Nonsolvable signalizer functors revisited

Abstract

In an earlier paper [7], the authors proved three interdependent odd prime nonsolvable signalizer functor theorems, the third (and most technical) of which was designed for specific application to the analysis of the local structure of finite simple groups of characteristic 2 type (subsequently published in [S]). Shortly thereafter, Patrick McBride [IS, lo] established the authoritative result, namely, a completely general nonsolvable signalizer functor theorem, which we were able to quote directly in [6], thereby entirely by-passing [7]. Our present research Goint with Ronald Solomon) is aimed at constructing a simplified, conceptually more coherent “second generation” classification of the finite simple groups. Our intention is to limit drastically the number of results from the literature which we quote, using only the following: (a) standard “textbook” results, (b) the odd order theorem of Feit and Thompson and the classification of split (B, N)-pairs of rank 1, the proofs of both of which have undergone considerable recent revision and simplification, and (c) the existence and uniqueness of the sporadic groups along with many of their structural properties, a major chapter of simple group theory not presently in satisfactory shape. It is thus reasonable to seek a quicker-perhaps ad hoc-way around the nonsolvable signalizer functor theorem than quoting McBride’s dificult and rather long papers. The purpose of this note is to present such an alternative: improved versions (and proofs) of [7, Theorems 2 and 31, which when combined