The theory of groups with finite classes of conjugate elements

Abstract

In recent investigations concerning the structure of infinite groups, several important results were obtained for groups which are in one sense or another close to abelian groups resp. to finite groups. Such a category of groups is, for instance, the grouPs in which every element has a finite number of conjugates. In what follows such groups will be called briefly FC-groups. Trivial examples of FC-gr0ups are the abelian groups, the finite groups, and any direct product of an arbitrary set of abelian or finite groups. But we shall see in 9 8 below that there exist FC-groups, even ones with two generators~ which cannot be obtained by such a trivial construction. The theory of FC-groups is due to R. BAER and B. H. NEUMANN and is developed in [2] and [6]. 1 The present paper has arisen from the observation that NEUMANN'S very clear and excellent treatment in [6] admits some simplifications. In particular, in the present exposition of NEUMANN'S theory I have succeeded in avoiding the use of a deep result of SCHREmR in [7] and the extension of the group under consideration by constructing a direct product with amalgamation (this plays an imporfant role in NEUMANN'S treatment). Thus the development of the theory of FC-groups becomes thoroughly elementary, so that the reading of this paper supposes merely the: knowledge of the simplest basic concepts of group theory. In addition to NEUMANN'S theory, we give here also a new application and we make some remarks on the converses of the results. The main results of the theory of FC-groups are the following. In an FC-group the elements of finite order form a (characteristic) subgroup whose factor group in the whole group is an abelian torsion-free group (9 4). A group is always an FC-group if its center has a finite index, or if its derived group is finite (9 2, 9 6). However, we shall see in 9 8 that noneof