Orbit of Subgroup under Coset Action is Coset Space
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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.
$\blacksquare$
Also see
- Subset Product Action is Group Action
- Stabilizer of Subset Product Action on Power Set
- Stabilizer of Coset Action on Set of Subgroups
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.15$