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.

Let G be a  nite group and Z(G) be the center of G. For a subset A of G, we de ne kG(A), the number of conjugacy classes of G that intersect A non-trivially. In this paper, we ver-ify the structure of all  nite groups G which satisfy the property kG(G 􀀀 Z(G)) = 5; and classify them.

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.

Suppose G is a  nite group, A and B are conjugacy classes of G, and  (AB) denotes the number of conjugacy classes contained in AB. The set of all  (AB), such that A; B run over conjugacy classes of G, is denoted by  (G). The aim of this paper is to compute  (G), for G ∈ {D2n; T4n; U6n; V8n; SD8n} or G is a decomposable group of order 2pq, a group of order 4p or p3, where p and q are primes.