# On group such that the group of inner automorphisms of it is isomorphic to $S_{3}$.

Let $\frac{G}{Z(G)}\cong S_{3}$, such that $S_{3}$ is permutation group on 3 letters and $Z(G)$ is non trivial central subgroup of $G$. Then does there exist an automorphism $\alpha$ of $G$ such that $\alpha(g)\neq g$ for $g\in G-Z(G)$? Thanks.

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Is $\alpha$ supposed to be an inner automorphism? Do you ask about $\forall g\exists \alpha$ or $\exists\alpha\forall g$? – Hagen von Eitzen Nov 12 '12 at 7:22
@HagenvonEitzen: By considration in question $\alpha$ can not be inner automorphism. – elham Nov 12 '12 at 7:58
It depends on $G$. – Derek Holt Nov 12 '12 at 9:26
@DerekHolt: Thank you. Please explain more for me. – user49122 Nov 12 '12 at 10:01
I mean there exist examples of groups $G$ with $G/Z(G) \cong S_3$ for which there exists $\alpha \in {\rm Aut}(G)$ with $\alpha(g) \ne g$ for all $g \in G \setminus Z(G)$, and there also exist examples of groups $G$ with $G/Z(G) \cong S_3$ for which there does NOT exist $\alpha \in {\rm Aut}(G)$ with $\alpha(g) \ne g$ for all $g \in G \setminus Z(G)$. – Derek Holt Nov 12 '12 at 11:46