The Sylow Theorems

The Sylow theorems are the sharpest general tools for the structure of a finite group: they guarantee subgroups of every prime-power order dividing $|G|$, control how many there are, and how they sit together. They are the engine behind the classification of groups of small order and the proof that many orders force non-simplicity. This page uses the group-action technology (orbit–stabiliser, the class equation) and feeds series-solvable.md and simple-groups.md.

$p$-groups

A $p$-group is a finite group of order $p^n$ for a prime $p$. Two facts from the class equation drive everything:

• A non-trivial $p$-group has non-trivial centre: $p\mid |Z(G)|$.
• Cauchy's theorem: if a prime $p$ divides $|G|$, then $G$ has an element (hence a subgroup) of order $p$.

A Sylow $p$-subgroup of a finite group $G$ with $|G|=p^{a}m$ (where $p\nmid m$) is a subgroup of order $p^a$ — a $p$-subgroup of maximal possible order.

The three theorems

Sylow I (existence). For each prime $p$ with $p^a\mid |G|$ (maximal), $G$ has a Sylow $p$-subgroup of order $p^a$. (More: $G$ has a subgroup of every order $p^b$, $b\le a$.)

Sylow II (conjugacy). Any two Sylow $p$-subgroups are conjugate; every $p$-subgroup lies inside some Sylow $p$-subgroup.

Sylow III (counting). The number $n_p$ of Sylow $p$-subgroups satisfies $$n_p\equiv 1(\mathrm{mod}p)\text{and}{n}_p\mid m=|G|/p^a.$$ Moreover $n_p=[G:N_G(P)]$, the index of the normaliser of a Sylow subgroup.

Key corollary. A Sylow $p$-subgroup $P$ is normal iff $n_p=1$ (it is then the unique Sylow $p$-subgroup, since all are conjugate). So whenever the arithmetic of Sylow III forces $n_p=1$, the group has a normal subgroup — and cannot be simple.

Proof idea. All three follow from letting $G$ act by conjugation on the set of its $p$-subgroups (or on cosets) and applying orbit–stabiliser; the congruence $n_p\equiv 1(\mathrm{mod}p)$ comes from counting fixed points of a Sylow subgroup acting on the set of all Sylow subgroups.

Worked classifications

Sylow's theorems classify groups of many small orders by cornering the counts.

• Order $pq$ ($p<q$ primes). Sylow III forces $n_q\equiv 1(\mathrm{mod}q)$ and $n_q\mid p$, so $n_q=1$: the Sylow $q$-subgroup is normal. Then $G={\mathbb{Z}}_q⋊{\mathbb{Z}}_p$. If $q\not\equiv 1(\mathrm{mod}p)$ the action is trivial and $G\cong {\mathbb{Z}}_{pq}$ is cyclic; if $q\equiv 1(\mathrm{mod}p)$ there is also one non-abelian type. (Example: order $6$ gives ${\mathbb{Z}}_6$ and $S_3$.)
• Order $15=3\cdot 5$. $n_5\equiv 1(\mathrm{mod}5)$, $n_5\mid 3\Rightarrow n_5=1$; $n_3\mid 5$, $n_3\equiv 1(\mathrm{mod}3)\Rightarrow n_3=1$. Both Sylow subgroups normal and intersecting trivially, so $G\cong {\mathbb{Z}}_3\times {\mathbb{Z}}_5\cong {\mathbb{Z}}_{15}$ — the only group of order $15$.
• Order $12$. $n_3\in \{1,4\}$ and $n_2\in \{1,3\}$; a short argument shows at least one is $1$, so every group of order $12$ has a normal Sylow subgroup. The five types are ${\mathbb{Z}}_{12}$, ${\mathbb{Z}}_2\times {\mathbb{Z}}_6$, $A_4$, $D_6$, and ${\mathrm{Dic}}_3$ (the dicyclic group).
• No simple group of order $<60$ except the cyclic ${\mathbb{Z}}_p$: Sylow counting eliminates every composite order below $60$, and $|A_5|=60$ is the first non-abelian simple group (see simple-groups.md).

Why it matters

Sylow's theorems are the reason finite group theory is tractable: they reduce the existence of substructure to arithmetic (divisibility and congruences on $n_p$), and they identify the primes at which a group can possibly be simple. The normality they so often force is the first step of building a composition series and testing solvability.

References

• Dummit & Foote, Abstract Algebra, §4.5.
• Rotman, An Introduction to the Theory of Groups, Ch. 4.
• Isaacs, Finite Group Theory, Ch. 1.