Some topics in the theory of generalized fc-groups

Abstract

A finiteness condition is a group-theoretical property which is possessed by all finite groups: thus it is a generalization of finiteness. This embraces an immensely wide collection of properties like, for example, finiteness, finitely generated, the maximal condition and so on. There are also numerous finiteness conditions which restrict, in some way, a set of conjugates or a set of commutators in a group. Sometimes these restrictions are strong enough to impose a recognizable structure on the group. R. Baer and B.H. Neumann were the first authors to discuss groups in which there is a limitation on the number of conjugates which an element may have. An element x of a group G is called FC-element of G if x has only a finite number of conjugates in G, that is to say, if |G : CG(x)| is finite or, equivalently, if the factor group G/CG(⟨x⟩G) is finite. It is a basic fact that the FC-elements always form a characteristic subgroup. An FC-element may be thought as a generalization of an element of the center of the group, because the elements of the latter type have just one conjugate. For this reason the subgroup of all FC-elements is called the FC-center and, clearly, always contains the center. A group G is called an FC-group if it equals its FC-center, in other words, every conjugacy class of G is finite. Prominent among the FC-groups are groups with center of finite index: in such a group each centralizer must be of finite index, because it contains the center. Of course in particular all abelian groups and all finite groups are FC-groups. Further examples of FC-groups can be obtained by noting that the class of FC-groups is closed with respect to forming subgroups, images and direct products. The theory of FC-groups had a strong development in the second half of the last century and relevant contributions have been given by several important authors including R. Baer, B.H. Neumann, Y.M. Gorcakov, Chernikov,L.A. Kurdachenko, and many others. We shall use the monographs , as a general reference for results on FC-groups. The study of FC-groups can be considered as a natural investigation on the properties common to both finite groups and abelian groups. A particular interest has been devoted to groups having many FC-subgroups or many FC-elements. [edited by the author]