Inversion Mapping is Automorphism iff Group is Abelian

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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.


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.


\(\displaystyle \forall x, y \in G: \map \iota {x \circ y}\) \(=\) \(\displaystyle \map \iota x \circ \map \iota y\) 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.


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.



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