This is a review of some elementary criteria for a group to be abelian.

**Lemma.** *Let be a group. Then the following are equivalent:*

- is abelian,
- The map given by is a group homomorphism;
- The map given by is a group homomorphism;
- The diagonal is normal.

*Proof.* We prove that each criterion is equivalent to (1).

For (2), note that , which equals if and only if .

For (3), note that , which equals if and only if .

For (4), clearly is normal if is abelian. Conversely, note that , which is in the diagonal if and only if .