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:
 * 1) The length of every orbit of this action is divisible by $k$
 * 2) Each orbit whose length is not divisible by $p$ contains exactly one Sylow $p$-subgroup
 * 3) Each orbit whose length is divisible by $p$ contains no Sylow $p$-subgroups.

Proof

 * First we show that the length of every orbit of this action is divisible by $k$:

From the Orbit-Stabilizer Theorem, $\order G = \card {\Orb S} \times \order {\Stab S}$.

From Group Action on Prime Power Order Subset, $\Stab S$ is a $p$-subgroup of $G$.

Therefore $k \nmid \order {\Stab S}$ and therefore $k \divides \card {\Orb S}$.


 * Next, each orbit whose length is not divisible by $p$ contains at least one Sylow $p$-subgroup:

From the reasoning in the First Sylow Theorem, we know that:
 * at least one $S \in \mathbb S$ is such that $p \nmid \card {\Orb S}$

and that for such a set:
 * $\Stab S$ is a Sylow $p$-subgroup of $G$.

From Group Action on Prime Power Order Subset:
 * $\forall s \in S: \Stab S s = S$

Therefore:
 * $\Stab S = S s^{-1}$

Thus for any $s \in S$, $S s^{-1}$ is a Sylow $p$-subgroup of $G$.

It also follows that $s^{-1} \Stab S s$ is also a subgroup of $G$ with the same number of elements as $\Stab S$.

Thus $s^{-1} \Stab S s = s^{-1} \paren {S s^{-1} } s = s^{-1} S$ is a Sylow $p$-subgroup of $G$ in the orbit of $S$.

From the Orbit-Stabilizer Theorem:
 * $\card {\Orb S} = k$

as $\order {\Stab S} = p^n$.

So if an orbit has length not divisible by $p$, then this orbit contains a Sylow $p$-subgroup of $G$ (that is, $\Stab S$ and also $s^{-1} \Stab S s$), and the length of the orbit is $k$.


 * Next we show that every Sylow $p$-subgroup of $G$ has to be in one of the orbits whose length is not divisible by $p$.

Let $H$ be a Sylow $p$-subgroup of $G$.

By the definition of $\mathbb S$, and because $\order H = p^n$:
 * $H \in \mathbb S$

By Group Action on Sets with k Elements:
 * $g \in G \implies g * H = g H$

which is a left coset of $G$.

However, we know that $g H = H \iff g \in H$.

Now:
 * $g \in \Stab H \iff g * H = H \iff g \in H$

Thus:
 * $\Stab H = H$

From the Orbit-Stabilizer Theorem:
 * $\card {\Orb H} = k$

that is, not divisible by $p$, and of course $H \in \Orb H$.


 * Now we show that each orbit whose length is not divisible by $p$ contains no more than one Sylow $p$-subgroup:

Let $H$ be a Sylow $p$-subgroup of $G$.

Now $\Orb H$ consists of all $g * H = g H$, that is, all the right cosets of $H$.

But the only right coset of $H$ which is a subgroup of $G$ is in fact $H$ itself.

None of the rest of the elements of $\mathbb S$ can actually be a subgroup of $G$.

Thus, no orbit can contain more than one Sylow $p$-subgroup.