Second Sylow Theorem

Introduction

The Second Sylow Theorem establishes that all Sylow $p$ -subgroups of a finite group are conjugate to one another. This is a powerful result that shows the uniqueness of Sylow subgroups up to conjugation.

Statement

Second Sylow Theorem (Conjugacy): All Sylow $p$ -subgroups of $G$ are conjugate to one another. That is, if $P_{1}$ and $P_{2}$ are Sylow $p$ -subgroups, then there exists a $g \in G$ such that $P_{1} = g P_{2} g^{- 1}$ . This also implies that any $p$ -subgroup is contained within some Sylow $p$ -subgroup.

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

Proof Sketch

The proof uses group actions:

Define the action: Let a Sylow $p$ -subgroup $P$ act on the set of left cosets of another Sylow $p$ -subgroup $Q$ .
Apply orbit-stabilizer: Show that there must be a fixed point under this action.
Conclude conjugacy: The existence of a fixed point implies that $P$ is a conjugate of $Q$ .

Key Steps

The action of $P$ on $G / Q$ has orbits of size dividing $| P | = p^{n}$
Since $[G : Q] = m$ is not divisible by $p$ , there must be a fixed point
A fixed point corresponds to $g Q$ where $P \subseteq g Q g^{- 1}$
Since both have the same order, $P = g Q g^{- 1}$

Examples

Example 1: Groups of Order 12

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

All Sylow 2-subgroups (order 4) are conjugate
All Sylow 3-subgroups (order 3) are conjugate

Example 2: Groups of Order 30

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

All Sylow 2-subgroups (order 2) are conjugate
All Sylow 3-subgroups (order 3) are conjugate
All Sylow 5-subgroups (order 5) are conjugate

Example 3: Symmetric Groups

In $S_{4}$ (order 24):

All Sylow 2-subgroups (order 8) are conjugate
All Sylow 3-subgroups (order 3) are conjugate

Applications

Application 1: Normal Sylow Subgroups

If a group has a unique Sylow $p$ -subgroup (i.e., $n_{p} = 1$ ), then that subgroup is normal, since it is its only conjugate.

Application 2: Group Classification

The Second Sylow Theorem helps classify groups by understanding the conjugacy of their Sylow subgroups.

Application 3: Simplicity Tests

If a group has multiple Sylow $p$ -subgroups, they must be conjugate, which can help determine if the group is simple.

Consequences

Uniqueness Up to Conjugation

The theorem shows that Sylow $p$ -subgroups are unique up to conjugation, providing a strong structural result.

Containment of p-Subgroups

Any $p$ -subgroup is contained within some Sylow $p$ -subgroup, giving a complete picture of $p$ -subgroup structure.