External Direct Product of Abelian Groups is Abelian Group

Theorem
The group direct product $G \times H$ is abelian iff both $G$ and $H$ are abelian.

Proof
Let $G$ and $H$ be groups whose identities are $e_G$ and $e_H$ respectively.


 * Suppose $G$ and $H$ are both abelian.

Then from External Direct Product Commutativity, $G \times H$ is also abelian.


 * Now suppose that $G \times H$ is abelian.

Then:


 * $\left({g_1 g_2, e_H}\right) = \left({g_1, e_H}\right) \left({g_2, e_H}\right) = \left({g_2, e_H}\right) \left({g_1, e_H}\right) = \left({g_2 g_1, e_H}\right)$

Thus $g_1 g_2 = g_2 g_1$ and $G$ is seen to be abelian.

A similar argument holds for $H$.