Groups all proper quotient groups of which have Chernikov conjugacy classes

Abstract

Numerous problems in group theory require the investigation of the influence of certain systems of proper quotient groups (quotient groups modulo nonidentity normal subgroups) on the structure of the entire group. Thus, the family of all finite quotient groups plays an important role in the investigation of algorithmic problems in finitely defined groups (Maltsev [1]) and in the theory of manifolds [2]. The important Robinson theorem on the nilpotency of a finitely generated soluble group all finite quotient groups of which are nilpotent (see Theorem 10.51 in [3]) also demonstrates the role of the family of finite quotient groups. Quotient groups modulo centralizers of principal quotients also play an important role in the theory of formations. On the other hand, the investigation of the structure of a group with the given family of all proper quotient groups has a comparatively long history. For the first time, problems of this type in the theory of infinite groups were considered by Newman in [4, 5], where the groups with Abelian proper quotient groups were studied. Later, investigations in this direction were continued for various group-theoretic properties: McCarthy [6, 7] and Wilson [8] studied groups all proper quotient groups of which are finite, Franciosi and de Giovanni [9] and Kurdachenko, Goretskii, and Pylaev [10] studied groups with Chernikov proper quotient groups, Groves [11] and Robinson and Wilson [ 12] investigated groups with proper polycyclic quotient groups, and Robinson and Zhang [ 13] considered groups all proper quotient groups of which are finite over the centers or have finite commutants. The investigations of Robinson and Zhang were continued by Franciosi, de Giovanni, and Kurdachenko in [14], where the groups all proper quotient groups of which are FC-groups were considered. At present, there exist various generalizations of FC-groups defined as follows: Let Y be a class of groups. We say that G is a ,~C-group (or that G has con-