Kernel of Inner Automorphism Group is Center

Theorem
Let the mapping $\kappa: G \to \Inn G$ from a group $G$ to its inner automorphism group $\Inn G$ be defined as:
 * $\map \kappa a = \kappa_a$

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 $G$:


 * $\map \ker \kappa = \map Z G$

Proof
Let $\kappa: G \to \Aut G$ be a mapping defined by $\map \kappa x = \kappa_x$.

It is clear that $\Img \kappa = \Inn G$.

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

Note that $\forall \kappa_x \in \Inn G: \exists x \in G: \map \kappa x = \kappa_x$.

Thus $\kappa: G \to \Inn G$ is a surjection and therefore an group epimorphism.

Now we investigate the kernel of $\kappa$:

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