\begin{definition}[Definition:Pointwise Inverse]
Let $\struct {G, \oplus}$ be a group whose identity is $e_G$.
Let $S$ be a set.
Let $\struct {G^S, \oplus}$ be the structure on $G^S$ induced by $\oplus$.
Let $f \in G^S$.
The '''pointwise inverse''' of $f$ in $G^S$ is denoted $f^*$ and defined as:
:$\forall x \in S: \map {f^*} x = \paren {\map f x}^{-1}$
\end{definition}