First Sylow Theorem

Theorem
Let $p$ be a prime number.

Let $G$ be a group such that:
 * $\order G = k p^n$

where:
 * $\order G$ denotes the order of $G$
 * $p$ is not a divisor of $k$.

Then $G$ has at least one Sylow $p$-subgroup.

Also see

 * Sylow Theorems

Note
By Orbits of Group Action on Sets with Power of Prime Size, it was clear that $k \divides \size {\Orb S}$.

However here, since it is established that $\size {\Stab S} = p^n$, and by Orbit-Stabilizer Theorem, we also have $k = \size {\Orb S}$.