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

I am aware that any automorphism of $\mathbb{P}^1$ is given by a 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 subgroups of real orthogonal group $\operatorname{SO}(3, \mathbb{R})$.

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    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...).2012-10-13

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