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
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 54 \gamma$