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.

I would like to know all finite groups of $\operatorname{Aut}(\mathbb{P}^1)$.

I am aware of that any automorphism of $\mathbb{P}^1$ is given by Möbius transformation $$ z\mapsto\frac{az+b}{cz+d} $$ and thus there is an identification $$ \operatorname{Aut}(\mathbb{P}^1)\cong \operatorname{PSL}(2,\mathbb{C})\cong \operatorname{SO}(3, \mathbb{C}). $$ I thought this solved the question, but what I know is the classification of finite groups of real orthogonal group $\operatorname{SO}(3, \mathbb{R})$.

share|improve this question
1  
I think the third groupe of your line of isomorphism is in fact $SO(3,\mathbb{R})$, not $SO(3,\mathbb{C})$ and hence your problem is solved (you can check that $SO(3,\mathbb{C})$ doesn't have the good dimension as a real manifold if you're not covinced...). –  Simon Henry Oct 13 '12 at 11:22
add comment

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.