Stabilizer of Conjugacy Action on Subgroup is Normalizer
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Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $X$ be the set of all subgroups of $G$.
Let $*$ be the conjugacy action on $H$:
- $\forall g \in G, H \in X: g * H = g \circ H \circ g^{-1}$
Then the stabilizer of $H$ in $\powerset G$ is given by:
- $\Stab H = \map {N_G} H$
where $\map {N_G} H$ is the normalizer of $H$ in $G$.
Proof
We have that:
- $\Stab H = \set {g \in G: g \circ H \circ g^{-1} = H}$
which is precisely how the normalizer is defined.
$\blacksquare$
Also see
- Conjugacy Action on Subgroups is Group Action
- Orbit of Conjugacy Action on Subgroup is Set of Conjugate Subgroups
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.6$. Stabilizers: Example $110$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.10$