# Structure Induced by Abelian Group Operation is Abelian Group

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## Theorem

Let $\struct {G, \circ}$ be an abelian group whose identity is $e$.

Let $S$ be a set.

Let $\struct {G^S, \oplus}$ be the structure on $G^S$ induced by $\circ$.

Then $\struct {G^S, \oplus}$ is an abelian group.

## Proof

From Structure Induced by Group Operation is Group, $\struct {G^S, \oplus}$ is a group.

From Structure Induced by Commutative Operation is Commutative, so is the pointwise operation it induces on $G^S$.

Hence the result.

$\blacksquare$