Cheryl E. Praeger

A classification of the maximal subgroups of the finite alternating and symmetric groups

Following the classification of finite simple groups, one of the major problems in finite group theory today is the determination of the maximal subgroups of the almost simple groups-that is, of groups X such that X0 u X< Aut X, for some finite non-abelian simple group X0. The problem has been solved for most sporadic groups and groups of Lie type of low rank (see [ 19, Sects. 3, 43 for discussion and references). This paper is a contribution to the case where X0 is an alternating group A, (so that for n # 6, X is A, or S,). The maximal subgroups of A, and S, are known for several classes of degrees n:

Vertex-primitive graphs of order a product of two distinct primes

Abstract Let k and p be odd primes with k p . All vertex-primitive graphs of order kp are classified, and those which are symmetric, or are edge-transitive but not symmetric. or are not Cayley graphs are identified. In addition, a classification is given of all vertex-primitive, edge-transitive lantisymmetric) directed graphs or order kp . The classification of the vertex-primitive symmetric graphs of order kp is used by the authors and others to complete the classification of symmetric graphs or order kp . The classification of the vertex-primitive non-Cayley graphs of order kp is used by B.D. McKay and the first author in an investigation of vertex-transitive non-Cayley graphs.

Linear groups with orders having certain large prime divisors

In this paper we obtain a classification of those subgroups of the finite general linear group GLd (q) with orders divisible by a primitive prime divisor of qe − 1 for some . In the course of the analysis, we obtain new results on modular representations of finite almost simple groups. In particular, in the last section, we obtain substantial extensions of the results of Landazuri and Seitz on small cross-characteristic representations of some of the finite classical groups. 1991 Mathematics Subject Classification: primary 20G40; secondary 20C20, 20C33, 20C34, 20E99.

Finite transitive permutation groups and bipartite vertex-transitive graphs

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:

Analysing Finite Locally s-arc Transitive Graphs

We present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms G and are either locally (G, s)-arc transitive for s > 2 or G-locally primitive. Such graphs are bipartite with the two parts of the bipartition being the orbits of G. Given a normal subgroup N which is intransitive on both parts of the bipartition, we show that taking quotients with respect to the orbits of N preserves both local primitivity and local s-arc transitivity and leads us to study graphs where G acts faithfully on both orbits and quasiprimitively on at least one. We determine the possible quasiprimitive types for G in these two cases and give new constructions of examples for each possible type. The analysis raises several open problems which are discussed in the final section.

On transitive Cayley graphs of groups and semigroups

We investigate Cayley graphs of semigroups and show that they sometimes enjoy properties analogous to those of the Cayley graphs of groups.

Symmetric Graphs of Order a Product of Two Distinct Primes

A simple undirected graph ? is said to be symmetric if its automorphism group Aut ? is transitive on the set of ordered pairs of adjacent vertices of ?, and ? is said to be imprimitive if Aut ? acts imprimitively on the vertices of ?. Let k and p be distinct primes with k < p. This paper gives a classification of all imprimitive symmetric graphs on kp vertices for k ? 5. The cases k < 5 have been treated previously by Cheng and Oxley (k = 2) and the second and third authors (k = 3), and the classification of primitive symmetric graphs on kp vertices with k ? 5 was done by the first and third authors.

Vertex-transitive graphs which are not Cayley graphs, I

The Petersen graph on 10 vertices is the smallest example of a vertex-transitive graph which is not a Cayley graph. We consider the problem of determining the orders of such graphs. In this, the first of a series of papers, we present a sequence of constructions which solve the problem for many orders. In particular, such graphs exist for all orders divisible by a fourth power, and all even orders which are divisible by a square. 1991 Mathematics subject classification (Amer. Math. Soc): primary 05 C 25; secondary 20 B 25.

Tetravalent Graphs Admitting Half-Transitive Group Actions

In this paper we study finite, connected, 4-valent graphsXwhich admit an action of a groupGwhich is transitive on vertices and edges, but not transitive on the arcs ofX. Such a graphXis said to be (G,1/2)-transitive. The groupGinduces an orientation of the edges ofX, and a certain class of cycles ofX(called alternating cycles) determined by the groupGis identified as having an important influence on the structure ofX. The alternating cycles are those in which consecutive edges have opposite orientations. It is shown thatXis a cover of a finite, connected, 4-valent, (G,1/2)-transitive graph for which the alternating cycles have one of three additional special properties, namely they aretightly attached, loosely attached, orantipodally attached.We give examples with each of these special attachment properties, and moreover we complete the classification (begun in a separate paper by the first author) of the tightly attached examples.

Remarks on path-transitivity in finite graphs

This paper deals with graphs the automorphism groups of which are transitive on vertices and on undirected paths (but not necessarily on directed walks) of some fixed length. In particular, it is shown that if the automorphism groupGof a graph Γ is transitive on vertices and on undirected paths of lengthk+1in Γ, for somek≥1, thenGis also transitive onk-arcs in Γ. Further details are given for the casek=1, for the case of cubic graphs, and for the casek>4.