Length of Orbit of Subgroup Action on Left Coset Space

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Theorem

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 length of the orbit of $g H$ is $\index K {K \cap H^g}$.


Proof

\(\ds \card {\Orb {g H} }\) \(=\) \(\ds \index K {\Stab {g H} }\) Orbit-Stabilizer Theorem
\(\ds \) \(=\) \(\ds \index K {K \cap H^g}\) Stabilizer of Subgroup Action on Left Coset Space

$\blacksquare$


Sources