The Sylow Theorems (Full Version)

The Sylow theorems describe the structure of $p$ -subgroups of finite groups, providing the most powerful general tool for analyzing the structure of finite groups.

Statement

Theorem10.5Sylow theorems

Let $G$ be a finite group with $|G| = p^a m$ where $p \nmid m$ and $a \geq 1$ .

(Existence) $G$ has a subgroup of order $p^a$ (a Sylow $p$ -subgroup).
(Conjugacy) Any two Sylow $p$ -subgroups are conjugate in $G$ .
(Counting) The number $n_p$ of Sylow $p$ -subgroups satisfies $n_p \equiv 1 \pmod{p}$ and $n_p \mid m$ .

Proof of Existence

Proof

We use the action of $G$ on subsets. Consider $\mathcal{S} = \{S \subseteq G : |S| = p^a\}$ , with $G$ acting by left multiplication: $g \cdot S = gS$ . We have $|\mathcal{S}| = \binom{p^a m}{p^a}$ . By a counting argument, $v_p(\binom{p^a m}{p^a}) = 0$ (where $v_p$ is the $p$ -adic valuation), so $p \nmid |\mathcal{S}|$ .

The orbits partition $\mathcal{S}$ : $|\mathcal{S}| = \sum |\mathrm{Orb}(S_i)|$ . Since $p \nmid |\mathcal{S}|$ , some orbit has $p \nmid |\mathrm{Orb}(S_0)|$ . The stabilizer of $S_0$ is $\mathrm{Stab}(S_0) = \{g : gS_0 = S_0\}$ , and $|\mathrm{Orb}(S_0)| = [G:\mathrm{Stab}(S_0)]$ .

Since $\mathrm{Stab}(S_0) \leq G$ acts on $S_0$ by left multiplication (faithfully restricted to $S_0$ ), $|\mathrm{Stab}(S_0)| \leq |S_0| = p^a$ . From $|G| = |\mathrm{Orb}(S_0)| \cdot |\mathrm{Stab}(S_0)|$ and $p \nmid |\mathrm{Orb}(S_0)|$ : $p^a \mid |\mathrm{Stab}(S_0)|$ . Combined with $|\mathrm{Stab}(S_0)| \leq p^a$ : $|\mathrm{Stab}(S_0)| = p^a$ . So $\mathrm{Stab}(S_0)$ is a Sylow $p$ -subgroup. $\blacksquare$

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Applications

ExampleUsing Sylow theorems

Groups of order 15: $|G| = 3 \cdot 5$ . $n_3 \mid 5$ and $n_3 \equiv 1 \pmod{3}$ : $n_3 = 1$ . $n_5 \mid 3$ and $n_5 \equiv 1 \pmod{5}$ : $n_5 = 1$ . So both Sylow subgroups are normal, $G \cong \mathbb{Z}/3 \times \mathbb{Z}/5 \cong \mathbb{Z}/15$ .

Groups of order 12: $n_3 \in \{1, 4\}$ and $n_2 \in \{1, 3\}$ . Analysis of cases yields exactly 5 groups: $\mathbb{Z}/12, \mathbb{Z}/2 \times \mathbb{Z}/6, A_4, D_6, \mathrm{Dic}_{12}$ .

RemarkPower of the Sylow theorems

For many small orders, the Sylow theorems alone suffice to classify all groups. When $n_p = 1$ , the Sylow $p$ -subgroup is normal, constraining the group structure. The Sylow theorems are often the first tool applied when studying the structure of an unknown finite group.