Let $n \neq 0$, $1$ be an integer. A group $G$ is said to be $n$-abelian if the mapping $f_{n} \colon x \to x^{n}$ is an endomorphism of $G$. Then $(xy)^{n} = x^{n}y^{n}$ for all $x$, $y \in G$, from which it follows $[x^{n}, y] = [x, y]^{n} = [x; y^{n}]$. In this paper we investigate groups $G$ such that $f_{n}$ is a monomorphism or an epimorphism of $G$. We also deal with the connections between $n$-abelian groups and groups satisfying the identity $[x^{n}, y] = [x, y]^{n} = [x; y^{n}]$. Finally, we provide an arithmetic description of the set of all integers $n$ such that $f_{n}$ is an automorphism of a given group $G$.