If $H$ is a $p$-subgroup of $G$ with order of $p^{a}$ and $K$ is a Sylow $p$-subgroup of $G$ with order of $p^{b}$. $X$ is the set of left coset of $K$.
Let $H$ acts on $X$, what is the order of orbits? I have a feeling that it might be related to the prime $p$. But I don't know how to get that.
(I am trying to use this result to establish that $H$ is contained in $K$. )
Thanks!