Let $G$ a group. Show that the set:
$Aut_C(G)=\{ \phi \in Aut(G) : a^{-1}\phi(a) \in Z(G), \ \forall a \in G \}$
is a normal subgroup in $Aut (G)$. Particularly, if $Z(G)=\{e\}$, then $Aut_C(G)=\{I\}$.
Note: $Z(G)$ is the center of $G$; $Aut (G)$ is the set of automorphisms of $G$.