Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $G$ be a group and $H$ be a subgroup of $G$. When is $\rm{Aut}(H)$ a subgroup of $\rm{Aut}(G)$?

share|improve this question
9  
Generally speaking a question of this type has a shot at a reasonable answer if there is a canonical map $\text{Aut}(H) \to \text{Aut}(G)$ and you want to ask when this map is injective. But in this case there is no such map. Instead there is a canonical map the other way from the subgroup of automorphisms of $G$ preserving $H$ to the group of automorphisms of $H$. So it's not clear to me what you want: do you just want to know when there exists, abstractly, an injection of $\text{Aut}(H)$ into $\text{Aut}(G)$? Do you want this injection to land only in the subgroup of $\text{Aut}(G)$... –  Qiaochu Yuan Jan 20 '12 at 6:30
5  
...which preserves $H$ and to respect the action on $H$? (That would require that you could somehow extend automorphisms of $H$ to automorphisms of $G$ in some reasonable way.) There are lots of things you could want and I doubt there is an easy answer to any of the possible forms of this question. –  Qiaochu Yuan Jan 20 '12 at 6:31
2  
@Ali Gholamian: Please consider registering in the site; that will make it easier to keep track of your questions. –  Arturo Magidin Jan 20 '12 at 16:17
    
@Ali Gholamian: As above, there is a lot of thing this qestion brings here. You certainly know that $\mathbb{Z}_{6}$ and $\mathbb{Z}_{3}$ know just one group as their Automorphism Group. –  Babak S. Jan 21 '12 at 15:14
    
This question on Mathoverflow is related: mathoverflow.net/questions/9749/… –  Bryan Jul 18 at 23:18

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.