2
$\begingroup$

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.

  • 2
    Is $\alpha$ supposed to be an *inner* automorphism? Do you ask about $\forall g\exists \alpha$ or $\exists\alpha\forall g$?2012-11-12
  • 0
    @HagenvonEitzen: By considration in question $\alpha$ can not be inner automorphism.2012-11-12
  • 0
    It depends on $G$.2012-11-12
  • 0
    @DerekHolt: Thank you. Please explain more for me.2012-11-12
  • 0
    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 $g \in G \setminus Z(G)$.2012-11-12

0 Answers 0