Inversion Mapping is Permutation/Proof 1

Theorem

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

Let $\iota: G \to G$ be the inversion mapping on $G$.

Then $\iota$ is a permutation on $G$.

Proof

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

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

where $g^{-1}$ is the inverse or $g$.

By Inversion Mapping is Involution, $\iota$ is an involution:

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

The result follows from Involution is Permutation.

$\blacksquare$