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 $$H$$.