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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 $G$.

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to use the definition... – Jarbas Dantas Silva Oct 16 '12 at 22:28

1 Answer 1

Let $\phi\in Aut_C(G)$ and $\psi\in Aut(G)$. Since $\psi(a)\in G$, $\left(\psi(a)\right)^{-1}\phi(\psi(a))=\psi(a^{-1})(\phi\psi)(a)\in Z(G)$. Thus $$\psi^{-1}\left(\psi(a^{-1})(\phi\psi)(a)\right)\in \psi^{-1}(Z(G))$$ $$(\psi^{-1}\psi)(a^{-1})(\psi^{-1}\phi\psi)(a)\in \psi^{-1}(Z(G))$$ $$a^{-1}(\psi^{-1}\phi\psi)(a)\in \psi^{-1}(Z(G))$$

$Z(G)$ is characteristic in $G$ so $\psi^{-1}(Z(G))=Z(G)$, whence $$a^{-1}(\psi^{-1}\phi\psi)(a)\in Z(G)$$ so $\psi^{-1}\phi\psi \in Aut_C(G)$.

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