Locally elusive classical groups

TL;DR

This paper investigates the property of $r$-elusivity in almost simple classical groups, completing the classification for non-geometric actions by employing representation theory techniques.

Contribution

It extends previous work by analyzing non-geometric actions of classical groups, providing a comprehensive classification of $r$-elusivity in this context.

Findings

Complete classification of $r$-elusive almost simple classical groups for non-geometric actions.

Application of representation theory to determine derangement properties.

Clarification of the structure of groups lacking derangements of a given prime order.

Abstract

Let $G$ be a transitive permutation group of degree $n$ with point stabiliser $H$ and let $r$ be a prime divisor of $n$. We say that $G$ is $r$-elusive if it does not contain a derangement of order $r$. The problem of determining the $r$-elusive primitive groups can be reduced to the almost simple case, and the purpose of this paper is to complete the study of $r$-elusivity for almost simple classical groups. Building on our earlier work for geometric actions of classical groups, in this paper we handle the remaining non-geometric actions where $H$ is almost simple and irreducible. This requires a completely different approach, using tools from the representation theory of quasisimple groups.