On n-Centralizer CA-Groups

Let G be a nite non-abelian group and m = jGj jZ(G)j . In this paper we prove that if G is a nite non-abelian m-centralizer CA-group, then there exists an integer 1 r such that m = 2r. It is also prove that if jG′j = 2, then G is an m-centralizer group.