Groups with iterated restrictions on conjugacy classes

Let be a group class (such as the class of all finite groups). Starting from , we can define the class C of all groups G such that, for any g G, the co-centralizer G/CG((g)G) of g in G is an -group; of course, if = , these are the well-known FC-groups. Iterating this request, we define the class C2 of groups whose co-centralizers are C-groups, and so on. We generically refer to these groups as groups with -iterated conjugacy classes. Of course, if is quotient closed, then any group G such that G/ζk(G) , for some k ≥ 0, has -iterated conjugacy classes, and actually these concepts are almost always equivalent in the universe of linear groups. For = , this type of restrictions have recently been investigated, and the aim of this paper is to study the general theory of groups with -iterated conjugacy classes, paying particular attention to the case in which is the class C of Cernikov groups: we extend (and improve) results concerning groups with -iterated conjugacy classes. The main focus is on Sylow theory, serial subgroups and groups with many proper subgroups having C-iterated conjugacy classes.