Let $G$ be $S L_{2}(\mathbb{R})$, the groups of real $2 \times 2$ matrices of determinant 1 , acting on $\mathbb{C} \cup\{\infty\}$ by Möbius transformations.

For each of the points $0, i,-i$, compute its stabilizer and its orbit under the action of $G$. Show that $G$ has exactly 3 orbits in all.

Compute the orbit of $i$ under the subgroup

$$H=\left\{\left(\begin{array}{ll} a & b \\ 0 & d \end{array}\right) \mid a, b, d \in \mathbb{R}, a d=1\right\} \subset G .$$

Deduce that every element $g$ of $G$ may be expressed in the form $g=h k$ where $h \in H$ and for some $\theta \in \mathbb{R}$,

$$k=\left(\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$$

How many ways are there of writing $g$ in this form?