Inversion Mapping is Involution

Theorem

Let $G$ be a group, and let $\iota: G \to G$ be the inversion mapping.

Then $\iota$ is an involution.

That is:

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

Proof

Let $g \in G$.

Then:

\(\ds \map \iota {\map \iota g}\)

\(=\)

\(\ds \paren {g^{-1} }^{-1}\)

Definition of Inversion Mapping

\(\ds \)

\(=\)

\(\ds g\)

Inverse of Group Inverse

which establishes the result.

$\blacksquare$