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