On fixed-point-free involutions in actions of finite exceptional groups of Lie type
Timothy C. Burness, Mikko Korhonen
TL;DR
This work classifies, for finite almost simple groups of Lie type with socle an exceptional group, when the action on a coset space $G/H$ is $2$-elusive, i.e., when every involution in the socle $T$ fixes some point. The authors develop a unified framework combining algebraic-group theory, involution classification, and computational tools (notably Magma) to study the intersection of $T$-conjugacy classes of involutions with maximal subgroups, organized into parabolic, subfield/twisted, and almost-simple families. Using feasible characters and cohomological criteria, they isolate all $2$-elusive instances, obtaining complete results for $T$ in the low-rank or classical-exceptional cases and a comprehensive list for the higher-rank exceptional groups, with a finite, well-described set of undetermined $E_8(q)$ cases tied to the existence of certain almost-simple maximal subgroups. The outcomes yield a thorough map of when almost simple exceptional groups admit $2$-elusive primitive actions, with implications for understanding derangements and fixed-point phenomena in Lie-type permutation actions. The findings advance the classification program initiated by Burness–Giudici–Wilson and provide a resource for researchers studying derangements in groups of Lie type and their local subgroups, aided by explicit tables of exceptional configurations.
Abstract
Let $G$ be a nontrivial transitive permutation group on a finite set $Ω$. By a classical theorem of Jordan, $G$ contains a derangement, which is an element with no fixed points on $Ω$. Given a prime divisor $r$ of $|Ω|$, we say that $G$ is $r$-elusive if it does not contain a derangement of order $r$. In a paper from 2011, Burness, Giudici and Wilson essentially reduce the classification of the $r$-elusive primitive groups to the case where $G$ is an almost simple group of Lie type. The classical groups with an $r$-elusive socle have been determined by Burness and Giudici, and in this paper we consider the analogous problem for the exceptional groups of Lie type, focussing on the special case $r=2$. Our main theorem describes all the almost simple primitive exceptional groups with a $2$-elusive socle. In other words, we determine the pairs $(G,M)$, where $G$ is an almost simple exceptional group of Lie type with socle $T$ and $M$ is a core-free maximal subgroup that intersects every conjugacy class of involutions in $T$. Our results are conclusive, with the exception of a finite list of undetermined cases for $T = E_8(q)$, which depend on the existence (or otherwise) of certain almost simple maximal subgroups of $G$ that have not yet been completely classified.