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 $G$.

Also known as
When the elements of $\struct {G, \circ}$ are numbers, and $\circ$ is (or derives from) addition, the product inverse is usually called subtraction.

Similarly, if $\circ$ is (or derives from) numerical multiplication, the product inverse is usually called division.

However, no general term has been uncovered in the literature designed to encompass the arbitrary abstract group concept.

Hence the term product inverse operation.

Also see

 * Definition:B-Algebra