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.
$\blacksquare$