ON THE SUBGROUP STRUCTURE OF EXCEPTIONAL GROUPS OF LIE TYPE

We study finite subgroups of exceptional groups of Lie type, in partic ular maximal subgroups. Reduction theorems allow us to concentrate on almost simple subgroups, the main case being those with socle X(q) of Lie type in the natural characteristic. Our approach is to show that f or sufficiently large q (usually q > 9 suffices), X(q) is contained in a subgroup of positive dimension in the corresponding exceptional alg ebraic group, stabilizing the same subspaces of the Lie algebra. Appli cations are given to the study of maximal subgroups of finite exceptio nal groups. For example, we show that all maximal subgroups of suffici ently large order arise as fixed point groups of maximal closed subgro ups of positive dimension.