Finite transitive permutation groups and finite vertex-transitive graphs

Abstract

The theory of vertex-transitive graphs has developed in parallel with the theory of transitive permutation groups. The chapter explores the way the two theories have influenced each other. Examples are drawn from the enumeration of vertex-transitive graphs of small order, the classification problem for finite distance transitive graphs, and the investigations of finite 2-arc transitive graphs, finite primitive and quasiprimitive permutation groups, and finite locally primitive graphs. The nature of the group theoretic techniques used range from elementary ones to some involving the finite simple group classification. In particular the theorem of O’Nan and Scott for finite primitive permutation groups, and a generalisation of it for finite quasiprimitive permutation groups is discussed.