Almost elusive classical groups

Abstract

Let $G$ be a transitive permutation group acting on a finite set $Ω$ with $|Ω|\geqslant 2$. An element of $G$ is said to be a derangement if it has no fixed points on $Ω$, and by a theorem of Jordan from 1872, $G$ always contains such an element. In particular by a theorem of Fein, Kantor and Schacher $G$ contains a derangement of prime power order. Nevertheless there exist groups in which there are no derangements of prime order, these groups are called elusive groups. Defining an natural extension of this we say $G$ is almost elusive if it contains a unique conjugacy class of derangements of prime order. In recent work with Burness, we reduced the problem of determining the almost elusive quasiprimitive groups to the almost simple and 2-transitive affine cases. Additionally we classified the primitive almost elusive almost simple groups with socle an alternating group, a sporadic group or a group of Lie type with (twisted) Lie rank equal to 1. In this paper we complete the classification of the primitive almost elusive almost simple classical groups.