Subgroups of Maximal Rank in Finite Exceptional Groups of Lie Type

The purpose of this paper is to investigate a large and natural class of maximal subgroups of the finite exceptional groups of Lie type, which we call subgroups of maximal rank. These subgroups play a prominent role both in the classification of local maximal subgroups in [9, Theorem 1] and in the reduction theorem for subgroups in [28, Theorem 2]: in [9] it is shown that any local maximal subgroup of a finite exceptional group of Lie type is either a subgroup of maximal rank or one of a small list of exceptions; and in [28], using this result, it is proved that any maximal subgroup is either of maximal rank, or almost simple, or in a known list. To describe these subgroups, we require the following notation. Let G be a simple adjoint algebraic group over an algebraically closed field K of characteristic p>0, and let o be an endomorphism of G such that L = (Ga)' is a finite exceptional simple group of Lie type over Fq, where q =p . Let A!" be a group such that F*(X) = L. The group Aut L is generated by Ga> together with field and graph automorphisms (see [5,38]), all of which extend to morphisms of the abstract group G commuting with o. Thus there is a subgroup X of CAut(C)(a) such that X = X/(o), and so X acts on the set of a-stable subsets of G. For a a-stable subset Y, we write NX(Y) for the stabilizer in X of Y. If D is a a-stable closed connected reductive subgroup of G containing a maximal torus T of G, and M = NX(D), we call M a subgroup of maximal rank in X. In this paper we determine the structure and conjugacy classes of those subgroups of maximal rank which are maximal in X.