Introduction

Groups with Černikov conjugacy classes or CC-groups were considered by Polovickn [11] as an extension of the concept of an FC-group, that is, a group in which every element has only a finite number of conjugates. A group G is said to be a CC-group if G/CG(XG) is a Cernikov group for each x ∈ G (see also [12, 4.36]).

In [1] and [7] a classical Sylow theory for CC-groups was initiated. The local conjugacy of the Sylow p-subgroups of a CC-group G was established as was the characterization of when they are conjugate in terms of their number and of internal properties of the group G. We now present a summary of the generalization of the above results to an arbitrary set π of primes. We also consider the usual ingredients of a more complete Sylow theory such as Sylow bases or Carter subgroups, thought of as an extension of the FC-case: see [4], [15], [16], [17] and [18]. The details of the contents of this note can be found in [9] and this, being a survey article, does not contain proofs, except as an illustration.

Throughout our notation is standard and is taken from [12] and [18], to which we refer for the basic definitions and the setting of the problems we are considering. Some parts of the proofs of our results, as we present them here, depend on the interpretation of a residually Černikov group as a co-Černikov group, following the topological approach set out and developed by Dixon [2].