# Orbit of Subgroup under Coset Action is Coset Space

## Theorem

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $\powerset G$ be the power set of $G$.

Let $H \in \powerset G$ 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 $\powerset G$ is the (left) coset space of $H$:

$\Orb H = G / H^l$

## Proof

From the definition of orbit:

$\Orb H = \set {y \in G: \exists g \in G: y = g \circ H}$

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

$\blacksquare$