Stabilizer of Subset Product Action on Power Set

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 $10.10$