On the involution fixity of simple groups
Timothy C. Burness, Elisa Covato

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 be a finite permutation group of degree and let be the involution fixity of , 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 is an alternating or sporadic group; our main result classifies the groups of this form with . This builds on earlier work of Burness and Thomas, who studied the case where is an exceptional group of Lie type, and it strengthens the bound (with prescribed exceptions), which was proved by Liebeck and Shalev in 2015. A similar result for classical groups will be established in a sequel.
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