Permutation groups and derangements of odd prime order

TL;DR

This paper investigates the structure of certain permutation groups that lack derangements of odd prime order, extending previous classifications and applying findings to automorphisms of prime-valency graphs.

Contribution

It characterizes quasiprimitive and biquasiprimitive 2'-elusive groups, expanding understanding of elusive groups and their automorphism properties.

Findings

Classified quasiprimitive 2'-elusive groups

Extended previous work on elusive groups

Applied results to automorphisms of prime-valency graphs

Abstract

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.