Group Actions and Sylow Theorems - Examples and Constructions

The Sylow theorems are among the most powerful results in finite group theory, guaranteeing the existence of subgroups of prime power order and constraining their number.

TheoremSylow's First Theorem

Let $G$ be a finite group of order $|G| = p^n m$ where $p$ is prime and $\gcd(p, m) = 1$ . Then $G$ contains a subgroup of order $p^k$ for each $0 \leq k \leq n$ .

In particular, $G$ has a subgroup of order $p^n$ , called a Sylow $p$ -subgroup.

This partially answers the converse of Lagrange's theorem: for prime power divisors, subgroups always exist. A Sylow $p$ -subgroup is a maximal $p$ -subgroup—it is not properly contained in any larger $p$ -subgroup.

TheoremSylow's Second Theorem

All Sylow $p$ -subgroups of $G$ are conjugate to each other. That is, if $P$ and $Q$ are Sylow $p$ -subgroups, there exists $g \in G$ such that $Q = gPg^{-1}$ .

This means all Sylow $p$ -subgroups are isomorphic and occupy equivalent positions in the group structure. The group acts transitively on its Sylow $p$ -subgroups by conjugation.

TheoremSylow's Third Theorem

Let $n_p$ denote the number of Sylow $p$ -subgroups of $G$ . Then:

$n_p \equiv 1 \pmod{p}$
$n_p$ divides $|G|/p^n = m$

Furthermore, $n_p = [G : N_G(P)]$ where $N_G(P)$ is the normalizer of any Sylow $p$ -subgroup $P$ .

ExampleGroups of Order 12

Let $|G| = 12 = 2^2 \cdot 3$ . For Sylow 3-subgroups:

$n_3 \equiv 1 \pmod{3}$ and $n_3 \mid 4$
Possible values: $n_3 \in \{1, 4\}$

For Sylow 2-subgroups:

$n_2 \equiv 1 \pmod{2}$ and $n_2 \mid 3$
Possible values: $n_2 \in \{1, 3\}$

If $n_3 = 1$ , the unique Sylow 3-subgroup is normal. This constrains the structure of $G$ significantly.

Remark

The Sylow theorems provide a systematic method for analyzing finite groups. By examining Sylow subgroups for each prime divisor of $|G|$ , we can often determine the group's structure completely. For many orders, the Sylow theorems force $G$ to be a direct or semidirect product of its Sylow subgroups.

The Sylow theorems extend Cauchy's theorem (which guarantees elements of prime order) and provide the foundation for classifying finite groups.