# Inversion Mapping is Automorphism iff Group is Abelian

## Theorem

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

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

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

Then $\iota$ is an automorphism if and only if $G$ is abelian.

## Proof

From Inversion Mapping is Permutation, $\iota$ is a permutation.

It remains to be shown that $\iota$ has the morphism property if and only if $G$ is abelian.

### Sufficient Condition

Suppose $\iota$ is an automorphism.

Then:

 $\displaystyle \forall x, y \in G: \map \iota {x \circ y}$ $=$ $\displaystyle \map \iota x \circ \map \iota y$ Definition of Morphism Property $\displaystyle \leadsto \ \$ $\displaystyle \paren {x \circ y}^{-1}$ $=$ $\displaystyle x^{-1} \circ y^{-1}$ Definition of $\iota$

Thus from Inverse of Commuting Pair, $x$ commutes with $y$.

This holds for all $x, y \in G$.

So $\struct {G, \circ}$ is abelian by definition.

$\Box$

### Necessary Condition

Let $\struct {G, \circ}$ be abelian.

 $\displaystyle \forall x, y \in G: \paren {x \circ y}^{-1}$ $=$ $\displaystyle x^{-1} \circ y^{-1}$ Inverse of Commuting Pair $\displaystyle \leadsto \ \$ $\displaystyle \map \iota {x \circ y}$ $=$ $\displaystyle \map \iota x \circ \map \iota y$ Definition of $\iota$

Thus $\iota$ has the morphism property and is therefore an automorphism.

$\blacksquare$

## Sources

except this source requests only that the morphism property is demonstrated, and not the bijectivity.
except this source proves only the necessary condition.