Stabilizer of Coset under Group Action on Coset Space
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Theorem
Let $G$ be a group whose identity is $e$.
Let $H$ be a subgroup of $G$.
Let $*: G \times G / H \to G / H$ be the action on the (left) coset space:
- $\forall g \in G, \forall g' H \in G / H: g * \paren {g' H} := \paren {g g'} H$
Then the stabilizer of $a H$ under $*$ is given by:
- $\Stab {a H} = a H a^{-1}$
Proof
It is established in Action of Group on Coset Space is Group Action that $*$ is a group action.
Then:
\(\ds \Stab {a H}\) | \(=\) | \(\ds \set {g \in G: g * a H = a H}\) | Definition of Stabilizer | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {g \in G: \paren {g a} H = a H}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {g \in G: g H = a H a^{-1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a H a^{-1}\) |
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $6$: Cosets: Exercise $9$