Group Action on Prime Power Order Subset/Stabilizer is p-Subgroup

Lemma

Let $G$ be a finite group.

Let $\mathbb S = \set {S \subseteq G: \card S = p^n}$ where $p$ is prime.

That is, the set of all subsets of $G$ whose cardinality is the power of a prime number.

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:

$\Stab S$ is a $p$-subgroup of $G$.

Proof

First we show that $\Stab S$ is a $p$-subgroup of $G$:

From Group Action on Sets with k Elements:

$\forall S \in \mathbb S: \order {\Stab S} \divides \card S$

So:

$\order {\Stab S} \divides p^\alpha$

Thus $\Stab S$ is a $p$-group

Thus by Stabilizer is Subgroup, $\Stab S$ is a $p$-subgroup of $G$.

$\blacksquare$