# Inner Automorphism is Automorphism

## Theorem

Let $G$ be a group.

Let $x \in G$.

Let $\kappa_x$ be the inner automorphism of $x$ in $G$.

Then $\kappa_x$ is an automorphism of $G$.

## Proof

By definition, $\kappa_x: G \to G$ is a mapping defined as:

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

We need to show that $\kappa_x$ is an automorphism.

First we show $\kappa_x$ is a homomorphism.

 $\ds \forall g, h \in G: \,$ $\ds \map {\kappa_x} g \map {\kappa_x} h$ $=$ $\ds \paren {x g x^{-1} } \paren {x h x^{-1} }$ Definition of $\kappa_x$ $\ds$ $=$ $\ds x g \paren {x^{-1} x} h x^{-1}$ Group Axiom $\text G 1$: Associativity $\ds$ $=$ $\ds x \paren {g e h} x^{-1}$ Group Axiom $\text G 3$: Existence of Inverse Element $\ds$ $=$ $\ds x \paren {g h} x^{-1}$ Group Axiom $\text G 2$: Existence of Identity Element $\ds$ $=$ $\ds \map {\kappa_x} {g h}$ Definition of $\kappa_x$

Thus the morphism property is demonstrated.

Next we show that $\kappa_x$ is injective.

 $\ds \map {\kappa_x} g$ $=$ $\ds \map {\kappa_x} h$ $\ds \leadsto \ \$ $\ds x g x^{-1}$ $=$ $\ds x h x^{-1}$ Definition of $\kappa_x$ $\ds \leadsto \ \$ $\ds g$ $=$ $\ds h$ Cancellation Laws

So $\kappa_x$ is injective.

Finally we show that $\kappa_x$ is surjective.

Note that $\forall h \in G: x^{-1} h x \in G$ from fact that $G$ is a group and therefore closed. So:

 $\ds \forall h \in G: \,$ $\ds \map {\kappa_x} {x^{-1} h x}$ $=$ $\ds x \paren {x^{-1} h x} x^{-1}$ Definition of $\kappa_x$ $\ds$ $=$ $\ds \paren {x x^{-1} } h \paren {x x^{-1} }$ Group Axiom $\text G 1$: Associativity $\ds$ $=$ $\ds e h e$ Group Axiom $\text G 3$: Existence of Inverse Element $\ds$ $=$ $\ds h$ Group Axiom $\text G 2$: Existence of Identity Element

Thus every element of $G$ is the image of some element of $G$ under $\kappa_x$ (that is, of $x^{-1} h x$), and surjectivity is proved.

$\blacksquare$