The $\pi$-layer of a finite group

Abstract

A fundamental property of a r-separable group H, r a set of primes, is the fact that H is v-constrained; that is, O(H/O,(H) contains its own centralizer in H/O,(H) (Theorem 6.3.2 of [3]'). This concept plays a basic role in al- most all of the general classification problems solved to date (see, for example, [1], [2], [4], [5], [6], [7], [8]). On the other hand, in the classification of groups with abelian Sylow 2-subgroups [9], large portions of this analysis involve non-constrained subgroups of the simple group under consideration. It is clear that the latter situation will be typical in more general classification problems.