On the subgroup structure of classical groups

Abstract

In this paper we establish a result on the subgroup structure of classical groups over algebraically closed ®elds, and use this to give a new proof of a fundamental theorem of M. Aschbacher on subgroups of ®nite classical groups. Let V be a ®nite-dimensional vector space over an algebraically closed ®eld K, and let G be one of the classical algebraic groups SL V †; Sp V † or SO V †. Our result is a reduction theorem concerning the subgroups of G: we de®ne a certain collection C of natural proper subgroups of G, and prove that any closed (®nite or in®nite) subgroup of G either lies in a member of C, or is, roughly speaking, a simple group acting irreducibly on V . Aschbacher's result is an analogous reduction theorem for subgroups of ®nite classical groups. We obtain this as a relatively easy consequence of our main result by taking ®xed points under the action of a Frobenius morphism, using a standard process involving Lang's theorem. The proof of the main result uses elementary linear algebra, together with a few basic facts from the theory of algebraic groups. Various complications which arise in the ®nite group setting in [As] become much more straightforward in the algebraic group setting; in particular, questions involving extension ®elds do not occur, and issues of conjugacy are easily settled. When we descend to ®nite Invent. math. 134, 427 ± 453 (1998)