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

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