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Let $n\ne 0$, $1$ be an integer. A group $G$ is said to be $n$-abelian if the mapping $f_n:x\to x^n$ 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 $G$.

A subgroup H of a group G is inert if |H: H ∩ H g| is finite for all g ∈ G and a group G is totally inert if every subgroup H of G is inert. We investigate the structure of minimal normal subgroups of totally inert groups and show that infinite locally graded simple groups cannot be totally inert.