First Sylow Theorem

Introduction

The First Sylow Theorem guarantees the existence of Sylow $p$ -subgroups for any prime factor $p$ of a finite group's order. This is a powerful existence result that provides a partial converse to Lagrange's Theorem.

Statement

First Sylow Theorem (Existence): For any prime factor $p$ of $| G |$ , Sylow $p$ -subgroups of $G$ exist.

Let $G$ be a finite group of order $| G | = p^{n} m$ , where $p$ is a prime and $p$ does not divide $m$ . A subgroup of order $p^{n}$ is called a Sylow $p$ -subgroup of $G$ .

Proof Sketch

The proof is an elegant application of group actions:

Define the action: Let $G$ act on the set of all subsets of size $p^{n}$ by left multiplication.
Combinatorial argument: Show that there must be an orbit whose size is not divisible by $p$ .
Stabilizer is the subgroup: The stabilizer of any element in this orbit is a subgroup of order $p^{n}$ .

Key Steps

The number of subsets of size $p^{n}$ is $(\binom{| G |}{p^{n}})$
By Lucas's theorem or direct calculation, this number is not divisible by $p$
Since the orbits partition the set, at least one orbit has size not divisible by $p$
The stabilizer of any element in such an orbit has order $p^{n}$

Examples

Example 1: Groups of Order 12

Let $G$ be a group of order 12. We have $12 = 2^{2} \cdot 3$ .

By the First Sylow Theorem, $G$ has Sylow 2-subgroups of order 4
$G$ also has Sylow 3-subgroups of order 3

Example 2: Groups of Order 30

Let $G$ be a group of order 30. We have $30 = 2 \cdot 3 \cdot 5$ .

By the First Sylow Theorem, $G$ has Sylow 2-subgroups of order 2
$G$ has Sylow 3-subgroups of order 3
$G$ has Sylow 5-subgroups of order 5

Example 3: Symmetric Groups

In $S_{4}$ (order 24), the First Sylow Theorem guarantees:

Sylow 2-subgroups of order 8
Sylow 3-subgroups of order 3

Applications

Application 1: Group Classification

The First Sylow Theorem is the first step in classifying groups of a given order, as it guarantees the existence of certain subgroups.

Application 2: Structure Analysis

Knowing that Sylow $p$ -subgroups exist helps us understand the structure of finite groups.

Application 3: Simplicity Tests

The existence of Sylow subgroups is crucial for testing whether a group is simple.