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 $\struct {G_1, \circ_1}, \struct {G_2, \circ_2}, \ldots, \struct {G_n, \circ_n}$ be abelian groups.

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

From External Direct Product of Groups is Group: Finite Product it follows that $\struct {G, \circ}$ 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, $\struct {G, \circ}$ is an abelian group.

$\blacksquare$