External Direct Product of Abelian Groups is Abelian Group

Theorem
Let $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ be groups.

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

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

From External Direct Product of Groups is Group, $\struct {G \times H, \circ}$ is indeed a group whose identity is $\tuple {e_G, e_H}$.

Suppose $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ are both abelian.

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

Now suppose that $\struct {G \times H, \circ}$ is abelian.

Then:

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

and $\struct {G, \circ_1}$ is seen to be abelian.

The same argument holds for $\struct {H, \circ_2}$.

Also see

 * External Direct Product of Groups is Group