Definition:Product Inverse Operation

Definition
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 $G$.