Inner automorphisms and the center of a group

#group_theory

Let $G$ be a group. For each $a \in G$ , let $γ_{a}$ denote the automorphism of $G$ defined by $γ_{a} (b) = a b a^{- 1}$ for all $b \in G$ . The set $Inn (G) = {γ_{a} : a \in G}$ is a subgroup of the automorphism group of $G$ , called the subgroup of inner automorphisms.

Prove that $Inn (G)$ is isomorphic to $G / Z (G)$ , where $Z (G)$ is the center of $G$ .

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Let $G$ be a group. For each $a\in G$, let $\gamma_a$ denote the automorphism of $G$ defined by $\gamma_a(b)=aba^{-1}$ for all $b\in G$. The set $\operatorname{Inn}(G)=\{\gamma_a:a\in G\}$ is a subgroup of the automorphism group of $G$, called the subgroup of {\bfseries inner automorphisms}.

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Prove that $\operatorname{Inn}(G)$ is isomorphic to $G/Z(G)$, where $Z(G)$ is the center of $G$.