Generalized FC-groups

It is well-known that groups with finite conjugacy classes (i.e., FC-groups) can be considered as the most natural tool in order to study properties which are common both to finite groups and abelian groups. The attempt to investigate wider classes of groups generalizing both finiteness and nilpotency suggests the introduction of the classes of FCn-groups (n positive integer) that generalize those of FC-groups. It turns out that this classes are intermediate to finiteby- nilpotent and FC-nilpotent groups. In this paper we will review the investigations pertaining to FCn-groups highlighting some critical aspects and open questions.