Permutation groups and derangements of odd prime order

Abstract

Let $G$ be a transitive permutation group of degree $n$. We say that $G$ is $2^{\prime }$-elusive if $n$ is divisible by an odd prime, but $G$ does not contain a derangement of odd prime order. In this paper we study the structure of quasiprimitive and biquasiprimitive $2^{\prime }$-elusive permutation groups, extending earlier work of Giudici and Xu on elusive groups. As an application, we use our results to investigate automorphisms of finite arc-transitive graphs of prime valency.

arXiv comment21 pages; to appear in J. Combin. Theory Ser. A