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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 $G$.

  • 0
    to use the definition...2012-10-16

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