Orbit of Conjugacy Action on Subgroup is Set of Conjugate Subgroups
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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 $H$ defined as:
- $\forall g \in G, H \in X: g * H = g \circ H \circ g^{-1}$
Then the orbit $\Orb H$ of $H$ in $\powerset G$ is the set of subgroups of $G$ conjugate to $H$.
Proof
We have that:
- $\Orb H = \set {g \circ H \circ g^{-1}: g \in G}$
from the definition.
The result follows by definition of conjugate subgroup.
$\blacksquare$
Also see
- Conjugacy Action on Subgroups is Group Action
- Stabilizer of Conjugacy Action on Subgroup is Normalizer
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.15$