# Square map is endomorphism iff abelian

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## Contents

This article describes an easy-to-prove fact about basic notions in group theory, that is not very well-known or important in itself
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## Statement

### Verbal statement

The square map on a group, viz the map sending each element to its square, is an endomorphism if and only if the group is Abelian.

### Symbolic statement

Let $G$ be a group and $\sigma:G \to G$ be the map defined as $\sigma(x) = x^2$. Then, $\sigma$ is an endomorphism if and only if $G$ is Abelian.

## Proof

### From endomorphism to Abelian

Suppose $\sigma = x \mapsto x^2$ is an endomorphism of the group $G$. Then for any $x,y \in G$ we want to show that $x$ and $y$ commute. This can be proved as follows:

$\sigma(xy) = \sigma(x)\sigma(y)$ becaus $\sigma$ is an endomorphism

Thus:

$xyxy = x^2y^2$

Cancelling the leftmost $x$ and the rightmost $y$, we get:

$yx = xy$

and hence $x,y$ commute.