On the maximal subgroups of finite simple groups

Abstract

In the course of their proof of the solvability of groups of odd order, W. Feit and J. G. Thompson [I] establish many deep properties of the maximal subgroups of a minimal simple group 8 of odd order. Perhaps the most important of these results is the following: if for some prime p, 6 possesses an elementary abelian subgroup of order p3, then there exists a unique maximal subgroup 9X of 6 containing a given Sylow p-subgroup $3 of 6; furthermore if si is any proper subgroup of 8 such that Rn $3 possesses an elementary abelian subgroup of order p3, then R is necessarily a subgroup of 92. In our study of finite groups 6 with dihedral Sylow 2-subgroups, we require the identical result for certain odd primes p dividing the order of the centralizer of an involution in 6. On the basis of this theorem, we can show that the unique maximal subgroup 911 of 6 containing a given Sylow p-subgroup actually contains the entire centralizer of some involution in 6 and has no normal subgroups of index 2. We are then able to apply to both ?JJI and 8 formulas for the order of an arbitrary group with dihedral Sylow 2-subgroup having no normal subgroups of index 2, as developed by means of character