Sylow Theorems

The Sylow theorems are a collection of results in the theory of finite groups. They give a partial converse to Lagrange's Theorem, and are one of the most important results in the field. They are named for P. Ludwig Sylow, who published their proof in 1872.

The Theorems

Throughout this article, $p$ will be a prime.

First, we show a lemma.

Lemma. Let $n=p^rm$ , where $r$ is an nonnegative integer and $m$ is a nonnegative integer not divisible by $p$ . Then $\binom{n}{p^r} \not\equiv 0 \pmod{p} .$

Proof. Let $G$ be a group of order $p^r$ (e.g., $\mathbb{Z}/p^r\mathbb{Z}$ , and let $S$ be a set of size $m$ . Let $G$ act on the set $G \times T$ by the law $g(\alpha, x) \mapsto (g\alpha, x)$ ; extend this action canonically to the subsets of $G \times S$ of size $p^r$ . Evidently, there are $\binom{n}{p^r}$ such subsets.

Evidently, a subset of $A \times T$ is stable under this action if and only if $A= G$ . Thus the fixed points of the action are exactly the subsets of the form $G \times \{x\}$ , for $x\in S$ . Then there are $m$ fixed points. Then $\binom{n}{p^r} \equiv m \not\equiv 0 \pmod{p},$ since the $p$ -group $G$ operates on a set of size $\binom{n}{p^r}$ with $m$ fixed points. $\blacksquare$

Let $G$ be a finite group.

Theorem 1. For every prime $p$ , every finite group contains a Sylow $p$ -subgroup.

Proof. Let $G$ be a finite group of order $n = p^r m$ , for some positive integer $m$ , not divisible by $p$ . Let $\mathfrak{P}$ denote the set of subsets of $G$ of size $p^r$ . Consider the action of $G$ by left translation on the elements of $\mathfrak{P}$ . There are $\binom{n}{p^r}$ such subsets. Since $\binom{n}{p^r} \not\equiv 0 \pmod{p},$ some orbit $\mathfrak{H}$ of $\mathfrak{P}$ must have cardinality not divisible by $p$ . Since $\lvert \mathfrak{H} \rvert \mid n$ , it follows that $\lvert \mathfrak{H} \rvert \mid m$ ; in particular, $\lvert \mathfrak{H} \rvert \le m$ . Since each element of $G$ must be contained in one of the elements of $\mathfrak{H}$ , it follows that the elements of $\mathfrak{H}$ must be disjoint.

Consider now the equivalence relation $R(x,y)$ on elements of $G$ , defined as " $x$ and $y$ are in the same element of $\mathfrak{H}$ ". Then $R$ is compatible with left translation by $G$ ; since the elements of $\mathfrak{H}$ are disjoint, $R$ is an equivalence relation. Thus the equivalence class of the identity is a subgroup $H$ of $G$ , which must have order $p^r$ . $\blacksquare$

Theorem 2. The Sylow $p$ -subgroups of $G$ are conjugates.

Theorem 3. The number of Sylow $p$ -subgroups of $G$ is equivalent to 1 (mod $p$ ).

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Sylow Theorems

The Theorems

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