\begin{definition}[Definition:Product Inverse Operation]
Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $\oplus: G \times G \to G$ be the operation on $G$ defined as:
:$\forall a, b \in G: a \oplus b := a \circ b^{-1}$
where $b^{-1}$ denotes the inverse of $b$ in $G$.
Then $\oplus$ is the '''product inverse (of $\circ$)''' on $G$.
\end{definition}