On non-generic finite subgroups of exceptional algebraic groups

The study of finite subgroups of a simple algebraic group G reduces in a sense to those which are almost simple. If an almost simple subgroup of G has a socle which is not isomorphic to a group of Lie type in the underlying characteristic of G, then the subgroup is called non-generic. This paper considers non-generic subgroups of simple algebraic groups of exceptional type in arbitrary characteristic. A finite subgroup is called Lie primitive if it lies in no proper subgroup of positive dimension. We prove here that many non-generic subgroup types, including the alternating and symmetric groups Altn, Symn for n ≥ 10, do not occur as Lie primitive subgroups of an exceptional algebraic group. A subgroup of G is called G-completely reducible if, whenever it lies in a parabolic subgroup of G, it lies in a conjugate of the corresponding Levi factor. Here, we derive a fairly short list of possible isomorphism types of non-G-completely reducible, non-generic simple subgroups. As an intermediate result, for each simply connected G of exceptional type, and each non-generic finite simple group H which embeds into G/Z(G), we derive a set of feasible characters, which restrict the possible composition factors of V ↓ S, whenever S is a subgroup of G with image H in G/Z(G), and V is either the Lie algebra of G or a non-trivial Weyl module for G of least dimension. This has implications for the subgroup structure of the finite groups of exceptional Lie type. For instance, we show that for n ≥ 10, Altn and Symn, as well as numerous other almost simple groups, cannot occur as a maximal subgroup of an almost simple group whose socle is a finite simple group of exceptional Lie type.