Almost elusive permutation groups

TL;DR

This paper investigates the structure of almost elusive permutation groups, classifying those that are almost simple or affine 2-transitive, and provides a detailed classification for certain primitive cases.

Contribution

It introduces the concept of almost elusive groups, proves their structure in quasiprimitive cases, and classifies all such groups with specific socles.

Findings

Every quasiprimitive almost elusive group is either almost simple or affine 2-transitive.

Classified all almost elusive groups with socles as alternating, sporadic, or rank one Lie type groups.

Identified the conditions under which groups contain a unique conjugacy class of prime order derangements.

Abstract

Let $G$ be a nontrivial transitive permutation group on a finite set $\Omega$. An element of $G$ is said to be a derangement if it has no fixed points on $\Omega$. From the orbit counting lemma, it follows that $G$ contains a derangement, and in fact $G$ contains a derangement of prime power order by a theorem of Fein, Kantor and Schacher. However, there are groups with no derangements of prime order; these are the so-called elusive groups and they have been widely studied in recent years. Extending this notion, we say that $G$ is almost elusive if it contains a unique conjugacy class of derangements of prime order. In this paper we first prove that every quasiprimitive almost elusive group is either almost simple or $2$-transitive of affine type. We then classify all the almost elusive groups that are almost simple and primitive with socle an alternating group, a sporadic group, or a rank one group of Lie type.