On group characterization by numbers of conjugate classes

Abstract

Let c(n,G) be a number of conjugate elements of order n in a group G.In the article we study the problem of recognition of finite group by the set ncl(G) that consists of numbers c(n,G). We prove that Abelian groups can be recognized by the set ncl(G) when the order of the group is known. We also describe some other types of groups that can be recognized. The examples of non-isomorphic groups with the same sets ncl(G) are given. Some theorems about a group recognition by partial conditions on c(n,G). are proved.