External Direct Product of Abelian Groups is Abelian Group

Theorem
Let $\left({G, \circ_1}\right)$ and $\left({H, \circ_2}\right)$ be groups.

Then the group direct product $\left({G \times H, \circ}\right)$ is abelian both $\left({G, \circ_1}\right)$ and $\left({H, \circ_2}\right)$ are abelian.

Proof
Let $\left({G, \circ_1}\right)$ and $\left({H, \circ_2}\right)$ be groups whose identities are $e_G$ and $e_H$ respectively.

Suppose $\left({G, \circ_1}\right)$ and $\left({H, \circ_2}\right)$ are both abelian.

Then from External Direct Product Commutativity, $\left({G \times H, \circ}\right)$ is also abelian.

Now suppose that $\left({G \times H, \circ}\right)$ is abelian.

Then:

Thus:
 * $g_1 \circ_1 g_2 = g_2 \circ_1 g_1$

and $\left({G, \circ_1}\right)$ is seen to be abelian.

The same argument holds for $\left({H, \circ_2}\right)$.

Also see

 * External Direct Product of Groups is Group