Inversion Mapping is Involution
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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.
$\blacksquare$