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Sylow theorems

Summary

In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow[1] that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups.

For a prime number ${\displaystyle p}$, a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group ${\displaystyle G}$ is a maximal ${\displaystyle p}$-subgroup of ${\displaystyle G}$, i.e., a subgroup of ${\displaystyle G}$ that is a p-group (meaning its cardinality is a power of ${\displaystyle p,}$ or equivalently, the order of every group element is a power of ${\displaystyle p}$) that is not a proper subgroup of any other ${\displaystyle p}$-subgroup of ${\displaystyle G}$. The set of all Sylow ${\displaystyle p}$-subgroups for a given prime ${\displaystyle p}$ is sometimes written ${\displaystyle {\text{Syl}}_{p}(G)}$.

The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group ${\displaystyle G}$ the order (number of elements) of every subgroup of ${\displaystyle G}$ divides the order of ${\displaystyle G}$. The Sylow theorems state that for every prime factor ${\displaystyle p}$ of the order of a finite group ${\displaystyle G}$, there exists a Sylow ${\displaystyle p}$-subgroup of ${\displaystyle G}$ of order ${\displaystyle p^{n}}$, the highest power of ${\displaystyle p}$ that divides the order of ${\displaystyle G}$. Moreover, every subgroup of order ${\displaystyle p^{n}}$ is a Sylow ${\displaystyle p}$-subgroup of ${\displaystyle G}$, and the Sylow ${\displaystyle p}$-subgroups of a group (for a given prime ${\displaystyle p}$) are conjugate to each other. Furthermore, the number of Sylow ${\displaystyle p}$-subgroups of a group for a given prime ${\displaystyle p}$ is congruent to ${\displaystyle 1{\text{ mod }}p}$.

Theorems

Motivation

The Sylow theorems are a powerful statement about the structure of groups in general, but are also powerful in applications of finite group theory. This is because they give a method for using the prime decomposition of the cardinality of a finite group ${\displaystyle G}$ to give statements about the structure of its subgroups: essentially, it gives a technique to transport basic number-theoretic information about a group to its group structure. From this observation, classifying finite groups becomes a game of finding which combinations/constructions of groups of smaller order can be applied to construct a group. For example, a typical application of these theorems is in the classification of finite groups of some fixed cardinality, e.g. ${\displaystyle |G|=60}$.[2]

Statement

Collections of subgroups that are each maximal in one sense or another are common in group theory. The surprising result here is that in the case of ${\displaystyle \operatorname {Syl} _{p}(G)}$, all members are actually isomorphic to each other and have the largest possible order: if ${\displaystyle |G|=p^{n}m}$ with ${\displaystyle n>0}$ where p does not divide m, then every Sylow p-subgroup P has order ${\displaystyle |P|=p^{n}}$. That is, P is a p-group and ${\displaystyle {\text{gcd}}(|G:P|,p)=1}$. These properties can be exploited to further analyze the structure of G.

The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in Mathematische Annalen.

Theorem (1) — For every prime factor p with multiplicity n of the order of a finite group G, there exists a Sylow p-subgroup of G, of order ${\displaystyle p^{n}}$.

The following weaker version of theorem 1 was first proved by Augustin-Louis Cauchy, and is known as Cauchy's theorem.

Corollary — Given a finite group G and a prime number p dividing the order of G, then there exists an element (and thus a cyclic subgroup generated by this element) of order p in G.[3]

Theorem (2) — Given a finite group G and a prime number p, all Sylow p-subgroups of G are conjugate to each other. That is, if H and K are Sylow p-subgroups of G, then there exists an element ${\displaystyle g\in G}$ with ${\displaystyle g^{-1}Hg=K}$.

Theorem (3) — Let p be a prime factor with multiplicity n of the order of a finite group G, so that the order of G can be written as ${\displaystyle p^{n}m}$, where ${\displaystyle n>0}$ and p does not divide m. Let ${\displaystyle n_{p}}$ be the number of Sylow p-subgroups of G. Then the following hold:

• ${\displaystyle n_{p}}$ divides m, which is the index of the Sylow p-subgroup in G.
• ${\displaystyle n_{p}\equiv 1{\bmod {p}}}$
• ${\displaystyle n_{p}=|G:N_{G}(P)|}$, where P is any Sylow p-subgroup of G and ${\displaystyle N_{G}}$ denotes the normalizer.

Consequences

The Sylow theorems imply that for a prime number ${\displaystyle p}$ every Sylow ${\displaystyle p}$-subgroup is of the same order, ${\displaystyle p^{n}}$. Conversely, if a subgroup has order ${\displaystyle p^{n}}$, then it is a Sylow ${\displaystyle p}$-subgroup, and so is isomorphic to every other Sylow ${\displaystyle p}$-subgroup. Due to the maximality condition, if ${\displaystyle H}$ is any ${\displaystyle p}$-subgroup of ${\displaystyle G}$, then ${\displaystyle H}$ is a subgroup of a ${\displaystyle p}$-subgroup of order ${\displaystyle p^{n}}$.

A very important consequence of Theorem 2 is that the condition ${\displaystyle n_{p}=1}$ is equivalent to saying that the Sylow ${\displaystyle p}$-subgroup of ${\displaystyle G}$ is a normal subgroup. However, there are groups that have normal subgroups but no normal Sylow subgroups, such as ${\displaystyle S_{4}}$.

Sylow theorems for infinite groups

There is an analogue of the Sylow theorems for infinite groups. One defines a Sylow p-subgroup in an infinite group to be a p-subgroup (that is, every element in it has p-power order) that is maximal for inclusion among all p-subgroups in the group. Such subgroups exist by Zorn's lemma. Let ${\displaystyle \operatorname {Cl} (K)}$ denote the set of conjugacy classes of a subgroup ${\displaystyle K\subset G}$

Theorem — If K is a Sylow p-subgroup of G, and ${\displaystyle n_{p}=|\operatorname {Cl} (K)|}$ is finite, then every Sylow p-subgroup is conjugate to K, and ${\displaystyle n_{p}\equiv 1{\bmod {p}}}$.

Examples

In D6 all reflections are conjugate, as reflections correspond to Sylow 2-subgroups.

A simple illustration of Sylow subgroups and the Sylow theorems are the dihedral group of the n-gon, D2n. For n odd, 2 = 21 is the highest power of 2 dividing the order, and thus subgroups of order 2 are Sylow subgroups. These are the groups generated by a reflection, of which there are n, and they are all conjugate under rotations; geometrically the axes of symmetry pass through a vertex and a side.

In D12 reflections no longer correspond to Sylow 2-subgroups, and fall into two conjugacy classes.

By contrast, if n is even, then 4 divides the order of the group, and the subgroups of order 2 are no longer Sylow subgroups, and in fact they fall into two conjugacy classes, geometrically according to whether they pass through two vertices or two faces. These are related by an outer automorphism, which can be represented by rotation through π/n, half the minimal rotation in the dihedral group.

Another example are the Sylow p-subgroups of GL2(Fq), where p and q are primes ≥ 3 and p ≡ 1 (mod q) , which are all abelian. The order of GL2(Fq) is (q2 − 1)(q2 − q) = (q)(q + 1)(q − 1)2. Since q = pnm + 1, the order of GL2(Fq) = p2n m. Thus by Theorem 1, the order of the Sylow p-subgroups is p2n.

One such subgroup P, is the set of diagonal matrices ${\displaystyle {\begin{bmatrix}x^{im}&0\\0&x^{jm}\end{bmatrix}}}$, x is any primitive root of Fq. Since the order of Fq is q − 1, its primitive roots have order q − 1, which implies that x(q − 1)/pn or xm and all its powers have an order which is a power of p. So, P is a subgroup where all its elements have orders which are powers of p. There are pn choices for both a and b, making |P| = p2n. This means P is a Sylow p-subgroup, which is abelian, as all diagonal matrices commute, and because Theorem 2 states that all Sylow p-subgroups are conjugate to each other, the Sylow p-subgroups of GL2(Fq) are all abelian.

Example applications

Since Sylow's theorem ensures the existence of p-subgroups of a finite group, it's worthwhile to study groups of prime power order more closely. Most of the examples use Sylow's theorem to prove that a group of a particular order is not simple. For groups of small order, the congruence condition of Sylow's theorem is often sufficient to force the existence of a normal subgroup.

Example-1
Groups of order pq, p and q primes with p < q.
Example-2
Group of order 30, groups of order 20, groups of order p2q, p and q distinct primes are some of the applications.
Example-3
(Groups of order 60): If the order |G| = 60 and G has more than one Sylow 5-subgroup, then G is simple.

Cyclic group orders

Some non-prime numbers n are such that every group of order n is cyclic. One can show that n = 15 is such a number using the Sylow theorems: Let G be a group of order 15 = 3 · 5 and n3 be the number of Sylow 3-subgroups. Then n3 ${\displaystyle \mid }$ 5 and n3 ≡ 1 (mod 3). The only value satisfying these constraints is 1; therefore, there is only one subgroup of order 3, and it must be normal (since it has no distinct conjugates). Similarly, n5 must divide 3, and n5 must equal 1 (mod 5); thus it must also have a single normal subgroup of order 5. Since 3 and 5 are coprime, the intersection of these two subgroups is trivial, and so G must be the internal direct product of groups of order 3 and 5, that is the cyclic group of order 15. Thus, there is only one group of order 15 (up to isomorphism).

Small groups are not simple

A more complex example involves the order of the smallest simple group that is not cyclic. Burnside's pa qb theorem states that if the order of a group is the product of one or two prime powers, then it is solvable, and so the group is not simple, or is of prime order and is cyclic. This rules out every group up to order 30 (= 2 · 3 · 5).

If G is simple, and |G| = 30, then n3 must divide 10 ( = 2 · 5), and n3 must equal 1 (mod 3). Therefore, n3 = 10, since neither 4 nor 7 divides 10, and if n3 = 1 then, as above, G would have a normal subgroup of order 3, and could not be simple. G then has 10 distinct cyclic subgroups of order 3, each of which has 2 elements of order 3 (plus the identity). This means G has at least 20 distinct elements of order 3.

As well, n5 = 6, since n5 must divide 6 ( = 2 · 3), and n5 must equal 1 (mod 5). So G also has 24 distinct elements of order 5. But the order of G is only 30, so a simple group of order 30 cannot exist.

Next, suppose |G| = 42 = 2 · 3 · 7. Here n7 must divide 6 ( = 2 · 3) and n7 must equal 1 (mod 7), so n7 = 1. So, as before, G can not be simple.

On the other hand, for |G| = 60 = 22 · 3 · 5, then n3 = 10 and n5 = 6 is perfectly possible. And in fact, the smallest simple non-cyclic group is A5, the alternating group over 5 elements. It has order 60, and has 24 cyclic permutations of order 5, and 20 of order 3.

Wilson's theorem

Part of Wilson's theorem states that

${\displaystyle (p-1)!\ \equiv \ -1{\pmod {p}}}$

for every prime p. One may easily prove this theorem by Sylow's third theorem. Indeed, observe that the number np of Sylow's p-subgroups in the symmetric group Sp is (p − 2)!. On the other hand, np ≡ 1 (mod p). Hence, (p − 2)! ≡ 1 (mod p). So, (p − 1)! ≡ −1 (mod p).

Fusion results

Frattini's argument shows that a Sylow subgroup of a normal subgroup provides a factorization of a finite group. A slight generalization known as Burnside's fusion theorem states that if G is a finite group with Sylow p-subgroup P and two subsets A and B normalized by P, then A and B are G-conjugate if and only if they are NG(P)-conjugate. The proof is a simple application of Sylow's theorem: If B=Ag, then the normalizer of B contains not only P but also Pg (since Pg is contained in the normalizer of Ag). By Sylow's theorem P and Pg are conjugate not only in G, but in the normalizer of B. Hence gh−1 normalizes P for some h that normalizes B, and then Agh−1 = Bh−1 = B, so that A and B are NG(P)-conjugate. Burnside's fusion theorem can be used to give a more powerful factorization called a semidirect product: if G is a finite group whose Sylow p-subgroup P is contained in the center of its normalizer, then G has a normal subgroup K of order coprime to P, G = PK and PK = {1}, that is, G is p-nilpotent.

Less trivial applications of the Sylow theorems include the focal subgroup theorem, which studies the control a Sylow p-subgroup of the derived subgroup has on the structure of the entire group. This control is exploited at several stages of the classification of finite simple groups, and for instance defines the case divisions used in the Alperin–Brauer–Gorenstein theorem classifying finite simple groups whose Sylow 2-subgroup is a quasi-dihedral group. These rely on J. L. Alperin's strengthening of the conjugacy portion of Sylow's theorem to control what sorts of elements are used in the conjugation.

Proof of the Sylow theorems

The Sylow theorems have been proved in a number of ways, and the history of the proofs themselves is the subject of many papers, including Waterhouse,[4] Scharlau,[5] Casadio and Zappa,[6] Gow,[7] and to some extent Meo.[8]

One proof of the Sylow theorems exploits the notion of group action in various creative ways. The group G acts on itself or on the set of its p-subgroups in various ways, and each such action can be exploited to prove one of the Sylow theorems. The following proofs are based on combinatorial arguments of Wielandt.[9] In the following, we use ${\displaystyle a\mid b}$ as notation for "a divides b" and ${\displaystyle a\nmid b}$ for the negation of this statement.

Theorem (1) — A finite group G whose order ${\displaystyle |G|}$ is divisible by a prime power pk has a subgroup of order pk.

Proof

Let |G| = pkm = pk+ru such that ${\displaystyle p\nmid u}$, and let Ω denote the set of subsets of G of size pk. G acts on Ω by left multiplication: for gG and ω ∈ Ω, gω = { gx | xω }. For a given set ω ∈ Ω, write Gω for its stabilizer subgroup { gG | gω = ω } and Gω for its orbit { gω | gG } in Ω.

The proof will show the existence of some ω ∈ Ω for which Gω has pk elements, providing the desired subgroup. This is the maximal possible size of a stabilizer subgroup Gω, since for any fixed element αωG, the right coset Gωα is contained in ω; therefore, |Gω| = |Gωα| ≤ |ω| = pk.

By the orbit-stabilizer theorem we have |Gω| |Gω| = |G| for each ω ∈ Ω, and therefore using the additive p-adic valuation νp, which counts the number of factors p, one has νp(|Gω|) + νp(|Gω|) = νp(|G|) = k + r. This means that for those ω with |Gω| = pk, the ones we are looking for, one has νp(|Gω|) = r, while for any other ω one has νp(|Gω|) > r (as 0 < |Gω| < pk implies νp(|Gω|) < k). Since |Ω| is the sum of |Gω| over all distinct orbits Gω, one can show the existence of ω of the former type by showing that νp(|Ω|) = r (if none existed, that valuation would exceed r). This is an instance of Kummer's theorem (since in base p notation the number |G| ends with precisely k + r digits zero, subtracting pk from it involves a carry in r places), and can also be shown by a simple computation:

${\displaystyle |\Omega |={p^{k}m \choose p^{k}}=\prod _{j=0}^{p^{k}-1}{\frac {p^{k}m-j}{p^{k}-j}}=m\prod _{j=1}^{p^{k}-1}{\frac {p^{k-\nu _{p}(j)}m-j/p^{\nu _{p}(j)}}{p^{k-\nu _{p}(j)}-j/p^{\nu _{p}(j)}}}}$

and no power of p remains in any of the factors inside the product on the right. Hence νp(|Ω|) = νp(m) = r, completing the proof.

It may be noted that conversely every subgroup H of order pk gives rise to sets ω ∈ Ω for which Gω = H, namely any one of the m distinct cosets Hg.

Lemma — Let H be a finite p-group, let Ω be a finite set acted on by H, and let Ω0 denote the set of points of Ω that are fixed under the action of H. Then |Ω| ≡ |Ω0| (mod p).

Proof

Any element x ∈ Ω not fixed by H will lie in an orbit of order |H|/|Hx| (where Hx denotes the stabilizer), which is a multiple of p by assumption. The result follows immediately by writing |Ω| as the sum of |Hx| over all distinct orbits Hx and reducing mod p.

Theorem (2) — If H is a p-subgroup of G and P is a Sylow p-subgroup of G, then there exists an element g in G such that g−1HgP. In particular, all Sylow p-subgroups of G are conjugate to each other (and therefore isomorphic), that is, if H and K are Sylow p-subgroups of G, then there exists an element g in G with g−1Hg = K.

Proof

Let Ω be the set of left cosets of P in G and let H act on Ω by left multiplication. Applying the Lemma to H on Ω, we see that |Ω0| ≡ |Ω| = [G : P] (mod p). Now ${\displaystyle p\nmid [G:P]}$ by definition so ${\displaystyle p\nmid |\Omega _{0}|}$, hence in particular |Ω0| ≠ 0 so there exists some gP ∈ Ω0. With this gP, we have hgP = gP for all hH, so g−1HgP = P and therefore g−1HgP. Furthermore, if H is a Sylow p-subgroup, then |g−1Hg| = |H| = |P| so that g−1Hg = P.

Theorem (3) — Let q denote the order of any Sylow p-subgroup P of a finite group G. Let np denote the number of Sylow p-subgroups of G. Then (a) np = [G : NG(P)] (where NG(P) is the normalizer of P), (b) np divides |G|/q, and (c) np ≡ 1 (mod p).

Proof

Let Ω be the set of all Sylow p-subgroups of G and let G act on Ω by conjugation. Let P ∈ Ω be a Sylow p-subgroup. By Theorem 2, the orbit of P has size np, so by the orbit-stabilizer theorem np = [G : GP]. For this group action, the stabilizer GP is given by {gG | gPg−1 = P} = NG(P), the normalizer of P in G. Thus, np = [G : NG(P)], and it follows that this number is a divisor of [G : P] = |G|/q.

Now let P act on Ω by conjugation, and again let Ω0 denote the set of fixed points of this action. Let Q ∈ Ω0 and observe that then Q = xQx−1 for all xP so that PNG(Q). By Theorem 2, P and Q are conjugate in NG(Q) in particular, and Q is normal in NG(Q), so then P = Q. It follows that Ω0 = {P} so that, by the Lemma, |Ω| ≡ |Ω0| = 1 (mod p).

Algorithms

The problem of finding a Sylow subgroup of a given group is an important problem in computational group theory.

One proof of the existence of Sylow p-subgroups is constructive: if H is a p-subgroup of G and the index [G:H] is divisible by p, then the normalizer N = NG(H) of H in G is also such that [N : H] is divisible by p. In other words, a polycyclic generating system of a Sylow p-subgroup can be found by starting from any p-subgroup H (including the identity) and taking elements of p-power order contained in the normalizer of H but not in H itself. The algorithmic version of this (and many improvements) is described in textbook form in Butler,[10] including the algorithm described in Cannon.[11] These versions are still used in the GAP computer algebra system.

In permutation groups, it has been proven, in Kantor[12][13][14] and Kantor and Taylor,[15] that a Sylow p-subgroup and its normalizer can be found in polynomial time of the input (the degree of the group times the number of generators). These algorithms are described in textbook form in Seress,[16] and are now becoming practical as the constructive recognition of finite simple groups becomes a reality. In particular, versions of this algorithm are used in the Magma computer algebra system.

Notes

1. ^ Sylow, L. (1872). "Théorèmes sur les groupes de substitutions". Math. Ann. (in French). 5 (4): 584–594. doi:10.1007/BF01442913. JFM 04.0056.02. S2CID 121928336.
2. ^ Gracia–Saz, Alfonso. "Classification of groups of order 60" (PDF). math.toronto.edu. Archived (PDF) from the original on 28 October 2020. Retrieved 8 May 2021.
3. ^ Fraleigh, John B. (2004). A First Course In Abstract Algebra. with contribution by Victor J. Katz. Pearson Education. p. 322. ISBN 9788178089973.
4. ^
5. ^
6. ^
7. ^
8. ^
9. ^
10. ^ Butler 1991, Chapter 16.
11. ^
12. ^
13. ^
14. ^
15. ^
16. ^

References

Proofs

• Casadio, Giuseppina; Zappa, Guido (1990). "History of the Sylow theorem and its proofs". Boll. Storia Sci. Mat. (in Italian). 10 (1): 29–75. ISSN 0392-4432. MR 1096350. Zbl 0721.01008.
• Gow, Rod (1994). "Sylow's proof of Sylow's theorem". Irish Math. Soc. Bull.. 0033 (33): 55–63. doi:10.33232/BIMS.0033.55.63. ISSN 0791-5578. MR 1313412. Zbl 0829.01011.
• Kammüller, Florian; Paulson, Lawrence C. (1999). "A formal proof of Sylow's theorem. An experiment in abstract algebra with Isabelle HOL" (PDF). J. Automat. Reason.. 23 (3): 235–264. doi:10.1023/A:1006269330992. ISSN 0168-7433. MR 1721912. S2CID 1449341. Zbl 0943.68149. Archived from the original (PDF) on 2006-01-03.
• Meo, M. (2004). "The mathematical life of Cauchy's group theorem". Historia Math. 31 (2): 196–221. doi:10.1016/S0315-0860(03)00003-X. ISSN 0315-0860. MR 2055642. Zbl 1065.01009.
• Scharlau, Winfried (1988). "Die Entdeckung der Sylow-Sätze". Historia Math. (in German). 15 (1): 40–52. doi:10.1016/0315-0860(88)90048-1. ISSN 0315-0860. MR 0931678. Zbl 0637.01006.
• Waterhouse, William C. (1980). "The early proofs of Sylow's theorem". Arch. Hist. Exact Sci.. 21 (3): 279–290. doi:10.1007/BF00327877. ISSN 0003-9519. MR 0575718. S2CID 123685226. Zbl 0436.01006.
• Wielandt, Helmut (1959). "Ein Beweis für die Existenz der Sylowgruppen". Arch. Math. (in German). 10 (1): 401–402. doi:10.1007/BF01240818. ISSN 0003-9268. MR 0147529. S2CID 119816392. Zbl 0092.02403.

Algorithms

• Butler, G. (1991). Fundamental Algorithms for Permutation Groups. Lecture Notes in Computer Science. Vol. 559. Berlin, New York City: Springer-Verlag. doi:10.1007/3-540-54955-2. ISBN 9783540549550. MR 1225579. S2CID 395110. Zbl 0785.20001.
• Cannon, John J. (1971). "Computing local structure of large finite groups". Computers in Algebra and Number Theory (Proc. SIAM-AMS Sympos. Appl. Math., New York City, 1970). SIAM-AMS Proc.. Vol. 4. Providence RI: AMS. pp. 161–176. ISSN 0160-7634. MR 0367027. Zbl 0253.20027.
• Kantor, William M. (1985a). "Polynomial-time algorithms for finding elements of prime order and Sylow subgroups" (PDF). J. Algorithms. 6 (4): 478–514. CiteSeerX 10.1.1.74.3690. doi:10.1016/0196-6774(85)90029-X. ISSN 0196-6774. MR 0813589. Zbl 0604.20001.
• Kantor, William M. (1985b). "Sylow's theorem in polynomial time". J. Comput. Syst. Sci.. 30 (3): 359–394. doi:10.1016/0022-0000(85)90052-2. ISSN 1090-2724. MR 0805654. Zbl 0573.20022.
• Kantor, William M.; Taylor, Donald E. (1988). "Polynomial-time versions of Sylow's theorem". J. Algorithms. 9 (1): 1–17. doi:10.1016/0196-6774(88)90002-8. ISSN 0196-6774. MR 0925595. Zbl 0642.20019.
• Kantor, William M. (1990). "Finding Sylow normalizers in polynomial time". J. Algorithms. 11 (4): 523–563. doi:10.1016/0196-6774(90)90009-4. ISSN 0196-6774. MR 1079450. Zbl 0731.20005.
• Seress, Ákos (2003). Permutation Group Algorithms. Cambridge Tracts in Mathematics. Vol. 152. Cambridge University Press. ISBN 9780521661034. MR 1970241. Zbl 1028.20002.