On a special normal subgroup of a group Let $G$ be a group such  $H$ is a normal subgroup of $G$ and $Z(H)=1$ and $Inn(H)=Aut(H)$. Then prove there exists a normal subgroup $K$ of $G$ such that $G=H\times K$.

First, define the map $\psi: G \to Inn(G)$ by $\psi_g(x) = gxg^{-1}$. Since $H$ is a normal subgroup, $\psi_g(H) = H$, so $Inn(G) = Aut(H) = Inn(H)$. From $Z(H) = 1$, we know that $Inn(H) \cong H$. Compose this isomorphism with $\psi$ to get a surjective map $G \to H$. The kernel of this map is $K$.
