External Direct Product of Abelian Groups is Abelian Group/General Result

Theorem
The external direct product of a finite sequence of abelian groups is itself an abelian group.

Proof
Let $\left({G_1, \circ_1}\right), \left({G_2, \circ_2}\right), \ldots, \left({G_n, \circ_n}\right)$ be abelian groups.

Let $\displaystyle \left({G, \circ}\right) = \prod_{k \mathop = 1}^n G_k$ be the external direct product of $\left({G_1, \circ_1}\right), \left({G_2, \circ_2}\right), \ldots, \left({G_n, \circ_n}\right)$.

From External Direct Product of Groups is Group: General Result it follows that $\left({G, \circ}\right)$ is a group.

By definition, each of $\circ_1, \circ_2, \ldots, \circ_n$ are commutative operations

From External Direct Product Commutativity: General Result it follows that $\circ$ is commutative.

Hence, by definition, $\left({G, \circ}\right)$ is an abelian group.