Balance and generation in finite groups

Abstract

In the study of finite simple groups, the analysis of the cores of the centralizers of involutions is of fundamental importance. The principal aim of this analysis is to prove that the centralizers of involutions have what is called semisimple 2-layers, a property which holds in all known simple groups. Thompson has termed this the B(G)-conjecture. At present two methods exist for treating this problem-the so-called signalixer functor method which has as its starting point the GorensteinGoldschmidt signalizer functor theorem [3-51 and the Bender method which emanates from his simplified proof of the uniqueness theorems of FeitThompson in minimal simple groups of odd order. Both methods have been used effectively in the applications, the former notably in the study of balanced groups [S] and in a number of characterization theorems concerning known simple groups and families of simple groups by the structure of their Sylow 2-subgroups; and the latter in Bender’s proof of the classification of simple groups with abelian Sylow 2-subgroups and Goldschmidt’s determination of all simple groups which possess a nontrivial abelian 2-subgroup which is strongly closed in a Sylow 2-subgroup (with respect to the given simple group). The prototype of a balanced group X is one in which the associated quasisimple components (if any) of the centralizers of the involutions of G are groups of Lie type over GF(2”) (with the exception of L,(4) and L,(4) and their covering groups, these being the only such groups which are not balanced quasisimple groups). In particular, X is balanced if the centralizer of every involution is 2-constrained (corresponding to the case in which no such components exist). The object of the present paper is to develop the