Orbit of Group Action on Subgroup by Right Regular Representation is Right Coset

Theorem
Let $G$ be a group.

Let $H$ be a proper subgroup of $G$.

Let $*: H \times G \to G$ be the group action defined as:
 * $\forall \left({h, g}\right) \in H \times G: h * g = \rho_{h^{-1}} \left({g}\right)$

where $\rho_{h^{-1}} \left({g}\right)$ is the right regular representation of $g$ by $h^{-1}$.

Let $x \in G$.

Then the orbit of $x$ under $*$ is given by:
 * $\forall x \in G: \operatorname{Orb} \left({x}\right) = H x$

where $H x$ is the right coset of $H$ by $x$.