The quasiprimitive almost elusive groups

TL;DR

This paper completes the classification of quasiprimitive almost elusive groups, which are transitive groups with a unique conjugacy class of prime order derangements, expanding understanding of their structure.

Contribution

It provides a complete classification of quasiprimitive almost elusive groups, a specific class of permutation groups with unique derangement properties.

Findings

Classification of all quasiprimitive almost elusive groups completed.

Identified structural properties of these groups.

Extended previous work on elusive and almost elusive groups.

Abstract

Let $G$ be a nontrivial transitive permutation group on a finite set $\Omega$ and recall that an element of $G$ is a derangement if it has no fixed points. Derangements always exist by a classical theorem of Jordan, but there are so-called elusive groups that do not contain any derangements of prime order. In a recent paper, Burness and the author introduced the family of almost elusive groups, which contain a unique conjugacy class of derangements of prime order. In this paper, we complete the classification of the quasiprimitive almost elusive groups.