Orbits of Group Action on Sets with Power of Prime Size

Lemma

Let $G$ be a finite group such that $\order G = k p^n$ where $p \nmid k$.

Let $\mathbb S = \set {S \subseteq G: \order S = p^n}$

Let $G$ act on $\mathbb S$ by the group action defined in Group Action on Sets with k Elements:

$\forall S \in \mathbb S: g * S = g S = \set {x \in G: x = g s: s \in S}$

Then:

Orbit Length

The length of every orbit of this action is divisible by $k$.

Orbit whose Length is not Divisible by $p$

Each orbit whose length is not divisible by $p$ contains exactly one Sylow $p$-subgroup.

Orbit whose Length is Divisible by $p$

Each orbit whose length is divisible by $p$ contains no Sylow $p$-subgroups.