Fitting classes of CC-groups

Abstract

The theory of Fitting classes is, by now, a well established part of the theory of finite soluble groups. In contrast, Fitting classes have received rather scant attention in infinite groups, although some recent work of Beidleman and Karbe [ 2 ] and Beidleman, Karbe and Tomkinson [ 3 ] suggest that one can obtain results in this direction. The paper [ 2 ], cited above, in fact generalizes earlier work of Tomkinson [ 9 ] to the class of locally soluble FC-groups. The present paper is concerned with the theory of Fitting classes in a class of groups somewhat similar to the class of FC-groups, namely the class of CC-groups, introduced by Polovickiǐ in [ 6 ]. A group G is a CC-group if G / C G ( x G ) is a Černikov group for all x ∈ G where, as in the rest of this paper, we use the standard group theoretic notation of [ 7 ]. Recently, Alcázar and Otal [ 1 ] have shown how to generalize results of B. H. Neumann [ 5 ] to the class of CC-groups. The main purpose of the present note is to illustrate further how one can handle CC-groups, in an analogous manner to FC-groups, by using techniques similar to those used in [ 1 ] and [ 4 ].