Universal power endomorphism

Definition
An endomorphism of a group is termed a universal power endomorphism if there exists an integer $$n$$ such that the endomorphism can be expressed as $$g \mapsto g^n$$ for all $$g$$ in the group.

Note that for $$n = 0,1$$, the $$n^{th}$$ power map is a universal power endomorphism. For other $$n$$, it is a universal power endomorphism if the group is abelian, otherwise it need not be.

If, for a particular group $$G$$, the $$n^{th}$$ power map is an endomorphism, then we may say that $$G$$ is a $$n$$-abelian group (see n-abelian group).

Facts
We say that a group is a n-abelian group if the $$n^{th}$$ power map is an endomorphism. Here are some related facts about $$n$$-abelian groups.