Sylow theorems - Math.SX

The Sylow theorems describe the existence, conjugacy, containment, and number of maximal prime-power subgroups of a finite group. They turn the arithmetic factorization of the group order into structural information and provide an important partial converse to Lagrange's theorem: although an arbitrary divisor of a group order need not occur as the order of a subgroup, every prime-power divisor does. ^{[@isaacs2008]}^[@sylow1872]

Definitions and notation

Let $$G$$ be a finite group, let $$p$$ be prime, and write uniquely

|G|=p^a m,\qquad a\geq0,\qquad p\nmid m.

The set of Sylow $$p$$ -subgroups is denoted by $\operatorname{Syl}_p(G)$ , and its cardinality by $$n_p(G)$$ or simply $$n_p$$ . Once the theorems have been proved, a Sylow $$p$$ -subgroup can equivalently be characterized as a maximal $$p$$ -subgroup under inclusion. Using that characterization prematurely, however, can make a proof of existence circular.

Statement

Textbooks differ in how they number the three parts. Isaacs calls existence, conjugacy, and containment Sylow E, C, and D respectively. The organization here follows logical content rather than treating those labels as part of the theorem. ^{[@isaacs2008]}

Wielandt's existence proof

The proof combines a binomial congruence with a nonconstructive group action. Wielandt published this short approach in 1954, and Isaacs uses it as his principal existence proof. ^{[@wielandt1954]}^{[@isaacs2008]}

Proof of Sylow existence

Let $\Omega$ be the set of all $$p^a$$ -element subsets of $$G$$ . Right multiplication gives an action of $$G$$ on $\Omega$ . The lemma yields

|\Omega|=\binom{p^a m}{p^a}\equiv m\not\equiv0\pmod p.

Since $\Omega$ is the disjoint union of its $$G$$ -orbits, at least one orbit $\mathcal O$ has size not divisible by $$p$$ . Choose $X\in\mathcal O$ and set $$H=G_X$$ . Orbit-stabilizer gives $|\mathcal O|=|G:H|$ . Because $$p^a$$ divides $$|G|$$ but not $$|G:H|$$ , it follows that $p^a\mid|H|$ .

On the other hand, $$H$$ stabilizes $$X$$ . For any $x\in X$ , the whole coset $$xH$$ lies in $$X$$ , so

|H|=|xH|\leq|X|=p^a.

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

To obtain the smaller orders $$p^b$$ , work inside a Sylow subgroup. A nontrivial finite $$p$$ -group has nontrivial center and admits a normal chain with successive index $$p$$ ; this produces subgroups of every required order. ^{[@isaacs2008]}

Containment and conjugacy

This also proves that every maximal $$p$$ -subgroup under inclusion is Sylow: otherwise it would be properly contained in one.

Counting Sylow subgroups

Conjugation makes $$G$$ act transitively on $\operatorname{Syl}_p(G)$ . The stabilizer of $$P$$ is $$N_G(P)$$ , so

\[ n_p(G)=|G:N_G(P)|. \]

Since $P\leq N_G(P)$ , this index divides $$|G:P|=m$$ .

Immediate consequences

A Sylow $$p$$ -subgroup is normal if and only if $$n_p(G)=1$$ .
A unique Sylow subgroup is characteristic, since automorphisms preserve subgroup order.
Cauchy's theorem follows: a nontrivial Sylow subgroup contains an element whose suitable power has order $$p$$ .
If $N\triangleleft G$ and $P\in\operatorname{Syl}_p(N)$ , then the Frattini argument gives

\[ G=N_G(P)N. \]

Indeed, $$P^g$$ is Sylow in $$N$$ and hence is conjugate to $$P$$ by an element of $$N$$ . ^{[@isaacs2008]}

Under a surjective homomorphism, Sylow subgroups map to Sylow subgroups of the image, and every Sylow subgroup of the image is obtained this way from some Sylow subgroup of the source.

Examples

Application to groups of order pq

Let $$p>q$$ be prime and $$|G|=pq$$ . Then $n_p\mid q$ and $n_p\equiv1\pmod p$ . Since $$q<p$$ , one must have $$n_p=1$$ , so the Sylow $$p$$ -subgroup is normal. Moreover, $$n_q$$ is either $$1$$ or $$p$$ . If $q\nmid p-1$ , then $p\not\equiv1\pmod q$ , forcing $$n_q=1$$ as well. The two normal Sylow subgroups intersect trivially and centralize each other, hence

G\cong C_p\times C_q\cong C_{pq}.

When $q\mid p-1$ , a nontrivial semidirect product $C_p\rtimes C_q$ may also occur. Thus Sylow counting reduces a classification problem to an automorphism-action problem. ^{[@isaacs2008]}

Similar arguments for orders $$p^2q$$ , $$p^3q$$ , $$30$$ , and $$56$$ often force a normal Sylow subgroup or rule out simplicity.

Results extracted from the exercises

McKay's proof of Cauchy's theorem

Suppose $p\mid|G|$ and define

\Omega=\{(x_1,\ldots,x_p)\in G^p:x_1x_2\cdots x_p=1\}.

The first $$p-1$$ entries are arbitrary and the final one is determined, so $|\Omega|=|G|^{p-1}$ is divisible by $$p$$ . The cyclic group $$C_p$$ acts by cyclically rotating coordinates. Its fixed points are precisely the constant tuples $(x,\ldots,x)$ with $$x^p=1$$ . All nonfixed orbits have size $$p$$ , so the number of fixed points is divisible by $$p$$ . The identity supplies one solution; consequently the number of nonidentity elements satisfying $$x^p=1$$ is congruent to $$-1$$ modulo $$p$$ , and in particular an element of order $$p$$ exists. This is the stronger counting statement developed in Problem 1A.8. ^{[@isaacs2008]}

Growing a p-subgroup using Cauchy's theorem

Let $$P$$ be a $$p$$ -subgroup with $p\mid|G:P|$ . Orbit counting on the right cosets of $$P$$ shows that $$p$$ divides $$|N_G(P):P|$$ . Cauchy's theorem applied to $$N_G(P)/P$$ gives a subgroup of order $$p$$ in the quotient. Its inverse image is a $$p$$ -subgroup $$Q$$ with $$P<Q$$ and $$|Q:P|=p$$ . Repetition reaches a Sylow subgroup and gives an alternative existence proof. ^{[@isaacs2008]}

Limitations and common errors

“Sylow $$p$$ -subgroup” does not imply uniqueness; uniqueness is equivalent to normality.
Sylow subgroups for distinct primes need not all be normal. If they are all normal, the finite group is their internal direct product and is nilpotent.
For infinite groups, defining Sylow $$p$$ -subgroups as maximal $$p$$ -subgroups does not preserve the finite conjugacy and counting conclusions in general.
Hall $\pi$ -subgroups generalize from one prime to a set of primes, but existence and conjugacy can fail outside finite solvable groups. See Hall subgroup.

History and formalization

Peter Ludvig Sylow published the original theorem in 1872. Wielandt later supplied the short proof based on the action on fixed-size subsets. Modern theorem-prover developments have formalized the dependency chain through finite groups, permutation groups, and Sylow theory, demonstrating that these arguments admit machine-checkable decompositions. ^[@sylow1872]^{[@wielandt1954]}^{[@russinoff2023]}

The page structure, historical pointers, and choice of examples were cross-checked against Wikipedia and the Encyclopedia of Mathematics; the mathematical exposition, proofs, and exercise-derived results were independently organized from the cited sources. The cited Wikipedia revision is attributed under CC BY-SA. ^{[@wikipedia-sylow-zh]}^[@eom-sylow]