Paper 3, Section II, D

Groups | Part IA, 2016

Define the Möbius group M\mathcal{M} and its action on the Riemann sphere C\mathbb{C}_{\infty}. [You are not required to verify the group axioms.] Show that there is a surjective group homomorphism ϕ:SL2(C)M\phi: S L_{2}(\mathbb{C}) \rightarrow \mathcal{M}, and find the kernel of ϕ.\phi .

Show that if a non-trivial element of M\mathcal{M} has finite order, then it fixes precisely two points in C\mathbb{C}_{\infty}. Hence show that any finite abelian subgroup of M\mathcal{M} is either cyclic or isomorphic to C2×C2C_{2} \times C_{2}.

[You may use standard properties of the Möbius group, provided that you state them clearly.]

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