# 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 $\mathsf{Pr} \infty \mathsf{fWiki}$ term product inverse operation.

## Also see

• Results about the product inverse operation can be found here.

## Historical Note

Seth Warner, in his Modern Algebra of $1965$, discusses this operation briefly in an exercise, but fails to give a name to it.

However, he helpfully provides the notation $\div$ for us, in order to guide us down the channels we acquired in elementary school.

## Linguistic Note

The term product inverse operation was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$.

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.