# Universal power endomorphism

This article defines a function property, viz a property of functions from a group to itself

## 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.

Value of $n$ (note that the condition for $n$ is the same as the condition for $1-n$) Characterization of $n$-abelian groups Proof Other related facts
0 all groups obvious
1 all groups obvious
2 abelian groups only 2-abelian iff abelian endomorphism sends more than three-fourths of elements to squares implies abelian
-1 abelian groups only -1-abelian iff abelian
3 3-abelian group means: 2-Engel group and derived subgroup has exponent dividing three Levi's characterization of 3-abelian groups cube map is surjective endomorphism implies abelian, cube map is endomorphism iff abelian (if order is not a multiple of 3), cube map is endomorphism implies class three
-2 same as for 3-abelian (based on n-abelian iff (1-n)-abelian)