On the subgroup structure of exceptional groups of Lie type

Abstract

We study finite subgroups of exceptional groups of Lie type, in particular maximal subgroups. Reduction theorems allow us to concentrate on almost simple subgroups, the main case being those with socle X ( q ) X(q) of Lie type in the natural characteristic. Our approach is to show that for sufficiently large q q (usually q > 9 q>9 suffices), X ( q ) X(q) is contained in a subgroup of positive dimension in the corresponding exceptional algebraic group, stabilizing the same subspaces of the Lie algebra. Applications are given to the study of maximal subgroups of finite exceptional groups. For example, we show that all maximal subgroups of sufficiently large order arise as fixed point groups of maximal closed subgroups of positive dimension.