On the involution fixity of simple groups

TL;DR

This paper classifies almost simple primitive groups with alternating or sporadic socles based on their involution fixity, extending previous bounds and providing a detailed understanding of involution fixed points in these groups.

Contribution

It provides a classification of groups with involution fixity at most n^{4/9}, advancing the understanding of involution fixity in almost simple primitive groups with specific socles.

Findings

Classified groups with involution fixity ≤ n^{4/9}

Extended previous bounds on involution fixity

Built groundwork for similar results on classical groups

Abstract

Let $G$ be a finite permutation group of degree $n$ and let ${\rm ifix}(G)$ be the involution fixity of $G$, which is the maximum number of fixed points of an involution. In this paper we study the involution fixity of almost simple primitive groups whose socle $T$ is an alternating or sporadic group; our main result classifies the groups of this form with ${\rm ifix}(T) \leqslant n^{4/9}$. This builds on earlier work of Burness and Thomas, who studied the case where $T$ is an exceptional group of Lie type, and it strengthens the bound ${\rm ifix}(T) > n^{1/6}$ (with prescribed exceptions), which was proved by Liebeck and Shalev in 2015. A similar result for classical groups will be established in a sequel.