Representations in Characteristiq

1. Maximal subgroups. I will assume in this section that the finite simple groups can be classified and even that their irreducible representations in characteristicp can someday be determined. What I wish to demonstrate is that this will carry us a long way toward the determination of their maximal subgroups. It should not come as a total surprise that such a problem is tractible, since the corresponding question for complex Lie algebras and connected Lie groups was solved some time ago by E. B. Dynkin [8]. Let us review Dynkin’s solution. First of all, Dynkin treated the exceptional types separately, and I surely expect that the same will be necessary in the finite group case, requiring internal classification theory type arguments rather than representation theory. This reduced Dynkin essentially to studying the maximal subgroups of SL(,, C), O(n, C), and Sp(n, C). Next, he reduced to the case of an irreducible subgroup by simply noting that any irreducible subspace has an obvious stabilizer (when there is a form around, the irreducibility forces the subspace to be either nonsingular or totally isotropic). This had the pleasant by-product for Dynkin of also reducing to the case of a semisimple subgroup, since any connected abelian normal subgroup would have to act by scalar multiplications; in the finite groups case the maximal local subgroups associated with primes distinct from the characteristic would still have to be determined. Next Dynkin reduced the problem from the semisimple to the simple’ case (i.e., the problem of finding all maximal connected subgroups which were irreducible and simple) by using the tensor product decomposition for irreducible representations of a direct product. Any product of two or more terms would