On fixed‐point‐free involutions in actions of finite exceptional groups of Lie type

Let G $G$ be a nontrivial transitive permutation group on a finite set Ω $\Omega$ . By a classical theorem of Jordan, G $G$ contains a derangement, which is an element with no fixed points on Ω $\Omega$ . Given a prime divisor r $r$ of | Ω | $|\Omega |$ , we say that G $G$ is r $r$ ‐elusive if it does not contain a derangement of order r $r$ . In a paper from 2011, Burness, Giudici, and Wilson essentially reduce the classification of the r $r$ ‐elusive primitive groups to the case where G $G$ is an almost simple group of Lie type. The classical groups with an r $r$ ‐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 $r=2$ . 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 ) $(G,M)$ , where G $G$ is an almost simple exceptional group of Lie type with socle T $T$ and M $M$ is a core‐free maximal subgroup that intersects every conjugacy class of involutions in T $T$ . Our results are conclusive, with the exception of a finite list of undetermined cases for T = E 8 ( q ) $T = E_8(q)$ , which depend on the existence (or otherwise) of certain almost simple maximal subgroups of G $G$ that have not yet been completely classified.