Some special classes of n-abelian groups

Abstract

‎‎Given an integer $n$‎, ‎we denote by $mathfrak B_n$ and $mathfrak C_n$ the classes of all groups $G$ for which the map $phi_{n}:gmapsto g^n$ is a monomorphism and an epimorphism of $G$‎, ‎respectively‎. ‎In this paper we give a characterization for groups in $mathfrak B_n$ and for groups in $mathfrak C_n$‎. ‎We also obtain an arithmetic description of the set of all integers $n$ such that a group $G$ is in $mathfrak B_ncapmathfrak C_n$‎.