# center automorphisms are all inner automorphisms?

Let $G$ be a group, $Z(G)$ be the center of $G$, $\theta: G\rightarrow G$ be an automorphism of $G$. $\theta$ is called center automorphism if $g^{-1}\cdot\theta(g)\in Z(G)$, $\forall g\in G$. I am having trouble to prove that each such $\theta$ is an inner automorphism of $G$. I can show that $\theta$ is center automorphism if and only if it is commutative with any inner automorphism of $G$. This suggests that center automorphisms are contained in the center of $Inn(G)$, but I have no way to prove that. Can anyone give me a hint? Thanks.

• I just realized that the statement is definitely not true for Abelian group since all inner automorphisms are trivial and all automorphisms are center automorphisms. How about non-abelian groups? – Roy Z Nov 22 '16 at 14:31
• Central automorphisms are precisely those which commute with all inner automorphisms. They are usually not inner. – Mariano Suárez-Álvarez Nov 22 '16 at 14:36
• I quite agree with you now. – Roy Z Nov 22 '16 at 15:21
• Well, that was not an opinion, really :-| – Mariano Suárez-Álvarez Nov 22 '16 at 16:44