Let $G$ be a group, $X$ a set on which $G$ acts transitively, $B$ the stabilizer of a point $x \in X$.

Show that if $g \in G$ stabilizes the point $y \in X$, then there exists an $h \in G$ with $h g h^{-1} \in B$.

Let $G=S L_{2}(\mathbb{C})$, acting on $\mathbb{C} \cup\{\infty\}$ by Möbius transformations. Compute $B=G_{\infty}$, the stabilizer of $\infty$. Given

$$g=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \in G$$

compute the set of fixed points $\{x \in \mathbb{C} \cup\{\infty\} \mid g x=x\} .$

Show that every element of $G$ is conjugate to an element of $B$.