Finite Simple Subgroups of Exceptional Algebraic Groups

Abstract

Let G = G(K) be a simple algebraic group over an algebraically closed field K of characteristic p ­­≥ 0. The study of subgroups of G splits naturally according to whether G is of classical or exceptional type, and according to whether the subgroups considered are finite or of positive dimension. This thesis considers finite subgroups of adjoint groups G of exceptional type. A finite subgroup of G is called Lie primitive if it lies in no proper, closed subgroup of positive dimension. This is a natural maximality condition and, when studying Lie primitive subgroups, a reduction theorem due to Borovik allows us to focus on those whose socle is a non-abelian finite simple group. The study then splits again according to whether or not this socle is a member of Lie(p), the simple groups of Lie type in characteristic p. For H = H(q) ∈ Lie(p), in [LS98b] Liebeck and Seitz prove, for all but finitely many q, that G cannot have a Lie primitive subgroup with socle H unless G and H are of the same Lie type. For H ∉ Lie(p), in [LS99] Liebeck and Seitz produce a complete (finite) list of those H which embed into an adjoint exceptional simple algebraic group, though conjugacy and Lie primitivity remain largely open. The first result of this thesis is to disprove the existence of Lie primitive embeddings of many simple groups H ∉ Lie(p). For example, for n ≥ 10 the alternating group Altn has no Lie primitive embeddings into an adjoint exceptional algebraic group, in any characteristic. This has implications for the subgroup structure of the nite groups of Lie type. In particular, it is deduced here that for n ≥ 11 the groups Altn and Symn never occur as a maximal subgroup of any nite almost-simple group of exceptional Lie type.