Sylow theorems

Collections of subgroups which are each maximal in one sense or another are not uncommon 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| = pⁿm with $n>0$ where p does not divide m, then any Sylow p-subgroup P has order |P| = pⁿ. That is, P is a p-group 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 any prime factor p with multiplicity n of the order of a finite group G, there exists a Sylow p-subgroup of G, of order pⁿ.

The following weaker version of theorem 1 was first proved by Cauchy.

Corollary: Given a finite group G and a prime number p dividing the order of G, then there exists an element of order p in G .

Theorem 2: Given a finite group G and a prime number p, all Sylow p-subgroups of G are conjugate (and therefore isomorphic) to each other, i.e. if H and K are Sylow p-subgroups of G, then there exists an element g in G with g⁻¹Hg = 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 pⁿ · m, where n > 0 and p does not divide m. Let n_p be the number of Sylow p-subgroups of G. Then the following hold:

n_p divides m, which is the index of the Sylow p-subgroup in G.
n_p ≡ 1 mod p.
n_p = |G : N_G(P)|, where P is any Sylow p-subgroup of G and N_G denotes the normalizer.

Consequences

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

A very important consequence of Theorem 3 is that the condition n_p = 1 is equivalent to saying that the Sylow p-subgroup of G is a normal subgroup. (There are groups which have normal subgroups but no normal Sylow subgroups, such as S₄.)

Sylow theorems for infinite groups

There is an analogue of the Sylow theorems for infinite groups. We define a Sylow p-subgroup in an infinite group to be a p-subgroup (that is, every element in it has p-power order) which is maximal for inclusion among all p-subgroups in the group. Such subgroups exist by Zorn's lemma.

Theorem: If K is a Sylow p-subgroup of G, and n_p = |Cl(K)| is finite, then every Sylow p-subgroup is conjugate to K, and n_p ≡ 1 mod p, where Cl(K) denotes the conjugacy class of K.

Examples

In $D_6,$ all reflections are conjugate, as reflections correspond to Sylow 2-subgroups.

In $D_{12},$ reflections no longer correspond to Sylow 2-subgroups, and fall into two conjugacy classes.

A simple illustration of Sylow subgroups and the Sylow theorems are the dihedral group of the n-gon, $D_{2n}.$ For n odd, $2 = 2^1$ is the higher 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 these 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 $\pi/n,$ half the minimal rotation in the dihedral group.

Example applications

Cyclic group orders

Some 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 $n_3\mid 5$ and $n_3\equiv1\pmod3$ . 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 which is not cyclic. Burnside's p^aq^b theorem states that if the order of a group is the product of 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.

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 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 power 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 P∩K = 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 are the subject of many papers including (Waterhouse 1979), (Scharlau 1988), (Casadio & Zappa 1990), (Gow 1994), and to some extent (Meo 2004).

One proof of the Sylow theorems exploit 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 1959). In the following, we use a | b as notation for "a divides b" and a $\nmid$ 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^km, and let Ω denote the set of subsets of G of size p^k. Clearly:

$|\Omega | ={p^km \choose p^k}\mathrm{.}$

Let ν_p(|Ω|)=ν_p(m)=r and G act on Ω by left multiplication. Then,

$|\Omega | =\sum_{[o],\ o\in\Omega}|Go|\mathrm{.}$

where the sum is taken over the equivalence classes of Ω under the action of G. Thus, there exists an element A∈ Ω, such that ν_p(|GA|) ≤ ν_p(|Ω|). Let its orbit θ = GA, and by construction p^r+1 $\nmid$ |θ|. Now |θ| = |GA| = [G : G_A] where G_A denotes the stabilizer subgroup of the set A, hence p^k | |G_A| so p^k ≤ |G_A|. On the other hand, fix an element a ∈ A . For any two g,h ∈ G_A we have that ga = ha implies g=h, because a ∈ A ⊆ G means that a^-1 ∈ G, whereupon one may cancel on the right. Thus the function [g ↦ ga] maps G_A to A injectively; therefore |A| ≥ |G_A|, hence |G_A| = p^k and G_A is a desired subgroup.

Lemma: Let G be a finite p-group, let G act on a finite set Ω, and let Ω₀ denote the set of points of Ω that are fixed under the action of G. Then |Ω| ≡ |Ω₀| mod p.

Proof: Write Ω as a disjoint sum of its orbits under G. Any element x ∈ Ω 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 p-subgroup of G and P is a Sylow p-subgroup of G, then there exists an element g in G such that g⁻¹Hg ≤ P. In particular, all Sylow p-subgroups of G are conjugate to each other (and therefore isomorphic), i.e. 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 p $\nmid$ [G : P] by definition so p $\nmid$ |Ω₀|, hence in particular |Ω₀| ≠ 0 so there exists some gP ∈ Ω₀. It follows that for some g ∈ G and ∀ h ∈ H we have hgP = gP so g⁻¹hgP ⊆ P and therefore g⁻¹Hg ≤ P. Now if H is a Sylow p-subgroup, |H| = |P| = |gPg⁻¹| so that H = gPg⁻¹ for some g ∈ G.

Theorem 3: Let q denote the order of any Sylow p-subgroup of a finite group G. Then n_p | |G|/q and n_p ≡ 1 mod p.

Proof: By Theorem 2, n_p = [G : N_G(P)], where P is any such subgroup, and N_G(P) denotes the normalizer of P in G, so this number is a divisor of |G|/q. Let Ω be the set of all Sylow p-subgroups of G, and let P act on Ω by conjugation. Let Q ∈ Ω₀ 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 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 = 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 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 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, 1988, 1990) 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 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.

Notes

References

Sylow, L. (1872), "Théorèmes sur les groupes de substitutions", Mathematische Annalen 5: 584–594, doi:10.1007/BF01442913

Proofs

Casadio, Giuseppina; Zappa, Guido (1990), "History of the Sylow theorem and its proofs", Bollettino di Storia delle Scienze Matematiche 10 (1): 29–75, MR 1096350, ISSN 0392-4432
Gow, Rod (1994), "Sylow's proof of Sylow's theorem", Irish Mathematical Society Bulletin (33): 55–63, MR 1313412, ISSN 0791-5578
Kammüller, Florian; Paulson, Lawrence C. (1999), "A formal proof of Sylow's theorem. An experiment in abstract algebra with Isabelle HOL", Journal of Automated Reasoning 23 (3): 235–264, doi:10.1023/A:1006269330992, MR 1721912, ISSN 0168-7433, http://www.cl.cam.ac.uk/users/lcp/papers/Kammueller/sylow.pdf
Meo, M. (2004), "The mathematical life of Cauchy's group theorem", Historia Mathematica 31 (2): 196–221, doi:10.1016/S0315-0860(03)00003-X, MR 2055642, ISSN 0315-0860
Scharlau, Winfried (1988), "Die Entdeckung der Sylow-Sätze", Historia Mathematica 15 (1): 40–52, doi:10.1016/0315-0860(88)90048-1, MR 931678, ISSN 0315-0860
Waterhouse, William C. (1979), "The early proofs of Sylow's theorem", Archive for History of Exact Sciences 21 (3): 279–290, doi:10.1007/BF00327877, MR 575718, ISSN 0003-9519
Wielandt, Helmut (1959), "Ein Beweis für die Existenz der Sylowgruppen", Archiv der Mathematik 10: 401–402, doi:10.1007/BF01240818, MR 0147529, ISSN 0003-9268

Algorithms

Butler, G. (1991), Fundamental algorithms for permutation groups, Lecture Notes in Computer Science, 559, Berlin, New York: Springer-Verlag, MR 1225579, ISBN 978-3-540-54955-0
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, 1970), Providence, R.I.: Amer. Math. Soc., pp. 161–176, MR 0367027
Kantor, William M. (1985), "Polynomial-time algorithms for finding elements of prime order and Sylow subgroups", Journal of Algorithms 6 (4): 478–514, MR 813589, ISSN 0196-6774
Kantor, William M. (1985), "Sylow's theorem in polynomial time", Journal of Computer and System Sciences 30 (3): 359–394, doi:10.1016/0022-0000(85)90052-2, MR 805654, ISSN 1090-2724
Kantor, William M.; Taylor, Donald E. (1988), "Polynomial-time versions of Sylow's theorem", Journal of Algorithms 9 (1): 1–17, MR 925595, ISSN 0196-6774
Kantor, William M. (1990), "Finding Sylow normalizers in polynomial time", Journal of Algorithms 11 (4): 523–563, MR 1079450, ISSN 0196-6774
Seress, Ákos (2003), Permutation group algorithms, Cambridge Tracts in Mathematics, 152, Cambridge University Press, MR 1970241, ISBN 978-0-521-66103-4

Sylow theorems

Contents

Sylow p-subgroups

Sylow theorems

Consequences

Sylow theorems for infinite groups

Examples

Example applications

Cyclic group orders

Small groups are not simple

Fusion results

Proof of the Sylow theorems

Algorithms

See also

Notes

References

Proofs

Algorithms