Square map is endomorphism iff abelian

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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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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.


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


xyxy = x^2y^2

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

yx = xy

and hence x,y commute.