Inner Automorphism Group is Isomorphic to Quotient Group with Center

Theorem
Let $$G$$ be a group.

Let $$\mathrm {Inn} \left({G}\right)$$ be the group of inner automorphisms of $$G$$.

Let $$Z \left({G}\right)$$ be the center of $$G$$.

Let $$G / Z \left({G}\right)$$ be the Quotient Group of $$G$$ by $$Z \left({G}\right)$$.

Then $$G / Z \left({G}\right) \cong \mathrm {Inn} \left({G}\right)$$.

Proof
Let $$G$$ be a group.

Let the mapping $$\kappa: G \to \mathrm{Inn} \left({G}\right)$$ be defined as:

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

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

From Kernel of Inner Automorphisms is Center, we have that $$\mathrm {ker} \left({\kappa}\right) = Z \left({G}\right)$$ and also that $$\mathrm {Im} \left({\kappa}\right) = \mathrm {Inn} \left({G}\right)$$.

From the First Isomorphism Theorem, $$\mathrm {Im} \left({\kappa}\right) \cong G / \mathrm {ker} \left({\kappa}\right)$$.

Thus as $$\mathrm {ker} \left({\kappa}\right) = Z \left({G}\right)$$ and $$\mathrm {Im} \left({\kappa}\right) = \mathrm {Inn} \left({G}\right)$$:

$$G / Z \left({G}\right) \cong \mathrm {Inn} \left({G}\right)$$.