List all the elements of the the subgroup $M_{\{0, 1, \infty\}}$ of the group of Möbius transformations, preserving the set $\{0, 1, \infty\}$ and give an explicit isomorphism $M_{\{0, 1, \infty\}} = S_3$. Using this, list all the elements of the subgroup $M_{\{0, 1 + i, \infty\}}$ of Möbius transformations preserving the set $\{0, 1 + i, \infty\}$, together with an explicit isomorphism.
I have done the beginning bit I am unsure how to map $M_{\{0, 1, \infty\}}$ and $M_{\{0, 1 + i, \infty\}}$ to each other.
There is a unique Möbius transformation $g$ which sends $0$ to $a$, $1$ to $b$, and $\infty$ to $c$ (provided $a$, $b$, and $c$ are distinct). Given $f \in M_{\{0, 1, \infty\}}$, note that $g\circ f\circ g^{-1} \in M_{\{a, b, c\}}$. So consider the map
\begin{align*} \Phi: M_{\{0, 1, \infty\}} &\to M_{\{a, b, c\}}\\ f &\mapsto g\circ f \circ g^{-1}. \end{align*}
You should show that $\Phi$ is a bijection. Also, depending on what you've seen, you may need to justify the existence of $g$.