Signalizer functors, proper 2-generated cores, and nonconnected groups

Abstract

As has been well understood since its inception, the signalizer functor method works most smoothly in finite groups satisfying some connectedness condition-in particular, in groups with a connected Sylow 2-subgroup. The original approach to nonconnectivity disposed of the problem ‘once and for all by completely classifying all groups of sectional 2-rank at most 4 and then determining those with a nonconnected Sylow 2-subgroup as a corollary (by a theorem of MacWilliams [ 141, every section of a nonconnected 2group can be generated by at most four elements). Even though this approach was very effective, it was extremely elaborate, for the actual proof of the classification of such groups runs to 464 pages [l 11, and, in addition, depends upon a number of other classification theorems concerning groups with specified types of Sylow 2-subgroups. Moreover, although the sectional 2-rank 4 theorem has been invoked at many other points in the classification of finite simple groups, the analysis of nonconnected groups has remained its primary application. Hence in attempting to simplify the existing classification of the finite simple groups, one of the first natural questions to consider is whether there exists a shorter, more direct way of treating nonconnectivity. Indeed, Harada has recently achieved just such a direct approach by a short, elementary fusion argument [ 121. His result can be stated as follows (here a K-group is any finite group whose composition factors are among the known simple groups):