Group Action on Prime Power Order Subset
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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:
Stabilizer is $p$-Subgroup
- $\Stab S$ is a $p$-subgroup of $G$.
Stabilizer of Maximal Power Order Subset
- If $p^n$ is the maximal power of $p$ dividing $\order G$, and if $p \nmid \card {\Orb S}$, then $\forall s \in S: \Stab S s = S$.