Let $G$ be the group of Möbius transformations of $\mathbb{C} \cup\{\infty\}$ and let $X=\{\alpha, \beta, \gamma\}$ be a set of three distinct points in $\mathbb{C} \cup\{\infty\}$.

(i) Show that there exists a $g \in G$ sending $\alpha$ to $0, \beta$ to 1 , and $\gamma$ to $\infty$.

(ii) Hence show that if $H=\{g \in G \mid g X=X\}$, then $H$ is isomorphic to $S_{3}$, the symmetric group on 3 letters.