Permutation Groups, Vertex-transitive Digraphs and Semiregular Automorphisms

A nonidentity element of a permutation group is said to be semiregular if all of its orbits have the same length. The work in this paper is linked to [6] where the problem of existence of semiregular automorphisms in vertex-transitive digraphs was posed. It was observed there that every vertex-transitive digraph of order pk or mp, where p is a prime, k 1 and m p are positive integers, has a semiregular automorphism. On the other hand, there are transitive permutation groups without semiregular elements [4]. In this paper, it is proved that every cubic vertex-transitive graph contains a semiregular automorphism, and moreover, it is shown that every vertex-transitive digraph of order 2p2, where p is a prime, contains a semiregular automorphism.