On Maximal Subgroups of Finite Classical Groups

In 1] M. Aschbacher described the structure of the possible maximal subgroups of a classical group G with natural module M. Each such subgroup is either a member of a canonical class of subgroups, or it is the normalizer of a quasi-simple, absolutely irreducible subgroup H of G. In the latter case suppose that H is a classical group whose deening characteristic is coprime to that of G. The aim of this article is to help classify those intermediate groups H K G, where K is again a classical group whose deening characteristic is equal to the one of G. If H is linear, then under mild hypotheses on the rank of H and the size of the eld Fq of deenition of H we show that there are no such proper K. 1. Introduction In 2] M. Aschbacher and L. Scott showed that the maximal subgroup problem for nite groups can be reduced to the following two problems: 1. Determine the maximal subgroups of the almost simple groups.