On the structure of certain p-solvable linear groups

Abstract

There are four sections in the paper. In the first section notation, past results and some preparations on p-solvable linear groups are presented. In the second section the inductive proof of the Theorem is begun and we reduce down to the case where G has a p-Sylow subgroup P of order p and a normal p-complement N, where N modulo the center is a nonabelian simple group. In Section 3, detailed information about some Sylow normalizers of N is obtained. In the last section the strong interplay between various Sylow normalizers of N and the subgroup %YN(P) is used in completing the proof. A recent result of Smith and Tyrer ([12]) helps dispose of some difficult cases that arise in Sections 3 and 4. The proof also apparently requires some deep classification theorems on finite simple groups in contrast to the earlier papers. Although we have not addressed ourselves to the problem of describing the structure of G when G is solvable, the following result can be extracted from the proof of the Theorem.