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 .

The map given by is a group homomorphism for .

Fascinating; I have never heard of this. Are you saying that this is true for any group ? Do you know where I can read more about this?

Hello, Remy ðŸ™‚

Yes, I meant that any group such that the three maps , , are three group endomorphisms must be an abelian group.

You can find conditions of this type among the exercises in Herstein’s algebra textbook (which is now a bit too old-fashioned, but contains a fantastic wealth of good exercises). For example you can prove that if is an endomorphism for three consecutive values of , then must be abelian; or that if and are both endomorphisms, then must be abelian.

I liked these exercises a lot as a student, so I was very pleased to see the topic addressed fully in this nice elementary paper:

Abelian Forcing Sets, Joseph A. Gallian and Michael Reid, The American Mathematical Monthly, Vol. 100, No. 6 (Jun. – Jul., 1993), pp. 580-582