# Square map is endomorphism iff abelian

From Groupprops

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 be a group and be the map defined as . Then, is an endomorphism if and only if is Abelian.

## Proof

### From endomorphism to Abelian

Suppose is an endomorphism of the group . Then for any we want to show that and commute. This can be proved as follows:

becaus is an endomorphism

Thus:

Cancelling the leftmost and the rightmost , we get:

and hence commute.