# Abelian implies universal power map is endomorphism

From Groupprops

(Redirected from Abelian implies n-abelian for all n)

## Statement

Let be an Abelian group, and be an integer. The map (i.e., the map done times when is positive and done times when is negative) is an endomorphism of .

## Proof

**Given**: An Abelian group , an integer .

**To prove**: The map is an endomorphism of : in other words, .

**Proof**: The proof basically follows from commutativity.

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