Show that every Möbius transformation can be expressed as a composition of maps of the forms: $S_{1}(z)=z+\alpha, S_{2}(z)=\lambda z$ and $S_{3}(z)=1 / z$, where $\alpha, \lambda \in \mathbb{C}$.

Show that if $z_{1}, z_{2}, z_{3}$ and $w_{1}, w_{2}, w_{3}$ are two triples of distinct points in $\mathbb{C} \cup\{\infty\}$, there exists a unique Möbius transformation that takes $z_{j}$ to $w_{j}(j=1,2,3)$.

Let $G$ be the group of those Möbius transformations which map the set $\{0,1, \infty\}$ to itself. Find all the elements of $G$. To which standard group is $G$ isomorphic?