On the involution fixity of simple groups

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