Inner Automorphisms form Normal Subgroup of Automorphism Group

Theorem
Let $$G$$ be a group.

Then the set $$\mathrm {Inn} \left({G}\right)$$ of all inner automorphisms of $$G$$ is a normal subgroup of the group of all automorphisms $$\mathrm{Aut} \left({G}\right)$$ of $$G$$:

$$\mathrm {Inn} \left({G}\right) \triangleleft \mathrm{Aut} \left({G}\right)$$