On the involution fixity of exceptional groups of Lie type

TL;DR

This paper investigates the maximum fixed points of involutions in primitive almost simple exceptional groups of Lie type, establishing bounds and identifying specific groups with low involution fixity, extending previous work in the area.

Contribution

It provides new bounds on involution fixity for exceptional groups of Lie type and characterizes groups with particularly low fixity, expanding understanding of their permutation actions.

Findings

If T is the socle, then ifix(T) > n^{1/3} or T = {}^2B_2(q) with ifix(T)=1.

The bounds are shown to be optimal.

Identifies groups with ifix(T) ≤ n^{4/9}.

Abstract

The involution fixity ${\rm ifix}(G)$ 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 ${\rm ifix}(T) > n^{1/3}$, or ${\rm ifix}(T) = 1$ and $T = {}^2B_2(q)$ 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 ${\rm ifix}(T) \leqslant n^{4/9}$. This extends recent work of Liebeck and Shalev, who established the bound ${\rm ifix}(T) > n^{1/6}$ 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.