Structure Induced by Abelian Group Operation is Abelian Group

Theorem
Let $\left({G, \circ}\right)$ be an abelian group whose identity is $e$.

Let $S$ be a set.

Let $\left({G^S, \oplus}\right)$ be the structure on $G^S$ induced by $\circ$.

Then $\left({G^S, \oplus}\right)$ is an abelian group.

Proof
From Structure Induced by Group Operation is Group, $\left({G^S, \oplus}\right)$ is a group.

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

Hence the result.