Permutation groups and derangements of odd prime order

Let G be a transitive permutation group of degree n. We say that G is 2′-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′-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.