# Definition:Inversion Mapping

## Definition

Let $\left({G, \circ}\right)$ be a group.

The inversion mapping on $G$ is the mapping $\iota: G \to G$ defined by:

$\forall g \in G: \iota \left({g}\right) = g^{-1}$

That is, $\iota$ assigns to an element of $G$ its inverse.

### Topological Group

Let $T = \left({G, \circ, \tau}\right)$ be a topological group.

Let $\phi: G \to G$ be the mapping defined as:

$\forall x \in G: \phi \left({x}\right) = x^{-1}$

Then $\phi$ is the inversion mapping of $T$.

## Also known as

Other notations for $\iota$ are $i$ and $(-)^{-1}$.