Definition:Inner Automorphism

Theorem
Let $$G$$ be a group.

Let $$x \in G$$.

Let the mapping $$\kappa_x: G \to G$$ be defined such that $$\forall g \in G: \kappa_x \left({g}\right) = x g x^{-1}$$.

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

$$\kappa_x$$ is called the inner automorphism of $$G$$ given by $$x$$.

The set of all inner automorphisms of $$G$$ is denoted $$\mathrm {Inn} \left({G}\right)$$.

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


 * First we show $$\kappa_x$$ is a homomorphism.

$$ $$ $$

Thus the morphism property is demonstrated.


 * Next we show that $$\kappa_x$$ is injective.

$$ $$ $$

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:

$$ $$

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.

Comment
Note that performing an inner automorphism of a subgroup results in a conjugate subgroup.