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In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
s named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
s of fixed
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
that a given
finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or ma ...
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 A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
p, a Sylow ''p''-subgroup (sometimes ''p''-Sylow subgroup) of a group G is a maximal p-subgroup of G, i.e., a subgroup of G that is a ''p''-group (meaning its cardinality is a power of p, or equivalently, the
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
of every group element is a power of p) that is not a proper subgroup of any other p-subgroup of G. The set of all Sylow p-subgroups for a given prime p is sometimes written \text_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 A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''p'' of the order of a finite group G, there exists a Sylow p-subgroup 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 ''p''-subgroup of G, and the Sylow p-subgroups of a group (for a given prime p) are conjugate to each other. Furthermore, the number of Sylow p-subgroups of a group for a given prime p is congruent to 1 (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 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. , G, = 60.


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 \operatorname_p(G), all members are actually
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to each other and have the largest possible order: if , G, =p^nm with n > 0 where does not divide , then every Sylow -subgroup has order , P, = p^n. That is, is a -group and \text(, G:P, , p) = 1. 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 ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...
''. The following weaker version of theorem 1 was first proved by
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. H ...
, and is known as Cauchy's theorem.


Consequences

The Sylow theorems imply that for a prime number p every Sylow p-subgroup is of the same order, p^n. Conversely, if a subgroup has order p^n, 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 p^n. A very important consequence of Theorem 2 is that the condition n_p = 1 is equivalent to saying that the Sylow p-subgroup of G is a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
. However, there are groups that have normal subgroups but no normal Sylow subgroups, such as S_4.


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. Let \operatorname(K) denote the set of conjugacy classes of a subgroup K \subset G


Examples

A simple illustration of Sylow subgroups and the Sylow theorems are the
dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...
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 Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
. 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 \beginx^ & 0 \\0 & x^ \end, ''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 Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
. For groups of small order, the congruence condition of Sylow's theorem is often sufficient to force the existence of a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
. ;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 ''p''2''q'', ''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 ''n''3 be the number of Sylow 3-subgroups. Then ''n''3 \mid 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 In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
, the intersection of these two subgroups is trivial, and so ''G'' must be the
internal direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
of groups of order 3 and 5, that is the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of order 15. Thus, there is only one group of order 15 (
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
isomorphism).


Small groups are not simple

A more complex example involves the order of the smallest simple group that is not
cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in so ...
. Burnside's ''pa qb'' theorem states that if the order of a group is the product of one or two
prime power In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17 ...
s, 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 In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic pr ...
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 In algebra and number theory, Wilson's theorem states that a natural number ''n'' > 1 is a prime number if and only if the product of all the positive integers less than ''n'' is one less than a multiple of ''n''. That is (using the notations of m ...
states that :(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 . 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 In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...
: 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'' = , 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 In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...
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 mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
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 a \mid b as notation for "a divides b" and a \nmid b for the negation of this statement. for each , and therefore using the additive p-adic valuation ''νp'', which counts the number of factors ''p'', one has . This means that for those with , the ones we are looking for, one has , while for any other one has (as implies . Since is the sum of } over all distinct orbits , one can show the existence of ω of the former type by showing that (if none existed, that valuation would exceed ''r''). This is an instance of Kummer's theorem (since in base ''p'' notation the number } 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: :, \Omega , = = \prod_^ \frac = m\prod_^ \frac and no power of ''p'' remains in any of the factors inside the product on the right. Hence , completing the proof. It may be noted that conversely every subgroup ''H'' of order ''pk'' gives rise to sets for which = ''H'', namely any one of the ''m'' distinct cosets ''Hg''. over all distinct orbits and reducing mod .


Algorithms

The problem of finding a Sylow subgroup of a given group is an important problem in
computational group theory In mathematics, computational group theory is the study of groups by means of computers. It is concerned with designing and analysing algorithms and data structures to compute information about groups. The subject has attracted interest because f ...
. 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 group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to ...
s, it has been proven, in Kantor and Kantor and Taylor, that a Sylow ''p''-subgroup and its normalizer can be found in
polynomial time In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...
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.


See also

* Frattini's argument *
Hall subgroup In mathematics, specifically group theory, a Hall subgroup of a finite group ''G'' is a subgroup whose order is coprime to its index. They were introduced by the group theorist . Definitions A Hall divisor (also called a unitary divisor) of ...
*
Maximal subgroup In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra. In group theory, a maximal subgroup ''H'' of a group ''G'' is a proper subgroup, such that no proper subgroup ''K'' contains ''H'' ...
*
p-group In mathematics, specifically group theory, given a prime number ''p'', a ''p''-group is a group in which the order of every element is a power of ''p''. That is, for each element ''g'' of a ''p''-group ''G'', there exists a nonnegative integer ...


Notes


References


Proofs

* * * * * * *


Algorithms

* * * * * * *


External links

* * * * {{MathWorld , title=Sylow Theorems , id=SylowTheorems Theorems about finite groups P-groups Articles containing proofs