and is also the case for the graphs in line 2. The graph 2k is the folded cube, the antipodal quotient of the k-cube graph; it may be defined as the graph with vertex set Ω = Z2k−1 with {x, y} an edge if and only if x − y has weight either 1 or k − 1. The graph Pm (a) (m ≥ 3, a ≥ 1) is defined in terms of the GF(2)Go -permutation module Zk2

for Go where PSL(m, 2a ) ≤ Go ≤ PΓL(m, 2a ) and k = (2ma − 1)/(2a − 1). Let W be

the GF(2)Go -submodule generated by the characteristic functions of all the complements of hyperplanes of the projective geometry PGm−1 (2a ). Then the vertex set of Pm (2a ) is the quotient module Ω := Zk2 /W ⊥ with edges as described above. The examples in line 4 probably do not exist (that is, they are probably not quasiprimitive examples). When q ≡ 3 module V =

Zk2

(mod 4), the submodule structure of the GF(2)Go -permutation

for the 2-transitive unitary groups Go , where PSU(3, q) ≤ Go ≤ PΓU(3, q)

and k = q 3 + 1, has not been completely determined. It is believed, but has not yet been 36

proved, that V has no faithful minimal G-submodule and therefore provides no quasiprimitive examples in Theorem 13.2. This has been proved by Jane McCorkindale [66] in her D. Phil. thesis in the case where q ≡ 3

(mod 8), q + 1 6≡ 0

(mod 3), and has also been verified

computationally when q = 3, 7, 11 by Andreas Brouwer (private communication). The graphs Γ(C23 ) and Γ(C22 ) are constructed in a similar manner from the GF(2)permutation modules for the 2-transitive groups M23 and M22 · 2 respectively. Descriptions of these graphs, including distance diagrams, may be found in [18, Sections 11.3.4 and 11.3.5] and in [47, 1.8]. We next turn to the classification of (G, 2)-arc transitive graphs with G quasiprimitive of type AS, say T ≤ G ≤ Aut(T ) for a finite nonabelian simple group T . In this case the approach of Section 11 is the most useful way both of constructing examples, and of classifying all possible examples for certain classes of almost simple groups. For each almost simple group G, we need to locate first all subgroups H not containing soc(G) which have a 2-transitive permutation representation of degree at least 3. Then we need to find the 2-elements g ∈ G such that H ∩ H g is a point stabiliser in this representation. For each such triple G, H, g, if the conditions of Theorem 11.1 hold then the graph Γ(G, H, HgH) is G-vertex-transitive and (G, 2)-arc transitive. The remaining problem then is to determine, where possible, any isomorphisms between the examples constructed. It seemed to the author that it should be possible to use this strategy to classify all (G, 2)arc transitive graphs for some classes of Lie type almost simple groups G of low Lie rank. This has been achieved for the groups PSL(2, q), Sz(q) and Ree(q) in [25, 26, 27, 35]. Each of these classifications involved the constructions of several new infinite families of examples. On the other hand it can be demonstrated that such an explicit classification is not possible for the family of finite alternating and symmetric groups. This was done in [79] by Jie Wang and the author, where a study was made of primitive permutation representations of G = An or Sn such that a point stabilizer Gα is 2-transitive on one of its orbits Γ(α). All examples were classified except those for which Gα was a primitive permutation group of degree n of primitive type AS and had a faithful 2-transitive permutation representation of degree |Γ(α)|. Although all possibilities for the 2-transitive group Gα are known (using the classification of the finite simple groups, see [20]) we still do not know all faithful primitive representations of all such groups. Thus it is not feasible at present to complete the classification in [79], and consequently it is not feasible to classify even the (G, 2)-arc transitive graphs with G primitive on vertices, and G = An or Sn . Robert Baddeley [9] has made a study of finite (G, 2)-arc transitive graphs with G quasiprimitive or primitive on vertices of type TW. There are many examples and his paper gives a general approach to their construction. The situation for finite (G, 2)-arc transitive 37

graphs with G quasiprimitive of type PA is not quite so clear. Examples may be constructed as follows (again using the approach described in Section 11). However we do not know what “typical” quasiprimitive graphs of this type are like. Example 13.3 Let ∆ = (V∆ , E∆ ) be a connected (H, 2)-arc transitive graph, where H is a nonabelian simple group. Then, for δ ∈ V∆ , there is a 2-element h ∈ H such that δ h is 2

adjacent to δ in ∆, δ h = δ, and H = hHδ , hi. Let G := H o Z2 = (H × H) · hπi, where π 2 = 1 and (h1 , h2 )π = (h2 , h1 ) for all hi ∈ H, and let K = {(x, x) | x ∈ Hδ } and g = (1H , h)π. Consider Γ := Γ(G, K, KgK). Now K ∩ K g = {(x, x) | x ∈ Hδ ∩ Hδh } and hK, gi = G, and it follows from Theorem 11.1 that Γ is a connected (G, 2)-arc transitive graph and G is quasiprimitive of type PA on vertices. Finally we make some remarks about the full automorphism groups of finite, quasiprimitive, 2-arc transitive graphs. Remark 13.4 Suppose that G < H ≤ Sym(Ω). It follows from the definition of a primitive permutation group that if G is primitive on Ω, then H will also be primitive. However, if G is an imprimitive, quasiprimitive permutation group on Ω, then H need not be quasiprimitive on Ω. For example, if G preserves a nontrivial partition P of Ω, we could take H to be the full stabiliser of P in Sym(Ω): H would be a wreath product of symmetric groups, and its base group would have the parts of P as orbits. Now consider the situation for graphs. Suppose that G < H ≤ AutΓ and that Γ = (Ω, E) is (G, 2)-arc transitive (or G-locally primitive) with G quasiprimitive on Ω. Certainly, for Γ (α)

α ∈ Ω, Gα1

Γ (α)

≤ Hα 1

so that Γ is (H, 2)-arc transitive (or H-locally primitive respectively)

on Γ. However it is not clear that H will be quasiprimitive on Ω. Thus, for example, it may be the case that the full automorphism group of a quasiprimitive, 2-arc transitive graph is not quasiprimtive. Robert Baddeley [9] gave the first example of a 2-arc transitive graph Γ = (Ω, E) and subgroup G ≤ AutΓ, with G quasiprimitive on Ω and transitive on 2-arcs of Γ, for which some over-group H of G in AutΓ was not quasiprimitive. In his example, the group G was quasiprimitive of type TW. Recently Cai Heng Li [53] has constructed an infinite family of such graphs Γ, where the group G is quasiprimitive of type AS, and AutΓ is not quasiprimitive on vertices. Further, Xin Gui Fang and the author [28] have made a study of over-groups in AutΓ of a quasiprimitive, locally primitive subgroup G, where G is of type AS. These considerations suggest some fundamental questions about the nature of over-groups in Sym(Ω) of a quasiprimitive permutation group G. For an imprimitive, quasiprimitive permutation group G on Ω, if G < H < Sym(Ω), then one of the following is true. 38

(1) H is not quasiprimitive on Ω (for example, H might be the full stabiliser of a nontrivial G-invariant partition of Ω); (2) H is quasiprimitive and imprimitive on Ω; (3) H is primitive on Ω. Several studies have been made: in [74] in the case where G is primitive, and in [11] in the case where G is imprimitive and quasiprimitive, and the over-group H is primitive. A more detailed discussion of the group theoretic problems which have arisen through attempts to give a precise description of the possibilities can be found in [77, Section 7].

14

Isomorphisms of Cayley graphs

This section was included because of the discussion in the lectures of Brian Alspach at this Advanced Studies Institute of certain problems about finite Cayley graphs. We shall consider the nature of isomorphisms between Cayley graphs for a given finite group G. Let Γ := Cay(G, S), for a Cayley subset S of G. Recall that the group G, acting by right ˆ multiplication on the vertex set G of Γ, induces a subgroup, which we shall denote by G, ˆ ∼ ˆ ≤ AutΓ ≤ Sym(G). We also have, as a subgroup of AutΓ. Thus we have G = G and G of Sym(G), the group Aut(G) acting in the natural way: σ ∈ Aut(G) maps g ∈ G to its ˆ in the image g σ under σ. This subgroup Aut(G) of Sym(G) intersects the subgroup G identity subgroup. Observe that each element σ ∈ Aut(G) induces a graph isomorphism from Γ = Cay(G, S) to Cay(G, S σ ). The graph Cay(G, S) is called a CI-graph of G if, whenever Cay(G, S) ∼ = Cay(G, T ) for some Cayley subset T of G, there is an element σ ∈ Aut(G) such that S σ = T. (CI stands for Cayley Invariant.) The general question arising here is: which Cayley graphs for a group G are CI-graphs? In the special case where all Cayley graphs for G are CI-graphs, the group G is said to be ´ am [4] that all a CI-group. Interest in this question stems from a conjecture in 1967 of Ad´ ´ am’s conjecture was disproved by Elspas and Turner finite cyclic groups are CI-groups. Ad´ [24] in 1970, and has received considerable attention in the literature (see for example [68, 72] and the references cited in these papers, and also the chapter by Alspach in the proceedings of this Advanced Studies Institute). In 1977, Babai [5] began a study of this question for arbitrary groups. In particular he observed in [5, Lemma 3.1], as did Alspach and Parsons in [3, Theorem 1], that there is a simple group theoretic criterion which determines whether or not a Cayley graph is a CI-graph. ˆ denote the subgroup of Theorem 14.1 Let Γ be a Cayley graph for a group G, and let G Sym(G) consisting of the permutations of G induced by the elements of G acting by right 39

multiplication. Then Γ is a CI-graph if and only if, whenever Gτ ≤ AutΓ, for τ ∈ Sym(G), there exists σ ∈ AutΓ such that Gσ = Gτ . It has very recently been proved by Cai Heng Li [51, 52] that all finite CI-groups are soluble. His result improves and completes results of Babai and Frankl [7] dating from 1978: they proved that, if G is a CI-group of odd order, then either G is abelian, or G has an abelian normal subgroup of index 3 and its Sylow 3-subgroup is either elementary abelian, or cyclic of order 9 or 27. Moreover, they showed in [8] that if G is an insoluble CI-group, then G = L × N with (|L|, |N |) = 1, where L ∼ = PSL(2, 5), SL(2, 5), PSL(2, 13) or SL(2, 13), and N is a direct product of elementary abelian groups. However, the lead-up to Li’s result was a comprehensive investigation of finite groups G for which all Cayley graphs Cay(G, S) with |S| bounded above by some constant m (m = 1, 2, 3, . . .) are CI-graphs. Such groups are called m-CI-groups. Similar questions have been posed and studied for Cayley di-graphs leading to investigations of so-called DCI-groups and m-DCI-groups (see [51, 54, 57, 58, 95]), but we shall confine our comments as far as possible to Cayley graphs rather than Cayley di-graphs. The study of m-CI-groups was suggested by M. Y. Xu [93] in 1988. Xu was influenced on the one hand by the results of Babai and Frankl, and on the other hand by work of Toida [88] on circulant graphs of small valency, proving that all Cayley graphs of valency 3 for cyclic groups are CI-graphs. We shall give an indication of the basic framework for this study by considering further the property of being an m-CI-group, for m small. Suppose that G is a finite 2-CI-group. If S = {x, y} is a Cayley subset of G, then S = S −1 so either y = x−1 or x and y are involutions (that is o(x) = o(y) = 2, where o(x) the order of an element x). Note that Cay(G, S) has |G|/n connected components, each of which is a cycle Cn of length n, where n = o(x) if y = x−1 6= x, and n = 2o(xy) if o(x) = o(y) = 2. By considering Cayley subsets of the form {x, x−1 } with o(x) ≥ 2 (noting that such a set consists of the single element x if o(x) = 2), we see that (a) for each n ≥ 2, and each pair x, y (if any) of elements of G of order n, x can be mapped to either y or y −1 by some element of Aut(G). If xσ = y for some σ ∈ Aut(G), then we say that x and y are fused under Aut(G), while if xσ = y −1 for some σ ∈ Aut(G), then we say that x and y are inverse fused under Aut(G). Thus, for each n ≥ 2, each pair of elements of G of order n is either fused or inverse fused under Aut(G). Such groups are called FIF-groups. So G is an FIF-group. Further, suppose that G has two Cayley subsets S1 = {x, y}, with o(x) = o(y) = 2 and o(xy) = n, and S2 = {z, z −1 }, with o(z) = 2n, for some n ≥ 2. Then Cay(G, Si ) ∼ = (|G|/2n) · C2n , for

40

i = 1, 2, but S1 cannot be mapped to S2 by any element of Aut(G), which is a contradiction. Thus G cannot contain a Cayley subset of both kinds, and so (b) for each n ≥ 2, if G contains a pair of involutions x, y such that o(xy) = n, then G contains no elements of order 2n. In the case where r = 2 this observation has the following important consequence. If G contains no elements of order 4, then a Sylow 2-subgroup of G is elementary abelian. On the other hand, if G does not contain a pair of commuting involutions, then a Sylow 2-subgroup of G is cyclic or generalised quaternion. Thus (c) a Sylow 2-subgroup of G is cyclic, elementary abelian, or generalised quaternion. These restrictions on a finite group G are very strong, and extra restrictions arise for m-CIgroups, with m ≥ 3. Jiping Zhang [95] made a study of finite groups G in which every pair of elements of the same order are fused under Aut(G). His techniques were extended by Li and the author and used to study finite FIF-groups (in [54] for simple groups, and in [55] for general finite groups). The following result for simple groups is the easiest to state. We note that it depends on the finite simple group classification. Theorem 14.2 ([54, Theorem 1.2]) A finite nonabelian simple group G is an FIF-group if and only if G is one of the following: A5 , A6 , PSL(2, 7), PSL(2, 8), PSL(3, 4), Sz(8), M11 or M23 . A detailed description of finite FIF-groups is given in [55]. We state below a corollary of the main result of that paper, namely a description of the structure of finite insoluble FIF-groups. Theorem 14.3 ([55]) Let G be a finite insoluble FIF-group. Then G = A × B, where |A|, |B| are coprime, A is a soluble FIF-group and B either is one of the simple groups in Theorem 14.2, or is SL(2, q) for q = 5, 7 or 9, or is (C × Sz(8)) o Z3s m , where m, s ≥ 1. Using these results to study m-CI-groups for small values of m ≥ 2, yielded the following theorem. We note that [56] contains a detailed description of the structure of arbitrary finite 4-CI-groups which, in particular, extends the results in [7, 8]. Theorem 14.4 ([54, Theorem 1.3] and [56]) Let G be a finite insoluble group. (a) If G is simple, then G is a 2-CI-group if and only if G = A5 or PSL(2, 8). (b) If G is simple, then G is a 3-CI-group if and only if G = A5 . 41

(c) If G is a 4-CI-group, then G = U × V , where (|U |, |V |) = 1, U is abelian, and V is A5 or SL(2, 5). Cai Heng Li was able to use this result, together with an ingenious graph construction, to show that all finite CI-groups are soluble. The proof strategy goes roughly as follows. Suppose that G is a finite insoluble CI-group. Then by Theorem 14.4, G = U × V where (|U |, |V |) = 1, U is abelian and V = A5 or SL(2, 5). Since V is a characteristic subgroup of G it follows that V is also a CI-group. Thus it is sufficient to show that neither A5 nor SL(2, 5) is a CI-group, and this is what Li did. Theorem 14.5 ([52]) The simple group A5 is not a 29-CI-group, and the group SL(2, 5) is not a 58-CI-group. Corollary 14.6 All finite CI-groups are soluble. The proof of Theorem 14.5 uses the criterion given in Theorem 14.1 for a Cayley graph to be a CI-graph. Li constructs Cayley graphs for A5 and SL(2, 5) of valencies 29 and 58 respectively, which are not CI-graphs. The former graph is constructed as an orbital graph of a certain transitive permutation representation of the group PSL(2, 29). We outline the ideas below. The lexicographic product Γ1 [Γ2 ] of graphs Γ1 = (Ω1 , E1 ) by Γ2 = (Ω2 , E2 ) is the graph (Ω1 × Ω2 , E) where {(α1 , α2 ), (β1 , β2 )} ∈ E (where αi , βi ∈ Ei , for i = 1, 2) if and only if either {α1 , β1 } ∈ E1 , or α1 = β1 and {α2 , β2 } ∈ E2 . Sketch of the proof of Theorem 14.5: Let G = PSL(2, 29). Then G has subgroups B, C isomorphic to A5 but not conjugate in G, see [23]. It was shown in [52, Theorem 1.1] that there exists an arc-transitive graph Γ = Γ(G, H, HgH) of valency 29, where H ∼ = Z29 oZ7 and g is an involution, such that Aut Γ ∼ = PSL(2, 29) × Z2 . (This used the construction method described in Section 11.) Therefore B and C are not conjugate in Aut Γ. Now (|H|, |A5 |) = 1 and |H||A5 | = |G|. It follows that H ∩ B = H ∩ C = 1 and hence that G = BH = CH. Since H is the stabiliser of a vertex of Γ, we deduce from G = BH = CH that both B and C are transitive on the vertices of Γ, and we deduce from H ∩ B = H ∩ C = 1 that B and C are both regular on vertices. Thus, by Lemma 5.2, we may identify the vertex set of Γ with B in such a way that Γ = Cay(B, S) for some Cayley subset S of B. Moreover, with this identification, we have both B and C as regular subgroups of Sym(B), contained in AutΓ. Since all regular permutation representations of any given group are permutationally equivalent, it follows that B and C are conjugate in Sym(B). It then follows from Theorem 14.1 that Γ = Cay(B, S) is not a CI-graph, and hence that B ∼ = A5 is not a 29-CI-group.

42

Next assume that F = SL(2, 5) and let F = F/Z(F ) ∼ = A5 . Since A5 is not a 29-CI-group, there exist two Cayley subsets S, T of F of cardinality 29 such that the graphs Γ1 = Cay(F , S) α and Γ2 = Cay(F , T ) are isomorphic, but S 6= T for any α ∈ Aut(F ). Let Z(F ) = hei ∼ = Z2 , and let S and T be the full preimages of S and T under F → F , respectively. (If s 7→ s then the full preimage of s is {s, se}.) Then |S| = 2|S| = 58. It is not difficult to show that Cay(F, S) = Γ1 [K 2 ] ∼ = Γ2 [K 2 ] = Cay(F, T ). If there is an element σ ∈ Aut(F ) such that σ

S σ = T , then S = T , where σ is the automorphism of F induced by σ. Since the latter cannot hold for any σ, it follows that Cay(F, S) is not a CI-graph of F , and so F is not a 58-CI-group.

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