Ergodic Theory with a view towards Number Theory, Chapter 9, Page 279, 280.
Lemma 9.1. The action of $\mathrm{PSL}_2(\mathbb{R}$) on $\mathbb{H}$ defined by (9.1) has the following properties.
(1) The action is isometric, meaning that $$\mathsf{d}(g(z_0),g(z_1))=\mathsf{d}(z_0,z_1)$$ for any $z_0,z_1\in\mathbb{H}$ and $g\in\mathrm{PSL}_2(\mathbb{R}$). Moreover, the action of $\mathrm{PSL}_2(\mathbb{R})$ on $\mathrm{T}\mathbb{H}$ defined by the derivative $\mathrm{D}g$ of the action of $g\in\mathrm{PSL}_2(\mathbb{R})$ on $\mathbb{H}$ preserves the Riemannian metric.
(2) The action is transitive on $\mathbb{H}$: given any two points $z_0,z_1\in\mathbb{H}$ there is a matrix $g\in\mathrm{PSL}_2(\mathbb{R})$ with $g(z_0)=z_1$.
(3) The stabilizer $$\mathrm{Stab}_{\mathrm{PSL}_2(\mathbb{R})}(\mathrm{i})=\{g\in\mathrm{PSL}_2(\mathbb{R})\mid g(\mathrm{i})=\mathrm {i}\}$$ of $\mathrm{i}\in\mathbb{H}$ is the projective special orthogonal group $$\mathrm{PSO}(2)=\mathrm{SO}(2)/\{\pm I_2\}$$ where $$\mathrm{SO}(2)=\left\{\left(\begin{array}{rr} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{array}\right)\mid\theta\in\mathbb{R}\right\}.$$ Notice that property (3) gives an identification $$\mathbb{H}\cong\mathrm{PSL}_2(\mathbb{R})/\mathrm{PSO}(2),$$ and under the identification the coset $g$ $\mathrm{PSO(2)}$ is sent to $g(\mathrm{i})$.
Where $$\mathbb{H}=\{x+iy \in \mathbb{C}, y>0\}.$$ How do we have that $$\mathbb{H} \cong\mathrm{PSL}_2(\mathbb{R}) / \mathrm{PSO}(2).$$
Is $\mathrm{PSO}(2)$ a normal subgroup of $\mathrm{PSL}_2(\mathbb{R})$? (Of course, it is a subgroup).
If $H\le G$ then the coset space $G/H$ carries a (left) $G$-action, $g(xH)=(gx)H$.
If $G$ is a topological group and $H$ is closed, there is a quotient topology on $G/H$, even a smooth structure if $G$ is a Lie group.
In any action $G\curvearrowright\Omega$, the orbit $\mathrm{Orb}(\omega)$ of a point $\omega\in\Omega$ is also a $G$-set contained in $\Omega$.
The orbit-stabilizer theorem says $\mathrm{Orb}(\omega)\cong G/\mathrm{Stab}(\omega)$ are equivalent $G$-sets for any $\omega\in\Omega$. Indeed, there is a $G$-equivariant one-to-one correspondence $g\omega\leftrightarrow g\mathrm{Stab}(\omega)$. Under nice conditions, this will also be a homeomorphism (indeed, diffeomorphism if applicable).
The OST applies here for $G=\mathrm{PSL}_2(\mathbb{R})$, $\Omega=\mathbb{H}$, $H=\mathrm{PSO}(2)$.
And no, $\mathrm{PSO}(2)$ is not normal in $\mathrm{PSL}_2(\mathbb{R})$ (can you show this?).