Group Actions and Sylow Theorems - Applications

The Sylow theorems provide powerful constraints on the structure of finite groups, enabling classification results and non-existence proofs.

TheoremSylow Theorems (Complete Statement)

Let $G$ be a finite group with $|G| = p^n m$ where $p$ is prime and $\gcd(p,m) = 1$ . Let $n_p$ be the number of Sylow $p$ -subgroups. Then:

Existence: Sylow $p$ -subgroups exist (of order $p^n$ )
Conjugacy: All Sylow $p$ -subgroups are conjugate
Counting: $n_p \equiv 1 \pmod{p}$ and $n_p \mid m$

These theorems transform the question "what subgroups must exist?" into concrete divisibility constraints that can often determine group structure completely.

ExampleNo Simple Group of Order 30

Suppose $G$ is simple with $|G| = 30 = 2 \cdot 3 \cdot 5$ . For Sylow 5-subgroups:

$n_5 \equiv 1 \pmod{5}$ and $n_5 \mid 6$
So $n_5 \in \{1, 6\}$

If $n_5 = 1$ , the unique Sylow 5-subgroup is normal, contradicting simplicity.

If $n_5 = 6$ , we have 6 subgroups of order 5, contributing $6 \cdot 4 = 24$ elements of order 5. Similarly analyzing Sylow 3-subgroups forces $n_3 = 10$ , contributing 20 more elements. But $24 + 20 > 30$ , a contradiction!

Therefore, no simple group of order 30 exists.

TheoremGroups of Order $p^2$

Every group of order $p^2$ (where $p$ is prime) is abelian. Moreover, there are exactly two such groups up to isomorphism: $\mathbb{Z}_{p^2}$ and $\mathbb{Z}_p \times \mathbb{Z}_p$ .

This follows from the class equation: $|G| = |Z(G)| + \sum [G:C_G(g_i)]$ . Since each term is a power of $p$ and $|Z(G)| \geq p$ , we must have $|Z(G)| \in \{p, p^2\}$ . If $|Z(G)| = p$ , the quotient $G/Z(G)$ has order $p$ , hence is cyclic, which forces $G$ to be abelian (contradicting $|Z(G)| = p$ ). Thus $|Z(G)| = p^2 = |G|$ .

Remark

The Sylow theorems enable a systematic approach to finite group theory. For small orders, we can often enumerate all possible groups by:

Decomposing $|G|$ into prime powers
Determining possible values of $n_p$ for each prime
Using semi direct products when unique Sylow subgroups don't exist

Combined with the classification of finite simple groups, the Sylow theorems provide a nearly complete understanding of finite group structure.