Sylow's Theorems

• Sylow's theorems guarantee that if prime power $p^k$ divides |G|, then G contains a subgroup of order $p^k$ (extends Lagrange's theorem)
• All maximal p-subgroups (Sylow p-subgroups) in a group are conjugate to each other, constraining structural diversity
• Number of Sylow p-subgroups $n_p$ satisfies $n_p\equiv 1(\mathrm{mod}p)$ and divides |G|/$p^n$, tightly limiting possibilities
• Critical tool for classifying small finite groups and proving non-simplicity of groups with specific orders
• Bridge between numerical properties of |G| and actual geometric/combinatorial group structure
• Published 1872, foundational results moving group theory from studying examples to universal structural principles

Sylow's Theorems

Overview

Sylow's theorems are fundamental results in finite group theory that provide detailed information about the existence and structure of subgroups whose order is a prime power. Named after the Norwegian mathematician Ludwig Sylow, these theorems answer a natural question arising from Lagrange's theorem: while Lagrange tells us that the order of any subgroup divides the order of the group, the converse fails in general. Sylow's theorems give precise conditions under which subgroups of particular orders must exist, and they characterize how many such subgroups there can be.

The power of Sylow's theorems lies in their ability to impose strong structural constraints on finite groups based solely on arithmetic properties of their order. They provide the primary tool for classifying groups of small order, for proving that groups of certain orders cannot be simple, and for understanding the internal architecture of arbitrary finite groups. The theorems establish a bridge between the purely numerical data encoded in $|G|$ and the geometric and combinatorial structure of $G$ itself.

Historically, these results marked a watershed moment in the development of group theory. Published in 1872, they represented one of the first deep theorems about the general structure of finite groups, moving beyond particular examples to establish universal principles. The Sylow theorems remain central to modern algebra, appearing in applications ranging from Galois theory to representation theory and even in combinatorial designs and graph theory.

If a prime power $p^k$ divides the order of a finite group $G$, then $G$ must contain a subgroup of order $p^k$. Moreover, all maximal such subgroups (Sylow $p$-subgroups) are conjugate, and their number is tightly constrained by congruence conditions modulo $p$.

Core Theory

Main Result

• $n_p\equiv 1(\mathrm{mod}p)$
• $n_p\mid m$

In particular, $n_p\in \{d\mid m:d\equiv 1(\mathrm{mod}p)\}$.

We prove each part in turn.

Let $\mathcal{S}$ denote the collection of all subsets of $G$ having exactly $p^n$ elements. Then $|\mathcal{S}|=\left(\genfrac{}{}{0ex}{}{|G|}{p^n}\right)=\left(\genfrac{}{}{0ex}{}{p^{n}m}{p^n}\right)$.

We analyze this binomial coefficient modulo $p$. Writing $\left(\genfrac{}{}{0ex}{}{p^{n}m}{p^n}\right)=\frac{(p^{n}m)!}{(p^n)!(p^{n}m-p^n)!}=\frac{(p^{n}m)!}{(p^n)!(p^n(m-1))!}$, we use the fact that the highest power of $p$ dividing $k!$ is ${\sum }_{i=1}^{\infty }\lfloor k/p^i\rfloor$.

For the numerator: $v_p((p^{n}m)!)={\sum }_{i=1}^{\infty }\lfloor p^{n}m/p^i\rfloor =p^{n-1}m+p^{n-2}m+\cdots +m+\lfloor m/p\rfloor +\cdots$

For the denominator: $v_p((p^n)!)=p^{n-1}+p^{n-2}+\cdots +1$ and $v_p((p^n(m-1))!)=p^{n-1}(m-1)+\cdots$

Computing $v_p((\genfrac{}{}{0ex}{}{p^{n}m}{p^n}))=v_p((p^{n}m)!)-v_p((p^n)!)-v_p((p^n(m-1))!)$, the $p^{n-1}$ terms give $p^{n-1}m-p^{n-1}-p^{n-1}(m-1)=0$, and similarly for all higher powers of $p$. The remaining terms involve only $m$ and $m-1$, which are coprime to $p$, so $v_p((\genfrac{}{}{0ex}{}{p^{n}m}{p^n}))=0$. Thus $p\nmid \left(\genfrac{}{}{0ex}{}{p^{n}m}{p^n}\right)$.

Now let $G$ act on $\mathcal{S}$ by left multiplication: $g\cdot S=\{gs:s\in S\}$. Since $p\nmid |\mathcal{S}|$, there exists at least one orbit whose size is not divisible by $p$. Let $S_0$ be a representative of such an orbit, and let $H=\{g\in G:g{S}_0=S_0\}$ be its stabilizer.

By the orbit-stabilizer theorem, $|G|=|\text{Orb}(S_0)|\cdot |H|$, so $p^{n}m=|\text{Orb}(S_0)|\cdot |H|$. Since $p\nmid |\text{Orb}(S_0)|$, we have $p^n\mid |H|$.

We claim $|H|\le p^n$. Note that $H\subseteq S_0$ (since if $h\in H$, then $h{S}_0=S_0$, and $e\in S_0$, so $h=he\in h{S}_0=S_0$). Thus $|H|\le |S_0|=p^n$.

Therefore $|H|=p^n$, and $H$ is a Sylow $p$-subgroup.

For the "moreover" clause, let $P$ be a Sylow $p$-subgroup and $p^k\mid |G|$ with $k\le n$. By Cauchy's theorem applied repeatedly, $P$ contains an element of order $p$, hence a subgroup of order $p$. Inductively, we can show $P$ contains a subgroup of order $p^k$ for each $k\le n$.

Let $P\in {\text{Syl}}_p(G)$ and let $Q$ be any $p$-subgroup of $G$ (not necessarily maximal). We show that $Q\subseteq gP{g}^{-1}$ for some $g\in G$.

Let $Q$ act on the set $G/P=\{gP:g\in G\}$ of left cosets by left multiplication. Since $|G/P|=m$ and $\gcd (p,m)=1$, and $Q$ is a $p$-group, there must exist a fixed point: some coset $gP$ such that $qgP=gP$ for all $q\in Q$.

This means $g^{-1}qg\in P$ for all $q\in Q$, so $Q\subseteq gP{g}^{-1}$.

Now if $Q$ is also a Sylow $p$-subgroup, then $|Q|=p^n=|P|=|gP{g}^{-1}|$, so $Q=gP{g}^{-1}$.

Let $P\in {\text{Syl}}_p(G)$. By (2), $G$ acts transitively on ${\text{Syl}}_p(G)$ by conjugation. By the orbit-stabilizer theorem, $n_p=[G:N_G(P)]$ where $N_G(P)=\{g\in G:gP{g}^{-1}=P\}$ is the normalizer of $P$.

Since $P\subseteq N_G(P)$, we have $|N_G(P)|=p^n\cdot k$ for some integer $k$, so $n_p=\frac{p^{n}m}{p^{n}k}=\frac{m}{k}$. Thus $n_p\mid m$.

For the congruence, let $P$ act on ${\text{Syl}}_p(G)$ by conjugation. Since $P$ is a $p$-group, each orbit has size a power of $p$. The orbit of $P$ itself is $\{P\}$ (size 1) since $pP{p}^{-1}=P$ for all $p\in P$.

If $Q\in {\text{Syl}}_p(G)$ with $Q\ne P$, we claim the orbit of $Q$ under $P$-conjugation has size divisible by $p$. The stabilizer is $\{p\in P:pQ{p}^{-1}=Q\}=P\cap N_G(Q)$. If this equals $P$, then $P\subseteq N_G(Q)$, so $P,Q\subseteq N_G(Q)$. But $P$ and $Q$ are both Sylow $p$-subgroups of $N_G(Q)$, so by conjugacy within $N_G(Q)$, there exists $n\in N_G(Q)$ with $Q=nP{n}^{-1}$. But $n\in N_G(Q)$ means $nQ{n}^{-1}=Q$, so $nP{n}^{-1}=Q$ gives $P=Q$, contradiction.

Thus $P\cap N_G(Q)⊊P$, so $|P\cap N_G(Q)|\le p^{n-1}$, and the orbit size $[P:P\cap N_G(Q)]\ge p$.

Therefore $n_p=1+(\text{sum of orbit sizes}\ge p)\equiv 1(\mathrm{mod}p)$.

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Examples and Applications

1

Let $|G|=15=3\cdot 5$. We prove $G\cong {\mathbb{Z}}_{15}$.

For Sylow 3-subgroups: $n_3\mid 5$ and $n_3\equiv 1(\mathrm{mod}3)$. The divisors of 5 are {1, 5}. Since $5\equiv 2(\mathrm{mod}3)$, we must have $n_3=1$.

For Sylow 5-subgroups: $n_5\mid 3$ and $n_5\equiv 1(\mathrm{mod}5)$. The divisors of 3 are {1, 3}. Since $3\equiv 3(\mathrm{mod}5)$, we must have $n_5=1$.

Let $P$ be the unique Sylow 3-subgroup and $Q$ be the unique Sylow 5-subgroup. Since they are unique, they are normal in $G$ (as $n_p=1$ implies $gP{g}^{-1}=P$ for all $g\in G$).

We have $P\cap Q=\{e\}$ (since $|P\cap Q|$ divides both 3 and 5, hence equals 1), and $|PQ|=\frac{|P||Q|}{|P\cap Q|}=\frac{3\cdot 5}{1}=15=|G|$, so $G=PQ$.

By Cauchy's theorem, $|P|=3$ (prime) implies $P\cong {\mathbb{Z}}_3$, and $|Q|=5$ implies $Q\cong {\mathbb{Z}}_5$. Since $P,Q$ are both normal and $P\cap Q=\{e\}$, we have $G\cong P\times Q\cong {\mathbb{Z}}_3\times {\mathbb{Z}}_5\cong {\mathbb{Z}}_{15}$ by the Chinese Remainder Theorem.

2

Let $G=A_5$, the group of even permutations on 5 elements. Then $|A_5|=60=2^2\cdot 3\cdot 5$.

A Sylow 5-subgroup has order 5, hence is cyclic, generated by a 5-cycle. The number of 5-cycles in $A_5$ is $\frac{5!}{5}=24$ (since there are $5!/5=24$ distinct 5-cycles, all even permutations). Each cyclic group of order 5 contains $\varphi (5)=4$ generators, so the number of Sylow 5-subgroups is $24/4=6$. Thus $n_5=6$.

This analysis is crucial for proving $A_5$ is simple: if $N◃A_5$ is normal and nontrivial, then by Sylow's second theorem (conjugacy), $N$ must contain entire conjugacy classes of Sylow subgroups, leading to contradictions with $|N|$ dividing 60.

Consider $G=S_3$ with $|G|=6=2\cdot 3$. There are three subgroups of order 2 (each generated by a transposition): $H_1=⟨(12)⟩$, $H_2=⟨(13)⟩$, $H_3=⟨(23)⟩$.

These are not conjugate to each other in the sense that all three exist simultaneously—but this doesn't violate Sylow's second theorem because they are all conjugate! Indeed: $(13)(12)(13)=(23)$ and $(12)(13)(12)=(23)$.

However, if we mistakenly tried to apply "conjugacy" to the non-maximal subgroup $\{e\}$ (order $p^0=1$), we'd trivially have only one such subgroup. The theorem's power is specifically for Sylow $p$-subgroups (maximal prime-power order).

A better counterexample: In ${\mathbb{Z}}_4\times {\mathbb{Z}}_2$ (order 8 = $2^3$), consider subgroups of order 2. We have $⟨(2,0)⟩$, $⟨(0,1)⟩$, and $⟨(2,1)⟩$—these are not all conjugate (the group is abelian!). But the unique Sylow 2-subgroup is the entire group, so there's no contradiction. Conjugacy applies only to Sylow subgroups, not arbitrary $p$-power subgroups.

The conjugacy theorem says that from the group's "symmetry perspective," all these maximal $p$-clumps look identical—they're related by the group's internal symmetries (conjugation). You can't distinguish between them using the group structure alone.

The counting theorem provides a "crystallographic restriction": the number of such clumps is constrained by both divisibility ($n_p\mid m$, the "$p$-free part") and by a periodicity condition ($n_p\equiv 1(\mathrm{mod}p)$). This is remarkably restrictive—often forcing $n_p=1$, which means that $p$-clump is actually frozen in place (normal), completely constraining the group's structure.

Historical Development

Ludwig Sylow published his theorems in 1872 in the paper "Théorèmes sur les groupes de substitutions" in Mathematische Annalen. At the time, group theory was still in its infancy, with groups primarily understood as permutation groups (substitution groups). Sylow's work built on earlier results by Augustin-Louis Cauchy, who had proven in 1845 that if a prime $p$ divides $|G|$, then $G$ contains an element of order $p$.

The context of Sylow's work was the broader program of understanding when polynomial equations are solvable by radicals, initiated by Évariste Galois. Sylow's theorems provided powerful tools for analyzing the structure of Galois groups, which are finite groups in many classical cases.

Initially, Sylow's proofs used detailed combinatorial arguments about permutation groups. The modern proof via group actions on coset spaces was developed later, particularly through the work of Ferdinand Georg Frobenius in the 1880s-1890s. Frobenius systematized the theory of group actions and the orbit-stabilizer theorem, providing the conceptual framework we use today.

The notation and terminology evolved over time. The term "Sylow $p$-subgroup" became standard in the early 20th century. William Burnside's 1897 treatise Theory of Groups of Finite Order was instrumental in disseminating these results and establishing them as foundational in finite group theory.

Connections to Other Areas

What You Need First

What This Opens Up

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