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 $$G$$:

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

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

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

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

$$ $$ $$

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

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


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

$$ $$ $$ $$ $$

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