Orbit of Subgroup under Coset Action is Coset Space

Theorem
Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Let $\mathcal P \left({G}\right)$ be the power set of $\left({G, \circ}\right)$.

Let $H \in \mathcal P \left({G}\right)$ be a subgroup of $G$.

Let $*$ be the group action on $H$ defined as:
 * $\forall g \in G: g * H = g \circ H$

where $g \circ H$ is the (left) coset of $g$ by $H$.

Then the orbit of $H$ in $\mathcal P \left({G}\right)$ is the (left) coset space of $H$:
 * $\operatorname{Orb} \left({H}\right) = G / H^l$

Proof
From the definition of orbit:


 * $\operatorname{Orb} \left({H}\right) = \left\{{y \in G: \exists g \in G: y = g \circ H}\right\}$

The result follows from the definition of (left) coset space.

Also see

 * Group Action on Subset of Group
 * Stabilizer of Subset Product Action on Power Set
 * Stabilizer of Coset Action on Power Set