# 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 if and only if both $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ are abelian.

### General Result

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

## 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:

\(\ds \tuple {g_1 \circ_1 g_2, e_H}\) | \(=\) | \(\ds \tuple {g_1 \circ_1 g_2, e_H \circ_2 e_H}\) | Definition of $e_H$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \tuple {g_1, e_H} \circ \tuple {g_2, e_H}\) | Definition of Group Direct Product | |||||||||||

\(\ds \) | \(=\) | \(\ds \tuple {g_2, e_H} \circ \tuple {g_1, e_H}\) | as $\struct {G \times H, \circ}$ is abelian | |||||||||||

\(\ds \) | \(=\) | \(\ds \tuple {g_2 \circ_1 g_1, e_H \circ_2 e_H}\) | Definition of Group Direct Product | |||||||||||

\(\ds \) | \(=\) | \(\ds \tuple {g_2 \circ_1 g_1, e_H}\) | Definition of $e_H$ |

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 $\struct {H, \circ_2}$.

$\blacksquare$

## Also see

## Sources

- 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $6$: An Introduction to Groups: Exercise $1$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $1$: Definitions and Examples: Exercise $6$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $13$: Direct products: Proposition $13.1 \ (1)$