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| In [[mathematics]], specifically in the field of [[finite group theory]], the '''Sylow theorems''' are a collection of [[theorem]]s named after the [[Norway|Norwegian]] [[mathematician]] [[Peter Ludwig Mejdell Sylow|Ludwig Sylow]] ([[#{{harvid|Sylow|1872}}|1872]]) that give detailed information about the number of [[subgroup]]s of fixed [[order of a group|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]].
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| For a [[prime number]] ''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|''p''-group]] (so that the [[order of a group element|order]] of any group element is a [[power (mathematics)|power]] of ''p''), and 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 Syl<sub>''p''</sub>(''G'').
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| The Sylow theorems assert a partial converse to [[Lagrange's theorem (group theory)|Lagrange's theorem]]. While 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 any [[prime factor]] ''p'' of the order of a finite group ''G'', there exists a Sylow ''p''-subgroup of ''G''. The order of a Sylow ''p''-subgroup of a finite group ''G'' is ''p<sup>n</sup>'', where ''n'' is the [[multiplicity of a prime factor|multiplicity]] of ''p'' in the order of ''G'', and any subgroup of order ''p<sup>n</sup>'' is a Sylow ''p''-subgroup of ''G''. The Sylow ''p''-subgroups of a group (for a given prime ''p'') are [[Conjugacy class|conjugate]] to each other. The number of Sylow ''p''-subgroups of a group for a given prime ''p'' is congruent to {{nowrap|1 mod ''p''.}}
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| == Theorems ==
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| 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<sub>''p''</sub>(''G''), all members are actually [[group isomorphism|isomorphic]] to each other and have the largest possible order: if |''G''| = ''p<sup>n</sup>m'' with ''n'' > 0 where ''p'' does not divide ''m'', then any Sylow ''p''-subgroup ''P'' has order |''P''| = ''p<sup>n</sup>''. 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''.
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| The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in ''[[Mathematische Annalen]]''.
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| '''Theorem 1''': For any [[prime factor]] ''p'' with [[multiplicity of a prime factor|multiplicity]] ''n'' of the order of a finite group ''G'', there exists a Sylow ''p''-subgroup of ''G'', of order ''p<sup>n</sup>''.
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| The following weaker version of theorem 1 was first proved by [[Augustin Cauchy|Cauchy]], it is known as [[Cauchy's theorem (group theory)|Cauchy's theorem]].
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| '''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''.<ref>Fraleigh, Victor J. Katz. A First Course In Abstract Algebra. p. 322. ISBN 9788178089973</ref>
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| '''Theorem 2''': Given a finite group ''G'' and a prime number ''p'', all Sylow ''p''-subgroups of ''G'' are [[conjugacy class|conjugate]] 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''<sup>−1</sup>''Hg'' = ''K''.
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| '''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 {{nowrap|''p<sup>n</sup>m''}}, where {{nowrap|''n'' > 0}} and ''p'' does not divide ''m''. Let ''n<sub>p</sub>'' be the number of Sylow ''p''-subgroups of ''G''. Then the following hold:
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| * ''n<sub>p</sub>'' divides ''m'', which is the [[index of a subgroup|index]] of the Sylow ''p''-subgroup in ''G''.
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| * ''n<sub>p</sub>'' ≡ 1 mod ''p''.
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| * ''n<sub>p</sub>'' = |''G'' : ''N<sub>G</sub>''(''P'')|, where ''P'' is any Sylow ''p''-subgroup of ''G'' and ''N<sub>G</sub>'' denotes the [[normalizer]].
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| === Consequences ===
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| The Sylow theorems imply that for a prime number ''p'' every Sylow ''p''-subgroup is of the same order, ''p<sup>n</sup>''. Conversely, if a subgroup has order ''p<sup>n</sup>'', 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<sup>n</sup>''.
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| A very important consequence of Theorem 3 is that the condition ''n<sub>p</sub>'' = 1 is equivalent to saying that the Sylow ''p''-subgroup of ''G'' is a [[normal subgroup]]
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| (there are groups that have normal subgroups but no normal Sylow subgroups, such as ''S''<sub>4</sub>).
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| === Sylow theorems for infinite groups ===
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| 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) that is maximal for inclusion among all ''p''-subgroups in the group. Such subgroups exist by [[Zorn's lemma]].
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| '''Theorem''': If ''K'' is a Sylow ''p''-subgroup of ''G'', and ''n<sub>p</sub>'' = |Cl(''K'')| is finite, then every Sylow ''p''-subgroup is conjugate to ''K'', and ''n<sub>p</sub>'' ≡ 1 mod ''p'', where Cl(''K'') denotes the conjugacy class of ''K''.
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| == Examples ==
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| [[File:Labeled Triangle Reflections.svg|thumb|In ''D''<sub>6</sub> all reflections are conjugate, as reflections correspond to Sylow 2-subgroups.]]
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| A simple illustration of Sylow subgroups and the Sylow theorems are the [[dihedral group]] of the ''n''-gon, ''D''<sub>2''n''</sub>. For ''n'' odd, 2 = 2<sup>1</sup> 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.
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| [[File:Hexagon Reflections.png|thumb|left|In ''D''<sub>12</sub> reflections no longer correspond to Sylow 2-subgroups, and fall into two conjugacy classes.]]
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| 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.
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| {{clear}}
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| ==Example applications==
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| Since Sylows theorem ensures the existence of p-subgroups of a finite group, its worthwhile to study groups of prime power order more closely. Most of the examples use Sylows theorem to prove that a group of a particular order is not [[Simple group|simple]]. For groups of small order, the congruence condition of Sylow's theorem is often sufficient to force the existence of a [[normal subgroup]].
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| Example-1: Groups of order pq, p and q primes with p<q.
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| Example-2 Group of order 30, groups of order 20, groups of order p2q, p and q distinct primes are some of the applications.
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| Example-3 (Groups of order 60): If o(G)=60 and G has more than one Sylow 5-subgroups, then G is simple.
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| === Cyclic group orders ===
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| 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 ''n''<sub>3</sub> be the number of Sylow 3-subgroups. Then ''n''<sub>3</sub> | 5 and ''n''<sub>3</sub> ≡ 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 subgroup|normal]] (since it has no distinct conjugates). Similarly, ''n''<sub>5</sub> must divide 3, and ''n''<sub>5</sub> 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).
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| === Small groups are not simple ===
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| A more complex example involves the order of the smallest [[simple group]] that is not [[cyclic group|cyclic]]. [[Burnside's theorem|Burnside's ''p<sup>a</sup> q<sup>b</sup>'' theorem]] states that if the order of a group is the product of one or two [[prime power]]s, then it is [[solvable group|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 {{nowrap|({{=}} 2 · 3 · 5)}}.
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| If ''G'' is simple, and |''G''| = 30, then ''n''<sub>3</sub> must divide 10 ( = 2 · 5), and ''n''<sub>3</sub> must equal 1 (mod 3). Therefore ''n''<sub>3</sub> = 10, since neither 4 nor 7 divides 10, and if ''n''<sub>3</sub> = 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.
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| As well, ''n''<sub>5</sub> = 6, since ''n''<sub>5</sub> must divide 6 ( = 2 · 3), and ''n''<sub>5</sub> 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.
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| Next, suppose |''G''| = 42 = 2 · 3 · 7. Here ''n''<sub>7</sub> must divide 6 ( = 2 · 3) and ''n''<sub>7</sub> must equal 1 (mod 7), so ''n''<sub>7</sub> = 1. So, as before, ''G'' can not be simple.
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| On the other hand for |''G''| = 60 = 2<sup>2</sup> · 3 · 5, then ''n''<sub>3</sub> = 10 and ''n''<sub>5</sub> = 6 is perfectly possible. And in fact, the smallest simple non-cyclic group is A<sub>5</sub>, the [[alternating group]] over 5 elements. It has order 60, and has 24 [[cyclic permutation]]s of order 5, and 20 of order 3.
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| === Wilson's theorem ===
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| Part of [[Wilson's theorem]] states that
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| :<math>(p-1)!\ \equiv\ -1 \pmod p</math>
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| for every prime ''p''. One may easily prove this theorem by Sylow's third theorem. Indeed,
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| observe that the number ''n<sub>p</sub>'' of Sylow's ''p''-subgroups
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| in the symmetric group ''S<sub>p</sub>'' is (p-2)!. On the other hand,
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| n<sub>p</sub> ≡ 1 mod p. Hence, (p-2)! ≡ 1 mod p. So, (p-1)! ≡ -1 mod p.
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| === Fusion results ===
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| [[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<sub>G</sub>''(''P'')-conjugate. The proof is a simple application of Sylow's theorem: If ''B''=''A<sup>g</sup>'', then the normalizer of ''B'' contains not only ''P'' but also ''P<sup>g</sup>'' (since ''P<sup>g</sup>'' is contained in the normalizer of ''A<sup>g</sup>''). By Sylow's theorem ''P'' and ''P<sup>g</sup>'' are conjugate not only in ''G'', but in the normalizer of ''B''. Hence ''gh''<sup>−1</sup> normalizes ''P'' for some ''h'' that normalizes ''B'', and then ''A''<sup>''gh''<sup>−1</sup></sup> = ''B''<sup>h<sup>−1</sup></sup> = ''B'', so that ''A'' and ''B'' are ''N<sub>G</sub>''(''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 group|''p''-nilpotent]].
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| 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 group]]s 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.
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| ==Proof of the Sylow theorems==
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| 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 {{harv|Waterhouse|1980}}, {{harv|Scharlau|1988}}, {{harv|Casadio|Zappa|1990}}, {{harv|Gow|1994}}, and to some extent {{harv|Meo|2004}}.
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| 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 {{harv|Wielandt|1959}}. In the following, we use ''a'' | ''b'' as notation for "a divides b" and ''a'' <math>\nmid</math> ''b'' for the negation of this statement.
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| <blockquote> '''Theorem 1''': A finite group ''G'' whose order |''G''| is divisible by a prime power ''p<sup>k</sup>'' has a subgroup of order ''p<sup>k</sup>''.</blockquote>
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| Proof: Let |''G''| = ''p<sup>k</sup>m = p<sup>k+r</sup>u'' such that ''p'' does not divide ''u'', and let Ω denote the set of subsets of ''G'' of size ''p<sup>k</sup>''. ''G'' [[Group action|acts]] on Ω by left multiplication. The [[Group action#Orbits and stabilizers|orbits]] ''G''ω = {''g''ω | ''g'' ∈ ''G''} of the ω ∈ Ω are the [[equivalence class]]es under the action of ''G''.
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| For any ω ∈ Ω consider its [[Group action#Orbits and stabilizers|stabilizer subgroup]] ''G''<sub>ω</sub>. For any fixed element α ∈ ω the function [''g'' ↦ ''g''α] maps ''G''<sub>ω</sub> to ω injectively: for any two ''g'', ''h'' ∈ ''G''<sub>ω</sub> we have that ''g''α = ''h''α implies ''g'' = ''h'', because α ∈ ω ⊆ ''G'' means that one may cancel on the right. Therefore '' p<sup>k</sup>'' = |ω| ≥ |''G''<sub>ω</sub>|.
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| On the other hand
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| :<math>|\Omega | ={p^km \choose p^k} = m \prod_{j=1}^{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)}} </math>
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| and no power of ''p'' remains in any of the factors inside the product on the right. Hence [[Additive p-adic valuation|''ν<sub>p</sub>'']](|Ω|) = ''ν<sub>p</sub>''(''m'') = ''r''.
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| Let ''R'' ⊆ Ω be a complete representation of all the equivalence classes under the action of ''G''. Then,
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| :<math>|\Omega | =\sum_{\omega\in R}|G\omega|\mathrm{.}</math>
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| Thus, there exists an element ω ∈ ''R'' such that ''s'' := ''ν<sub>p</sub>''(|''G''ω|) ≤ ''ν<sub>p</sub>''(|Ω|) = ''r''. Hence |''G''ω| = ''p<sup>s</sup>v'' where ''p'' does not divide ''v''. By the [[Group action#Orbits and stabilizers|stabilizer-orbit-theorem]] we have |''G''<sub>ω</sub>| = |''G''| / |''G''ω| = ''p<sup>k+r-s</sup>u/v''. Therefore ''p<sup>k</sup>'' | |''G''<sub>ω</sub>|, so ''p<sup>k</sup>'' ≤ |''G''<sub>ω</sub>| and ''G''<sub>ω</sub>'' is the desired subgroup.
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| <blockquote> '''Lemma''': Let ''G'' be a finite ''p''-group, let ''G'' act on a finite set Ω, and let Ω<sub>0</sub> denote the set of points of Ω that are fixed under the action of ''G''. Then |Ω| ≡ |Ω<sub>0</sub>| mod ''p''. </blockquote>
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| 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<sub>x</sub>''| (where ''G<sub>x</sub>'' denotes the [[Group action#Orbits and stabilizers|stabilizer]]), which is a multiple of ''p'' by assumption. The result follows immediately.
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| <blockquote>'''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''<sup>−1</sup>''Hg'' ≤ ''P''. In particular, all Sylow ''p''-subgroups of ''G'' are [[conjugacy class|conjugate]] to each other (and therefore [[isomorphism|isomorphic]]), i.e. if ''H'' and ''K'' are Sylow ''p''-subgroups of ''G'', then there exists an element ''g'' in ''G'' with ''g''<sup>−1</sup>''Hg'' = ''K''.</blockquote>
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| Proof: Let Ω be the set of left [[coset]]s of ''P'' in ''G'' and let ''H'' act on Ω by left multiplication. Applying the Lemma to ''H'' on Ω, we see that |Ω<sub>0</sub>| ≡ |Ω| = [''G'' : ''P''] mod ''p''. Now ''p'' <math>\nmid</math> [''G'' : ''P''] by definition so ''p'' <math>\nmid</math> |Ω<sub>0</sub>|, hence in particular |Ω<sub>0</sub>| ≠ 0 so there exists some ''gP'' ∈ Ω<sub>0</sub>. It follows that for some ''g'' ∈ ''G'' and ∀ ''h'' ∈ ''H'' we have ''hgP'' = ''gP'' so ''g''<sup>−1</sup>''HgP'' = ''P'' and therefore ''g''<sup>−1</sup>''Hg'' ≤ ''P''. Now if ''H'' is a Sylow ''p''-subgroup, |''H''| = |''P''| = |''gPg''<sup>−1</sup>| so that ''H'' = ''gPg''<sup>−1</sup> for some ''g'' ∈ ''G''.
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| <blockquote>'''Theorem 3''': Let ''q'' denote the order of any Sylow ''p''-subgroup of a finite group ''G''. Then ''n<sub>p</sub>'' | |''G''|/''q'' and ''n<sub>p</sub>'' ≡ 1 mod ''p''.</blockquote>
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| Proof: By Theorem 2, ''n<sub>p</sub>'' = [''G'' : ''N<sub>G</sub>''(''P'')], where ''P'' is any such subgroup, and ''N<sub>G</sub>''(''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'' ∈ Ω<sub>0</sub> and observe that then ''Q'' = ''xQx''<sup>−1</sup> for all ''x'' ∈ ''P'' so that ''P'' ≤ ''N<sub>G</sub>''(''Q''). By Theorem 2, ''P'' and ''Q'' are conjugate in ''N<sub>G</sub>''(''Q'') in particular, and ''Q'' is normal in ''N<sub>G</sub>''(''Q''), so then ''P'' = ''Q''. It follows that Ω<sub>0</sub> = {''P''} so that, by the Lemma, |Ω| ≡ |Ω<sub>0</sub>| = 1 mod ''p''.
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| == Algorithms ==
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| The problem of finding a Sylow subgroup of a given group is an important problem in [[computational group theory]].
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| 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<sub>G</sub>''(''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 {{harv|Butler|1991|loc=Chapter 16}}, including the algorithm described in {{harv|Cannon|1971}}. These versions are still used in the [[GAP computer algebra system]].
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| In [[permutation group]]s, it has been proven in ({{harvs|nb|last=Kantor|year1=1985a|year2=1985b|year3=1990}}; {{harvnb|Kantor|Taylor|1988}}) 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 {{harv|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]].
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| == See also ==
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| * [[Frattini's argument]]
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| * [[Hall subgroup]]
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| * [[Maximal subgroup]]
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| == Notes ==
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| {{Reflist}}
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| ==References==
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| * {{Citation | first=L. | last=Sylow | author-link=Peter Ludwig Mejdell Sylow | title=Théorèmes sur les groupes de substitutions | language=French | journal=[[Mathematische Annalen|Math. Ann.]] | volume=5 | issue=4 | pages=584–594 | year=1872 | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002242052 | doi=10.1007/BF01442913 | jfm=04.0056.02 }}
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| === Proofs ===
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| * {{Citation | last1=Casadio | first1=Giuseppina | last2=Zappa | first2=Guido | title=History of the Sylow theorem and its proofs | language=Italian | mr=1096350 | zbl = 0721.01008 | year=1990 | journal=<abbr title="Bollettino di Storia delle Scienze Matematiche">Boll. Storia Sci. Mat.</abbr> | issn=0392-4432 | volume=10 | issue=1 | pages=29–75}}
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| * {{Citation | last=Gow | first=Rod | title=Sylow's proof of Sylow's theorem | mr=1313412 | zbl = 0829.01011 | year=1994 | journal=<abbr title="Irish Mathematical Society Bulletin">Irish Math. Soc. Bull.</abbr> | issn=0791-5578 | issue=33 | pages=55–63}}
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| * {{Citation | last1=Kammüller | first1=Florian | last2=Paulson | first2=Lawrence C. | title=A formal proof of Sylow's theorem. An experiment in abstract algebra with Isabelle HOL | url=http://www.cl.cam.ac.uk/users/lcp/papers/Kammueller/sylow.pdf | doi=10.1023/A:1006269330992 | mr=1721912 | zbl = 0943.68149 | year=1999 | journal=<abbr title="Journal of Automated Reasoning">J. Automat. Reason.</abbr> | issn=0168-7433 | volume=23 | issue=3 | pages=235–264}}
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| *{{Citation | last=Meo | first=M. | title=The mathematical life of Cauchy's group theorem | doi=10.1016/S0315-0860(03)00003-X | mr=2055642 | zbl = 1065.01009 | year=2004 | journal=[[Historia Mathematica|Historia Math.]] | issn=0315-0860 | volume=31 | issue=2 | pages=196–221}}
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| * {{Citation | last=Scharlau | first=Winfried | title=Die Entdeckung der Sylow-Sätze | language=German | doi=10.1016/0315-0860(88)90048-1 | mr=931678 | zbl = 0637.01006 | year=1988 | journal=[[Historia Mathematica|Historia Math.]] | issn=0315-0860 | volume=15 | issue=1 | pages=40–52}}
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| * {{Citation | last=Waterhouse | first=William C. | authorlink = William C. Waterhouse | title=The early proofs of Sylow's theorem | doi=10.1007/BF00327877 | mr=575718 | zbl = 0436.01006 | year=1980 | journal=<abbr title="Archive for History of Exact Sciences">Arch. Hist. Exact Sci.</abbr> | issn=0003-9519 | volume=21 | issue=3 | pages=279–290}}
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| * {{Citation | last=Wielandt | first=Helmut | author-link=:de:Helmut Wielandt | title=Ein Beweis für die Existenz der Sylowgruppen | language=German | doi=10.1007/BF01240818 | mr=0147529 | zbl = 0092.02403 | year=1959 | journal=<abbr title="Archiv der Mathematik">Arch. Math.</abbr> | issn=0003-9268 | volume=10 | issue=1 | pages=401–402}}
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| === Algorithms ===
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| * {{Citation | last=Butler | first=G. | title=Fundamental Algorithms for Permutation Groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Lecture Notes in Computer Science]] | isbn=978-3-540-54955-0 | mr=1225579 | zbl = 0785.20001 | year=1991 | volume=559 | doi = 10.1007/3-540-54955-2}}
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| * {{Citation | last=Cannon | first=John J. | title=Computers in Algebra and Number Theory (<abbr title="Proceedings of a SIAM-AMS Symposium in Applied Mathematics">Proc. SIAM-AMS Sympos. Appl. Math.</abbr>, New York, 1970) | series=<abbr title="SIAM-AMS Proceedings">SIAM-AMS Proc.</abbr> | volume=4 | publisher=[[American Mathematical Society|AMS]] | location=Providence, RI | mr=0367027 | zbl = 0253.20027 | year=1971 | chapter=Computing local structure of large finite groups | pages=161–176 | issn=0160-7634}}
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| *{{Citation | last=Kantor | first=William M. | title=Polynomial-time algorithms for finding elements of prime order and Sylow subgroups | mr=813589 | zbl = 0604.20001 | year=1985a | journal=<abbr title="Journal of Algorithms">J. Algorithms</abbr> | issn=0196-6774 | volume=6 | issue=4 | pages=478–514 | doi=10.1016/0196-6774(85)90029-X}}
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| *{{Citation | last=Kantor | first=William M. | title=Sylow's theorem in polynomial time | doi=10.1016/0022-0000(85)90052-2 | mr=805654 | zbl = 0573.20022 | year=1985b | journal=<abbr title="Journal of Computer and System Sciences">J. Comput. System Sci.</abbr> | issn=1090-2724 | volume=30 | issue=3 | pages=359–394}}
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| *{{Citation | last1=Kantor | first1=William M. | last2=Taylor | first2=Donald E. | title=Polynomial-time versions of Sylow's theorem | mr=925595 | zbl = 0642.20019 | year=1988 | journal=<abbr title="Journal of Algorithms">J. Algorithms</abbr> | issn=0196-6774 | volume=9 | issue=1 | pages=1–17 | doi=10.1016/0196-6774(88)90002-8}}
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| *{{Citation | last=Kantor | first=William M. | title=Finding Sylow normalizers in polynomial time | mr=1079450 | zbl = 0731.20005 | year=1990 | journal=<abbr title="Journal of Algorithms">J. Algorithms</abbr> | issn=0196-6774 | volume=11 | issue=4 | pages=523–563 | doi=10.1016/0196-6774(90)90009-4}}
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| *{{Citation | last=Seress | first=Ákos | title=Permutation Group Algorithms | publisher=[[Cambridge University Press]] | series=Cambridge Tracts in Mathematics | isbn=978-0-521-66103-4 | mr=1970241 | zbl = 1028.20002 | year=2003 | volume=152}}
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| == External links ==
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| {{Sister project links|wikt=no|commons=no|b=Abstract_Algebra/Group_Theory/The_Sylow_Theorems|n=no|q=no|s=no|v=no|voy=no|species=no|d=no}}
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| [[Category:Finite groups]]
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| [[Category:P-groups]]
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| [[Category:Theorems in algebra]]
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| [[Category:Articles containing proofs]]
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