A group class Ӿ is said to be countably recognizable if a group belongs to Ӿ whenever all its countable subgroups lie in Ӿ. It is proved here that most of the relevant classes of groups de fined by restrictions on the conjugacy classes are countably recognizable.

Let n be a positive integer and let G be a group. We denote by , (G) a certain extension of the non-abelian tensor square G G by G ,G. Set T if the size of the conjugacy class , , , (G) = f g h j g,h 2 G g. We prove that , (G) x , , ,,n for every x 2 T (G), then the second derived subgroup , (G)′, ′,is , nite with n-bounded order. Moreover, we obtain a suffi, cient condition for a group to be a BFC-group.

We survey known results concerning how the conjugacy classes contained in a normal subgroup and their sizes exert an inuence on the normal structure of a finite group. The approach is mainly presented in the framework of graphs associated to the conjugacy classes, which have been introduced and developed in the past few years. We will see how the properties of these graphs, along with some extensions of the classic Landau's Theorem on conjugacy classes for normal subgroups, have been used in order to classify groups and normal subgroups satisfying certain conjugacy class numerical conditions.

We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. We also collect some problems that have only been partially solved.