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