Finite Transitive Permutation Groups and Finite 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: