Sylow theorems

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 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 formula_1, a Sylow "p"-subgroup (sometimes "p"-Sylow subgroup) of a group formula_2 is a maximal formula_1-subgroup of formula_2, i.e., a subgroup of formula_2 that is a "p"-group (so that the order of every group element is a power of formula_1) that is not a proper subgroup of any other formula_1-subgroup of formula_2. The set of all Sylow formula_1-subgroups for a given prime formula_1 is sometimes written formula_11.
The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group formula_2 the order (number of elements) of every subgroup of formula_2 divides the order of formula_2. The Sylow theorems state that for every prime factor "formula_1" of the order of a finite group formula_2, there exists a Sylow formula_1-subgroup of formula_2 of order formula_19, the highest power of formula_1 that divides the order of formula_2. Moreover, every subgroup of order "formula_19" is a Sylow "formula_1"-subgroup of formula_2, and the Sylow formula_1-subgroups of a group (for a given prime formula_1) are conjugate to each other. Furthermore, the number of Sylow formula_1-subgroups of a group for a given prime formula_1 is congruent to formula_29.
Theorems.
Motivation.
The Sylow theorems are a powerful statement about the structure of groups in general, but is 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 formula_2 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. formula_31.
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 formula_32, all members are actually isomorphic to each other and have the largest possible order: if formula_33 with formula_34 where does not divide , then every Sylow -subgroup has order formula_35. That is, is a -group and formula_36. These properties can be exploited to further analyze the structure of .
The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in "Mathematische Annalen".
The following weaker version of theorem 1 was first proved by Augustin-Louis Cauchy, and is known as Cauchy's theorem.
Consequences.
The Sylow theorems imply that for a prime number formula_1 every Sylow formula_1-subgroup is of the same order, formula_19. Conversely, if a subgroup has order formula_19, then it is a Sylow formula_1-subgroup, and so is isomorphic to every other Sylow formula_1-subgroup. Due to the maximality condition, if formula_43 is any formula_1-subgroup of formula_2, then formula_43 is a subgroup of a formula_1-subgroup of order formula_19.
A very important consequence of Theorem 2 is that the condition formula_49 is equivalent to saying that the Sylow formula_1-subgroup of formula_2 is a normal subgroup. However, there are groups that have normal subgroups but no normal Sylow subgroups, such as formula_52.
Sylow theorems for infinite groups.
There is an analogue of the Sylow theorems for infinite groups. One defines a Sylow -subgroup in an infinite group to be a "p"-subgroup (that is, every element in it has -power order) that is maximal for inclusion among all -subgroups in the group. Such subgroups exist by Zorn's lemma. Let formula_53 denote the set of conjugacy classes of a subgroup formula_54
Examples.
A simple illustration of Sylow subgroups and the Sylow theorems are the dihedral group of the "n"-gon, "D"2"n". 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.
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 "GL"2("F""q"), where "p" and "q" are primes ≥ 3 and , which are all abelian. The order of "GL"2("F""q") is . Since , the order of . Thus by Theorem 1, the order of the Sylow "p"-subgroups is "p"2"n".
One such subgroup "P", is the set of diagonal matrices formula_55, "x" is any primitive root of "F""q". Since the order of "F""q" is , its primitive roots have order "q" − 1, which implies that 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 "pn" choices for both "a" and "b", making |"P"| = "p"2"n". 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 "GL"2("F""q") 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. 
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 "n"3 be the number of Sylow 3-subgroups. Then "n"3 formula_56 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.
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 .
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 = 22 · 3 · 5, then "n"3 = 10 and "n"5 = 6 is perfectly possible. And in fact, the smallest simple non-cyclic 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.
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 "np" of Sylow's "p"-subgroups 
in the symmetric group "Sp" is . On the other hand, . Hence, . So, .
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 "A""gh"−1 = "B"h−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 "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 is the subject of many papers, including Waterhouse, Scharlau, Casadio and Zappa, Gow, and to some extent Meo.
One proof of the Sylow theorems exploits the notion of group action in various creative ways. The group 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. In the following, we use formula_58 as notation for "a divides b" and formula_59 for the negation of this statement.
and no power of "p" remains in any of the factors inside the product on the right. Hence , completing the proof.
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, including the algorithm described in Cannon. These versions are still used in the GAP computer algebra system.
In permutation groups, it has been proven, in Kantor and Kantor and Taylor, 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, 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.