Stabilizer of Coset Action on Set of Subgroups

Theorem

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

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

Let $\HH \subseteq \powerset G$ denote the set of subgroups of $G$.

Let $*$ be the subset product action on $\HH \subseteq \powerset G$ defined as:

$\forall g \in G: \forall H \in \HH: g * H = g \circ H$

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

Then the stabilizer of $H$ in $\powerset G$ is $H$ itself:

$\Stab H = H$

Proof

From the definition of Stabilizer of Subset Product Action on Power Set:

$\Stab H = H = \set {g \in G: g * H = H}$

The result follows from Left Coset Equals Subgroup iff Element in Subgroup.

$\blacksquare$

Also see

• Subset Product Action is Group Action
• Stabilizer of Subset Product Action on Power Set
• Orbit of Subgroup under Coset Action is Coset Space

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $5$: Subgroups: Exercise $19$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.10$