Splitting over nilpotent and hypercentral residuals

Abstract

It is a well known theorem of Gaschütz (4) and Schenkman (12) that if G is a finite group whose nilpotent residual A is Abelian, then G splits over A and the complements to A in G are conjugate. Following Robinson (10) we describe this situation by saying that G splits conjugately over A. A number of generalizations of this result have since been obtained, some of them being in the context of the formation theory of finite or locally finite groups (see, for example, (1), (3)) and others, for example, the recent and far-reaching results of Robinson (10, 11) being concerned with groups which are not necessarily periodic. Our results here are of the latter type.

Artinian Modules over Group Rings

“…The first results on the existence of the Z-decomposition in infinite modules were obtained by B. Hartley and M.J. Tomkinson [111]. The first results on the existence of the Z-decomposition in infinite modules were obtained by B. Hartley and M.J. Tomkinson [111].…”

A Class of infinite soluble groups with an A-group condition

“…is a saturated formation of finite soluble groups and G is a finite soluble group with abelian -residual G 8 , then G splits over G s and the complements are conjugate, being the $-normalizers of G [1]. This theorem has been extended to extensions of abelian Si-groups by hypercentral and hypercyclic groups in [3] and [10], where it is shown that if G is a group with hypercentral (hypercyclic) residual A such that G/A is hypercentral (hypercyclic) and A is an abelian ©i-group, then G splits over A and the complements are conjugate.…”

Anti-CC-Groups and Anti-PC-Groups

“…The background has been referred to [1,Section 4.3] for FC-groups, to [4,22,23] for CC-groups, and to [7] for PC-groups. General information on locally finite and locally nilpotent groups can be found in [10,14,24].…”