Nonsolvable finite groups all of whose local subgroups are solvable. II

Abstract

In this second paper, the bulk of the work is devoted to characterizing E 2 () and S*(3). These two groups are "almost" -groups and it is relevant to treat them separately. The actual characterizations (Theorems 8.1 and 9.1) are very technical but the hypotheses deal with the structure and embedding in a simple group of certain {2, 3}-subgroups. This paper is a continuation of an earlier paper. 1 The bibliographical references are to I. 7* Groups in which 1 is the only p-signalizer* DEFINITION 7.1. %f*(p) = {S3 | (i) 35 is a subgroup of of type (p, p). (ii) iV(S3) contains a S p -subgroup of .}. HYPOTHESIS 7.1. ( i ) p is a prime and if S3e^*(p), then no i^-subgroup of C(S3) normalizes any nonidentity ^'-subgroup of . (ii) The centralizer of every nonidentity p-subgroup of is psolvable. Lemmas 7.1, 7.2, 7.3 are proved under Hypothesis 7.1. LEMMA 7.1. (i) ^())g^(p). (See Definitions 2.8 and 2.10 of I). (ii) If p^5, then <%f*(p) S gf (p). (iii) Ifp = S and if no element of ^(3) centralizes a quaternion subgroup of , then ^*(3)Sg'(3).