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

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

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