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 $13.1 \ (1)$