Structure Induced by Group Operation is Group

Theorem
Let $\struct {G, \circ}$ be a 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 a group.

Proof
Taking the group axioms in turn:

$\text G 0$: Closure
Let $f, g \in G^S$.

As $\struct {G, \circ}$ is a group, it is closed by group axiom $G0$.

From Closure of Pointwise Operation on Algebraic Structure it follows that $\struct {G^S, \oplus}$ is likewise closed.

$\text G 1$: Associativity
As $\struct {G, \circ}$ is a group, $\circ$ is associative.

So from Structure Induced by Associative Operation is Associative, $\struct {G^S, \oplus}$ is also associative.

$\text G 2$: Identity
We have from Induced Structure Identity that the constant mapping $f_e: S \to T$ defined as:


 * $\forall x \in S: \map {f_e} x = e$

is the identity for $\struct {G^S, \oplus}$.

$\text G 3$: Inverses
Let $f \in G^S$.

Let $f^* \in G^S$ be defined as follows:


 * $\forall f \in G^S: \forall x \in S: \map {f^*} x = \paren {\map f x}^{-1}$

From Induced Structure Inverse, $f^*$ is the inverse of $f$ for the pointwise operation $\oplus$ induced on $G^S$ by $\circ$.

All the group axioms are thus seen to be fulfilled, and so $\struct {G^S, \oplus}$ is a group.