Stabilizer of Subgroup Action on Left Coset Space

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Let $G$ be a group.

Let $H$ and $K$ be subgroups of $G$.

Let $K$ act on the left coset space $G / H^l$ by:

$\forall \tuple {k, g H} \in K \times G / H^l: k * g H := \paren {k g} H$

The stabilizer of $g H$ is $K \cap H^g$, where $H^g$ denotes the $G$-conjugate of $H$ by $g$.


\(\ds \Stab {g H}\) \(=\) \(\ds \set {k \in K: \paren {k g} H = g H}\) Definition of Stabilizer
\(\ds \) \(=\) \(\ds \set {k \in K: g^{-1} \paren {k g} \in H}\) Left Cosets are Equal iff Product with Inverse in Subgroup
\(\ds \) \(=\) \(\ds \set {k \in K: k \in g H g^{-1} }\)
\(\ds \) \(=\) \(\ds K \cap H^g\) Definition of Conjugate of Set