The aim of this paper is to show how the number of conjugacy classes appearing in the product of classes affect the structure of a finite group. The aim of this paper was to show several results about solvability concerning the case in which the power of a conjugacy class is a union of one or two conjugacy classes. Moreover, we show that the above conditions can be determined through the character table of the group.

Let G be a finite group and let N be a normal subgroup of G. We attach to N two graphs Γ G (N ) and Γ * G (N ) related to the conjugacy classes of G contained in N and to the set of primes dividing the sizes of these classes, respectively. These graphs are subgraphs of the ordinary ones associated to the conjugacy classes of G, Γ(G) and Γ * (G), which have been widely studied by several authors. We prove that the number of connected components of both graphs is at most 2, we determine the best upper bounds for the diameters and characterize the structure of N when these graphs are disconnected.

We prove that if a finite group G contains a conjugacy class K whose square is of the form 1 ∪ D, where D is a conjugacy class of G, then K is a solvable proper normal subgroup of G and we completely determine its structure. We also obtain the structure of those groups in which the assumption above is true for all conjugacy classes and when every conjugacy class satisfies that its square is the union of all central conjugacy classes except at most one.

Suppose that G is a finite group and K a non-trivial conjugacy class of G such that KK −1 = 1 ∪ D ∪ D −1 with D a conjugacy class of G. We prove that G is not a non-abelian simple group. We also give arithmetical conditions on the class sizes determining the structure of K and D . Furthermore, if D = K is a non-real class, then K is p-elementary abelian for some odd prime p.