Kernel of Inner Automorphism Group is Center

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


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

Proof
Let $\kappa: G \to \operatorname{Aut} \left({G}\right)$ be a mapping defined by $\kappa \left({x}\right) = \kappa_x$.

It is clear that $\operatorname{Im} \left({\kappa}\right) = \operatorname{Inn} \left({G}\right)$.

It is also clear that $\kappa$ is a homomorphism:

Note that $\forall \kappa_x \in \operatorname {Inn} \left({G}\right): \exists x \in G: \kappa \left({x}\right) = \kappa_x$.

Thus $\kappa: G \to \operatorname {Inn} \left({G}\right)$ is a surjection and therefore an group epimorphism.


 * Now we investigate the kernel of $\kappa$:

So the kernel of $\kappa$ is the center of $G$.