# Orbits of Group Action on Sets with Power of Prime Size

Jump to navigation
Jump to search

## Contents

## 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

### 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.