Sylow theorems
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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 (1872) 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 p, a Sylow psubgroup (sometimes pSylow subgroup) of a group G is a maximal psubgroup of G, i.e., a subgroup of G that is a pgroup (so that the order of every group element is a power of p) that is not a proper subgroup of any other psubgroup of G. The set of all Sylow psubgroups for a given prime p is sometimes written Syl_{p}(G).
The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group G the order (number of elements) of every subgroup of G divides the order of G; the Sylow theorems state that for every prime factor p of the order of a finite group G, there exists a Sylow psubgroup of G of order p^{n}, the highest power of p that divides the order of G. Moreover, every subgroup of order p^{n} is a Sylow psubgroup of G, and the Sylow psubgroups of a group (for a given prime p) are conjugate to each other. Furthermore, the number of Sylow psubgroups of a group for a given prime p is congruent to 1 mod p.
Contents
Theorems[edit]
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 Syl_{p}(G), all members are actually isomorphic to each other and have the largest possible order: if G = p^{n}m with n > 0 where p does not divide m, then every Sylow psubgroup P has order P = p^{n}. That is, P is a pgroup and 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 psubgroup of G, of order p^{n}.
The following weaker version of theorem 1 was first proved by AugustinLouis 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 hence a subgroup) of order p in G.^{[1]}
Theorem 2: Given a finite group G and a prime number p, all Sylow psubgroups of G are conjugate to each other, i.e. if H and K are Sylow psubgroups of G, then there exists an element g in G with 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 p^{n}m, where n > 0 and p does not divide m. Let n_{p} be the number of Sylow psubgroups of G. Then the following hold:
 n_{p} divides m, which is the index of the Sylow psubgroup in G.
 n_{p} ≡ 1 (mod p).
 n_{p} = G : N_{G}(P), where P is any Sylow psubgroup of G and N_{G} denotes the normalizer.
Consequences[edit]
The Sylow theorems imply that for a prime number p every Sylow psubgroup is of the same order, p^{n}. Conversely, if a subgroup has order p^{n}, then it is a Sylow psubgroup, and so is isomorphic to every other Sylow psubgroup. Due to the maximality condition, if H is any psubgroup of G, then H is a subgroup of a psubgroup of order p^{n}.
A very important consequence of Theorem 3 is that the condition n_{p} = 1 is equivalent to saying that the Sylow psubgroup of G is a normal subgroup (there are groups that have normal subgroups but no normal Sylow subgroups, such as S_{4}).
Sylow theorems for infinite groups[edit]
There is an analogue of the Sylow theorems for infinite groups. We define a Sylow psubgroup in an infinite group to be a psubgroup (that is, every element in it has ppower order) that is maximal for inclusion among all psubgroups in the group; such subgroups exist by Zorn's lemma.
Theorem: If K is a Sylow psubgroup of G, and n_{p} = Cl(K) is finite, then every Sylow psubgroup is conjugate to K, and n_{p} ≡ 1 (mod p), where Cl(K) denotes the conjugacy class of K.
Examples[edit]
A simple illustration of Sylow subgroups and the Sylow theorems are the dihedral group of the ngon, D_{2n}. For n odd, 2 = 2^{1} 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.
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 psubgroups of GL_{2}(F_{q}), where p and q are primes ≥ 3 and p ≡ 1 (mod q) , which are all abelian. The order of GL_{2}(F_{q}) is (q^{2} − 1)(q^{2} − q) = (q)(q + 1)(q − 1)^{2}. Since q = p^{n}m + 1, the order of GL_{2}(F_{q}) = p^{2n} m′. Thus by Theorem 1, the order of the Sylow psubgroups is p^{2n}.
One such subgroup P, is the set of diagonal matrices , x is any primitive root of F_{q}. Since the order of F_{q} is q − 1, its primitive roots have order q − 1, which implies that x^{(q − 1)/pn} or x^{m} 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 p^{n} choices for both a and b, making P = p^{2n}. This means P is a Sylow psubgroup, which is abelian, as all diagonal matrices commute, and because Theorem 2 states that all Sylow psubgroups are conjugate to each other, the Sylow psubgroups of GL_{2}(F_{q}) are all abelian.
Example applications[edit]
Since Sylow's theorem ensures the existence of psubgroups of a finite group, its 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.
 Example1
 Groups of order pq, p and q primes with p < q.
 Example2
 Group of order 30, groups of order 20, groups of order p^{2}q, p and q distinct primes are some of the applications.
 Example3
 (Groups of order 60): If the order G = 60 and G has more than one Sylow 5subgroup, then G is simple.
Cyclic group orders[edit]
Some nonprime 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 n_{3} be the number of Sylow 3subgroups. Then n_{3} 5 and n_{3} ≡ 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, n_{5} must divide 3, and n_{5} 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[edit]
A more complex example involves the order of the smallest simple group that is not cyclic. Burnside's p^{a} q^{b} 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 n_{3} must divide 10 ( = 2 · 5), and n_{3} must equal 1 (mod 3). Therefore, n_{3} = 10, since neither 4 nor 7 divides 10, and if n_{3} = 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, n_{5} = 6, since n_{5} must divide 6 ( = 2 · 3), and n_{5} 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 n_{7} must divide 6 ( = 2 · 3) and n_{7} must equal 1 (mod 7), so n_{7} = 1. So, as before, G can not be simple.
On the other hand, for G = 60 = 2^{2} · 3 · 5, then n_{3} = 10 and n_{5} = 6 is perfectly possible. And in fact, the smallest simple noncyclic group is A_{5}, 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[edit]
Part of Wilson's theorem states that
for every prime p. One may easily prove this theorem by Sylow's third theorem. Indeed, observe that the number n_{p} of Sylow's psubgroups in the symmetric group S_{p} is (p − 2)!. On the other hand, n_{p} ≡ 1 (mod p). Hence, (p − 2)! ≡ 1 (mod p). So, (p − 1)! ≡ −1 (mod p).
Fusion results[edit]
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 psubgroup P and two subsets A and B normalized by P, then A and B are Gconjugate if and only if they are N_{G}(P)conjugate. The proof is a simple application of Sylow's theorem: If B=A^{g}, then the normalizer of B contains not only P but also P^{g} (since P^{g} is contained in the normalizer of A^{g}). By Sylow's theorem P and P^{g} 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 A^{gh−1} = B^{h−1} = B, so that A and B are N_{G}(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 psubgroup P is contained in the center of its normalizer, then G has a normal subgroup K of order coprime to P, G = PK and P∩K = {1}, that is, G is pnilpotent.
Less trivial applications of the Sylow theorems include the focal subgroup theorem, which studies the control a Sylow psubgroup 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 2subgroup is a quasidihedral 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[edit]
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 1980), (Scharlau 1988), (Casadio & Zappa 1990), (Gow 1994), and to some extent (Meo 2004).
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 psubgroups 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 1959). In the following, we use a b as notation for "a divides b" and a b for the negation of this statement.
Theorem 1: A finite group G whose order G is divisible by a prime power p^{k} has a subgroup of order p^{k}.
Proof: Let G = p^{k}m = p^{k+r}u such that p does not divide u, and let Ω denote the set of subsets of G of size p^{k}. G acts on Ω by left multiplication; the orbits Gω = {gω  g ∈ G} of the ω ∈ Ω are the equivalence classes under the action of G.
For any ω ∈ Ω consider its stabilizer subgroup G_{ω} = {g ∈ G  gω = ω}. For any fixed element α ∈ ω the function [g ↦ gα] maps G_{ω} to ω injectively: for any two g, h ∈ G_{ω} we have that gα = hα implies g = h, because α ∈ ω ⊆ G means that one may cancel on the right. Therefore, p^{k} = ω ≥ G_{ω}.
On the other hand,
and no power of p remains in any of the factors inside the product on the right. Hence ν_{p}(Ω) = ν_{p}(m) = r. Let R ⊆ Ω be a complete representation of all the equivalence classes under the action of G. Then,
Thus, there exists an element ω ∈ R such that s := ν_{p}(Gω) ≤ ν_{p}(Ω) = r. Hence Gω = p^{s}v where p does not divide v. By the orbitstabilizer theorem we have G_{ω} = G / Gω = p^{k+rs}u/v. Therefore, p^{k} G_{ω}, so p^{k} ≤ G_{ω} and G_{ω} is the desired subgroup.
Lemma: Let G be a finite pgroup, let Ω be a finite set, let Ω_{G} be the set generated by the action of G on all the elements of Ω, and let Ω_{0} denote the set of points of Ω_{G} that are fixed under the action of G. Then Ω_{G} ≡ Ω_{0} (mod p).
Proof: Write Ω_{G} as a disjoint sum of its orbits under G. Any element x ∈ Ω_{G} not fixed by G will lie in an orbit of order G/G_{x} (where G_{x} denotes the stabilizer), which is a multiple of p by assumption. The result follows immediately.
Theorem 2: If H is a psubgroup of G and P is a Sylow psubgroup of G, then there exists an element g in G such that g^{−1}Hg ≤ P. In particular, all Sylow psubgroups of G are conjugate to each other (and therefore isomorphic), that is, if H and K are Sylow psubgroups of G, then there exists an element g in G with g^{−1}Hg = 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 Ω_{0} ≡ Ω = [G : P] (mod p). Now p [G : P] by definition so p Ω_{0}, hence in particular Ω_{0} ≠ 0 so there exists some gP ∈ Ω_{0}. It follows that for some g ∈ G and ∀ h ∈ H we have hgP = gP so g^{−1}HgP = P and therefore g^{−1}Hg ≤ P. Now if H is a Sylow psubgroup, H = P = gPg^{−1} so that H = gPg^{−1} for some g ∈ G.
Theorem 3: Let q denote the order of any Sylow psubgroup P of a finite group G. Then n_{p} = G : N_{G}(P), n_{p} G/q and n_{p} ≡ 1 (mod p).
Proof: Let G act on P, a Sylow psubgroup, by conjugation. By the orbitstabilizer theorem, n_{p} = [G : Stab_{G}(P)]. Stab_{G}(P) = { g ∈ G  gPg^{−1} = P } = N_{G} (P), the normalizer of P in G. Thus, n_{p} = G : N_{G}(P), and it follows that this number is a divisor of G/q. Let Ω be the set of all Sylow psubgroups of G, and let P act on Ω by conjugation. Let Q ∈ Ω_{0} and observe that then Q = xQx^{−1} for all x ∈ P so that P ≤ N_{G}(Q). By Theorem 2, P and Q are conjugate in N_{G}(Q) in particular, and Q is normal in N_{G}(Q), so then P = Q. It follows that Ω_{0} = {P} so that, by the Lemma, Ω ≡ Ω_{0} = 1 (mod p).
Algorithms[edit]
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 psubgroups is constructive: if H is a psubgroup of G and the index [G:H] is divisible by p, then the normalizer N = N_{G}(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 psubgroup can be found by starting from any psubgroup H (including the identity) and taking elements of ppower 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 1991, Chapter 16), including the algorithm described in (Cannon 1971). These versions are still used in the GAP computer algebra system.
In permutation groups, it has been proven in (Kantor 1985a, 1985b, 1990; Kantor & Taylor 1988) that a Sylow psubgroup 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 2003), 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.
See also[edit]
Notes[edit]
 ^ Fraleigh, Victor J. Katz. A First Course In Abstract Algebra. p. 322. ISBN 9788178089973
References[edit]
 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
Proofs[edit]
 Casadio, Giuseppina; Zappa, Guido (1990), "History of the Sylow theorem and its proofs", Boll. Storia Sci. Mat. (in Italian), 10 (1): 29–75, ISSN 03924432, MR 1096350, Zbl 0721.01008
 Gow, Rod (1994), "Sylow's proof of Sylow's theorem", Irish Math. Soc. Bull. (33): 55–63, ISSN 07915578, 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 01687433, MR 1721912, Zbl 0943.68149, archived from the original (PDF) on 20060103
 Meo, M. (2004), "The mathematical life of Cauchy's group theorem", Historia Math., 31 (2): 196–221, doi:10.1016/S03150860(03)00003X, ISSN 03150860, MR 2055642, Zbl 1065.01009
 Scharlau, Winfried (1988), "Die Entdeckung der SylowSätze", Historia Math. (in German), 15 (1): 40–52, doi:10.1016/03150860(88)900481, ISSN 03150860, 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 00039519, MR 0575718, 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 00039268, MR 0147529, Zbl 0092.02403
Algorithms[edit]
 Butler, G. (1991), Fundamental Algorithms for Permutation Groups, Lecture Notes in Computer Science, 559, Berlin, New York: SpringerVerlag, doi:10.1007/3540549552, ISBN 9783540549550, MR 1225579, Zbl 0785.20001
 Cannon, John J. (1971), "Computing local structure of large finite groups", Computers in Algebra and Number Theory (Proc. SIAMAMS Sympos. Appl. Math., New York, 1970), SIAMAMS Proc., 4, Providence, RI: AMS, pp. 161–176, ISSN 01607634, MR 0367027, Zbl 0253.20027
 Kantor, William M. (1985a), "Polynomialtime algorithms for finding elements of prime order and Sylow subgroups" (PDF), J. Algorithms, 6 (4): 478–514, CiteSeerX 10.1.1.74.3690, doi:10.1016/01966774(85)90029X, ISSN 01966774, MR 0813589, Zbl 0604.20001
 Kantor, William M. (1985b), "Sylow's theorem in polynomial time", J. Comput. Syst. Sci., 30 (3): 359–394, doi:10.1016/00220000(85)900522, ISSN 10902724, MR 0805654, Zbl 0573.20022
 Kantor, William M.; Taylor, Donald E. (1988), "Polynomialtime versions of Sylow's theorem", J. Algorithms, 9 (1): 1–17, doi:10.1016/01966774(88)900028, ISSN 01966774, MR 0925595, Zbl 0642.20019
 Kantor, William M. (1990), "Finding Sylow normalizers in polynomial time", J. Algorithms, 11 (4): 523–563, doi:10.1016/01966774(90)900094, ISSN 01966774, MR 1079450, Zbl 0731.20005
 Seress, Ákos (2003), Permutation Group Algorithms, Cambridge Tracts in Mathematics, 152, Cambridge University Press, ISBN 9780521661034, MR 1970241, Zbl 1028.20002
External links[edit]
 Hazewinkel, Michiel, ed. (2001) [1994], "Sylow theorems", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 Abstract Algebra/Group Theory/The Sylow Theorems at Wikibooks
 Weisstein, Eric W. "Sylow pSubgroup". MathWorld.
 Weisstein, Eric W. "Sylow Theorems". MathWorld.