Groups with iterated restrictions on conjugacy classes

Abstract

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