Define the Möbius group, and describe how it acts on $\mathbb{C} \cup\{\infty\}$.

Show that the subgroup of the Möbius group consisting of transformations which fix 0 and $\infty$ is isomorphic to $\mathbb{C}^{*}=\mathbb{C} \backslash\{0\}$.

Now show that the subgroup of the Möbius group consisting of transformations which fix 0 and 1 is also isomorphic to $\mathbb{C}^{*}$.