$G/Z(G)$ is isomorphic to $\text{Inn}(G)$

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Problem ($G/Z(G)$ is isomorphic to $\text{Inn}(G)$)

Suppose that $G$ is a group. Let $f:G \to \text{Inn}(G)$ be defined by $f(x)=\phi_x$.

1. Show that $f$ is a homomorphism with kernel $Z(G)$.
2. Then prove that $G/Z(G)$ is isomorphic to $\text{Inn}(G)$.
3. Compute $\text{Inn}(G)$ for any Abelian group $G$ and then for $G=D_8$.
4. Why is $D_{10}$ isomorphic to $\text{Inn}(D_{10})$?

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