Stabilizer of Coset under Group Action on Coset Space

From ProofWiki
Jump to navigation Jump to search


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:

$\Stab {a H} = a H a^{-1}$


It is established in Action of Group on Coset Space is Group Action that $*$ is a group action.


\(\ds \Stab {a H}\) \(=\) \(\ds \set {g \in G: g * a H = a H}\) Definition of Stabilizer
\(\ds \) \(=\) \(\ds \set {g \in G: \paren {g a} H = a H}\)
\(\ds \) \(=\) \(\ds \set {g \in G: g H = a H a^{-1} }\)
\(\ds \) \(=\) \(\ds a H a^{-1}\)