On the involution fixity of simple groups

Let G be a finite permutation group of degree n and let 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 ifix(T) 6 n4/9. 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 ifix(T) > n1/6 (with prescribed exceptions), which was proved by Liebeck and Shalev in 2015. A similar result for classical groups will be established in a sequel.