Inner Automorphism Group is Isomorphic to Quotient Group with Center

Theorem
Let $G$ be a group.

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

Proof
Let $G$ be a group.

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


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

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

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

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

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


 * $G / Z \left({G}\right) \cong \operatorname {Inn} \left({G}\right)$