Length of Orbit of Subgroup Action on Left Coset Space

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 $\index K {K \cap H^g}$.