Criteria for groups to be abelian

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

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

  1. G is abelian,
  2. The map G \to G given by g \mapsto g^2 is a group homomorphism;
  3. The map G \to G given by g \mapsto g^{-1} is a group homomorphism;
  4. The diagonal G \subseteq G \times G is normal.

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

For (2), note that (gh)^2 = ghgh, which equals gghh if and only if gh = hg.

For (3), note that (gh)^{-1} = h^{-1}g^{-1}, which equals g^{-1}h^{-1} if and only if gh = hg.

For (4), clearly \Delta_G \colon G \hookrightarrow G \times G is normal if G is abelian. Conversely, note that (e,h)(g,g)(e,h^{-1}) = (g,hgh^{-1}), which is in the diagonal if and only if gh = hg. \qedsymbol

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