homogeneous space

Rewriting the half-plane as a homogeneous space: let \({\operatorname{SL}}_2({\mathbf{R}})\curvearrowright{\mathbb{H}}\) on the right by \(z\curvearrowleft{ \begin{bmatrix} {a} & {b} \\ {c} & {d} \end{bmatrix} }\coloneqq{ax + b\over cx+d}\), then the stabilizer at \(i\) is \({\operatorname{Stab}}_{{\operatorname{SL}}_2({\mathbf{R}})}(i) = \left\{{{ \begin{bmatrix} {a} & {b} \\ {-b} & {a} \end{bmatrix} }}\right\} = {\operatorname{SO}}_2({\mathbf{R}})\). So \begin{align*} {\mathbb{H}}\cong \dcoset{1}{{\operatorname{SL}}_2({\mathbf{R}})}{{\operatorname{Stab}}_{{\operatorname{SL}}_2({\mathbf{R}})}(i) } = \dcoset{1}{{\operatorname{SL}}_2({\mathbf{R}})}{{\operatorname{SO}}_2({\mathbf{R}}) } \end{align*}