Group Action on Prime Power Order Subset/Stabilizer is p-Subgroup
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Lemma
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 $G$.
$\blacksquare$