Automorphism of the subgroup

Asked Apr 24 '16 at 18:11

Active Apr 24 '16 at 18:56

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Let $G$ a finite group and $H<G$. Let $\varphi:H\rightarrow H$ be a nontrivial automorphism of $H$. My question is: it is possible to construct a nontrivial automorphism of $G$?

I had tried to extend the isomorphism of H, leaving invariant the elements of G that were not in H. But you can take the next case: if $a,b\in G$ such that $a,b\notin H$ and $ab\in H$, so $\varphi(ab)\in H$ but not necessarily $\varphi(ab)=ab$. So my idea is wrong.

I must demonstrate this using the non-trivial automorphism of $H$.

edited Apr 24 '16 at 18:56

Paolo

asked Apr 24 '16 at 18:11

Peter G

Do you want to show that there exists a nontrivial automorphism (in this case see the link above) or that $\varphi$ can be extended to $G$ ? – Captain Lama Apr 24 '16 at 18:15
I want to build an automorphism based – Peter G Apr 24 '16 at 18:20
It is a strange question, because you assume the existence of an automorphism $\varphi$ of $H$, but then you ask whether there exists an automorphism of $G$ that apparently has nothing to do with $\varphi$. But in fact the existence of $\varphi$ implies that $|H|>2$ and hence $|G|>2$, so the answer to the question is yes. – Derek Holt Apr 24 '16 at 18:51
I think they want an automorphism $\phi^*$ of $G$ such that its restriction to $H$ is $\phi$. – Apr 24 '16 at 18:57
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If the question is about whether all automorphisms of $H$ are restrictions of automorphisms of $G$, then see Are there automorphisms of $H$ which are not restriction of an automorphism of $G$? – Gregory Simon Apr 24 '16 at 19:00
Oh, I see now @GregorySimon. Thank you. – Peter G Apr 24 '16 at 19:03

Automorphism of the subgroup

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