Kernel of Inner Automorphism Group is Center

Theorem
Let the mapping $$\kappa: G \to \mathrm{Inn} \left({G}\right)$$ from a group $$G$$ to its group of inner automorphisms $$\mathrm{Inn} \left({G}\right)$$ be defined as:

$$\kappa \left({a}\right) = \kappa_a$$

...where $$\kappa_a$$ is the inner automorphism of $$G$$ given by $$a$$.

Then $$\kappa$$ is a group epimorphism, and its kernel is the center of $$G$$:

$$\mathrm {ker} \left({\kappa}\right) = Z \left({G}\right)$$