Definition:Inversion Mapping

Definition

Let $\struct {G, \circ}$ be a group.

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

$\forall g \in G: \map \iota g = g^{-1}$

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

Topological Group

Let $T = \struct {G, \circ, \tau}$ be a topological group.

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

$\forall x \in G: \map \iota x = x^{-1}$

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

Also known as

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

Also see

• Definition:Group

• Inversion Mapping is Mapping

• Results about inversion mappings can be found here.

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Morphisms