Prove that $\operatorname{Inn}(\operatorname{Aut}(G)) \cong \operatorname{Aut}(G)$ Suppose that $G$ is a group with trivial center. Prove that: $$\operatorname{Inn}(\operatorname{Aut}(G)) \cong \operatorname{Aut}(G)$$

**Hint:** Show that the map from $a$ to $f_a(x)=axa^{-1}$ from Aut$(G)$ to Inn$(\operatorname{Aut}(G))$ is an isomorphism