On the involution fixity of exceptional groups of Lie type

The involution fixity ifix(G) of a permutation group G of degree n is the maximum number of fixed points of an involution. In this paper we study the involution fixity of primitive almost simple exceptional groups of Lie type. We show that if T is the socle of such a group, then either ifix(T) > n1/3, or ifix(T) = 1 and T = 2B2(q) is a Suzuki group in its natural 2-transitive action of degree n = q 2 + 1. This bound is best possible and we present more detailed results for each family of exceptional groups, which allows us to determine the groups with ifix(T) 6 n 4/9. This extends recent work of Liebeck and Shalev, who established the bound ifix(T) > n1/6 for every almost simple primitive group of degree n with socle T (with a prescribed list of exceptions). Finally, by combining our results with the Lang-Weil estimates from algebraic geometry, we determine bounds on a natural analogue of involution fixity for primitive actions of exceptional algebraic groups over algebraically closed fields.