Permutation groups and derangements of odd prime order
Timothy C. Burness, Michael Giudici

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 be a transitive permutation group of degree . We say that is -elusive if is divisible by an odd prime, but does not contain a derangement of odd prime order. In this paper we study the structure of quasiprimitive and biquasiprimitive -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.
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