Stabilizer of Subset Product Action on Power Set
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Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $\powerset G$ be the power set of $\struct {G, \circ}$.
Let $*: G \times \powerset G \to \powerset G$ be the subset product action on $\powerset G$ defined as:
- $\forall g \in G: \forall S \in \powerset G: g * S = g \circ S$
where $g \circ S$ is the subset product $\set g \circ S$.
Then the stabilizer of $S$ in $\powerset G$ is the set:
- $\Stab S = S$
Proof
From the definition of stabilizer:
- $\Stab S = \set {g \in G: g * S = S}$
The result follows from the definition of the group action $*$ given.
$\blacksquare$
Also see
- Subset Product Action is Group Action
- Stabilizer of Coset Action on Set of Subgroups
- Orbit of Subgroup under Coset Action is Coset Space
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.6$. Stabilizers: Example $109$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.10$