Some special classes of n-abelian groups

Let n be an integer. A group G is said to be n-abelian if the map phi_n that sends g to g^n is an endomorphism of G. Then (xy) ^n=x^ny^n for all x،y in G، from which it follows [x^n،y] = [x،y] ^n= [x،y^n]. It is also easy to see that a group G is n-abelian if and only if it is (1-n) -abelian. If nneq 0،1 and G is an n-abelian group، then the quotient group G/Z (G) has finite exponent dividing n (n-1). This implies that every torsion-free n-abelian group is abelian. We denote by B_n and C_n the classes of all groups G for which phi_n is a monomorphism and an epimorphism of G، respectively. Then B_0=C_0 contains only the trivial group، B_1=C_1 is the class of all groups، and B_-1=C_-1 is the class of all abelian groups. Furthermore، with |n|>1، G is in B_n if and only if G is an n-abelian group having no elements of order dividing |n|. Similarly، G is in C_n if and only if G is n-abelian and for every g in G there exists an element x in G such that g=x^n. We also set A_n=B_ncap C_n. In this paper we give a characterization for groups in B_n and for groups in C_n. We also obtain an arithmetic description of the set of all integers n such that a group G is in A_n.