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{{short description|Theorems that help decompose a finite group based on prime factors of its order}}
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{{Use shortened footnotes|date=May 2021}}  
{{More footnotes|date=November 2018}}
In mathematics, specifically in the field of [[finite group theory]], the '''Sylow theorems''' are a collection of [[theorem]]s named after the Norwegian mathematician [[Peter Ludwig Mejdell Sylow|Peter Ludwig Sylow]]{{r|Sylow1872}} 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]].

For a [[prime number]] <math>p</math>, a '''Sylow ''p''-subgroup''' (sometimes '''''p''-Sylow subgroup''') of a finite group <math>G</math> is a [[Maximal subgroup|maximal]] <math>p</math>-subgroup of <math>G</math>, i.e., a subgroup of <math>G</math> that is a [[p-group|''p''-group]] (meaning its cardinality is a [[Power (mathematics)|power]] of <math>p;</math> or equivalently: For each group element, its [[Order of a group element|order]] is some power of <math>p</math>) that is not a proper subgroup of any other <math>p</math>-subgroup of <math>G</math>. The set of all Sylow <math>p</math>-subgroups for a given prime <math>p</math> is sometimes written <math>\text{Syl}_p(G)</math>.

The Sylow theorems assert a partial converse to [[Lagrange's theorem (group theory)|Lagrange's theorem]]. Lagrange's theorem states that for any finite group <math>G</math> the order (number of elements) of every subgroup of <math>G</math> divides the order of <math>G</math>. The Sylow theorems state that for every [[prime factor]] ''<math>p</math>'' of the order of a finite group <math>G</math>, there exists a Sylow <math>p</math>-subgroup of <math>G</math> of order <math>p^n</math>, the highest power of <math>p</math> that divides the order of <math>G</math>. Moreover, every subgroup of order ''<math>p^n</math>'' is a Sylow ''<math>p</math>''-subgroup of <math>G</math>, and the Sylow <math>p</math>-subgroups of a group (for a given prime <math>p</math>) are [[Conjugacy class|conjugate]] to each other. Furthermore, the number of Sylow <math>p</math>-subgroups of a group for a given prime <math>p</math> is congruent to 1 (mod <math>p</math>).

== 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 <math>G</math> 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. <math>|G| = 60</math>.{{r|GraciaSaz_WebPDF}}

=== 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 <math>\operatorname{Syl}_p(G)</math>, all members are actually [[group isomorphism|isomorphic]] to each other and have the largest possible order: if <math>|G|=p^nm</math> with <math>n > 0</math> where {{mvar|p}} does not divide {{mvar|m}}, then every Sylow {{mvar|p}}-subgroup {{mvar|P}} has order <math>|P| = p^n</math>. That is, {{mvar|P}} is a {{mvar|p}}-group and <math>\text{gcd}(|G:P|, p) = 1</math>. These properties can be exploited to further analyze the structure of {{mvar|G}}.

The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in ''[[Mathematische Annalen]]''.

{{math theorem|note=1|For every [[prime factor]] {{mvar|p}} with [[multiplicity of a prime factor|multiplicity]] {{mvar|n}} of the order of a finite group {{mvar|G}}, there exists a Sylow [[p-group|{{mvar|p}}-subgroup]] of {{mvar|G}}, of order ''<math>p^n</math>''.}}

The following weaker version of theorem 1 was first proved by [[Augustin-Louis Cauchy]], and is known as [[Cauchy's theorem (group theory)|Cauchy's theorem]].

{{math theorem|name=Corollary|Given a finite group {{mvar|G}} and a prime number {{mvar|p}} dividing the order of {{mvar|G}}, then there exists an element (and thus a cyclic subgroup generated by this element) of order {{mvar|p}} in ''{{mvar|G}}''.{{r|Fraleigh_2004_322}}}}

{{math theorem|note=2|Given a finite group {{mvar|G}} and a prime number {{mvar|p}}, all Sylow {{mvar|p}}-subgroups of {{mvar|G}} are [[conjugacy class|conjugate]] to each other. That is, if {{mvar|H}} and {{mvar|K}} are Sylow {{mvar|p}}-subgroups of {{mvar|G}}, then there exists an element <math>g \in G</math> with <math>g^{-1}Hg = K</math>.}}

{{math theorem|note=3|Let {{mvar|p}} be a prime factor with multiplicity {{mvar|n}} of the order of a finite group {{mvar|G}}, so that the order of {{mvar|G}} can be written as <math>p^nm</math>, where <math>n > 0</math> and {{mvar|p}} does not divide {{mvar|m}}. Let <math>n_p</math> be the number of Sylow {{mvar|p}}-subgroups of {{mvar|G}}. Then the following hold:
* <math>n_p</math> divides {{mvar|m}}, which is the [[index of a subgroup|index]] of the Sylow {{mvar|p}}-subgroup in {{mvar|G}}.
* <math>n_p \equiv 1 \pmod{p}</math>
* <math>n_p = |G:N_G(P)|</math>, where {{mvar|P}} is any Sylow {{mvar|p}}-subgroup of {{mvar|G}} and <math>N_G</math> denotes the [[normalizer]].
}}

=== Consequences ===
The Sylow theorems imply that for a prime number <math>p</math> every Sylow <math>p</math>-subgroup is of the same order, <math>p^n</math>. Conversely, if a subgroup has order <math>p^n</math>, then it is a Sylow <math>p</math>-subgroup, and so is conjugate to every other Sylow <math>p</math>-subgroup. Due to the maximality condition, if <math>H</math> is any <math>p</math>-subgroup of <math>G</math>, then <math>H</math> is a subgroup of a <math>p</math>-subgroup of order <math>p^n</math>.

An important consequence of Theorem 2 is that the condition <math>n_p = 1</math> is equivalent to the condition that the Sylow <math>p</math>-subgroup of <math>G</math> is a [[normal subgroup]] (Theorem 3 can often show <math>n_p = 1</math>). However, there are groups that have proper, non-trivial normal subgroups but no normal Sylow subgroups, such as <math>S_4</math>. Groups that are of prime-power order have no proper Sylow <math>p</math>-subgroups.

The third bullet point of the third theorem has as an immediate consequence that <math>n_p</math> [[divides]] <math>|G|</math>.

=== Sylow theorems for infinite groups ===
There is an analogue of the Sylow theorems for infinite groups. One defines a Sylow {{mvar|p}}-subgroup in an infinite group to be a ''p''-subgroup (that is, every element in it has {{mvar|p}}-power order) that is maximal for inclusion among all {{mvar|p}}-subgroups in the group.  Let <math>\operatorname{Cl}(K)</math> denote the set of conjugates of a subgroup <math>K \subset G</math>.

{{math theorem|If {{mvar|K}} is a Sylow {{mvar|p}}-subgroup of {{mvar|G}}, and <math>n_p = |\operatorname{Cl}(K)|</math> is finite, then every Sylow {{mvar|p}}-subgroup is conjugate to {{mvar|K}}, and <math>n_p \equiv 1\ (\mathrm{mod}\ p)</math>.}}

== Examples ==
[[File:Labeled Triangle Reflections.svg|thumb|In ''D''<sub>6</sub> all reflections are conjugate, as reflections correspond to Sylow 2-subgroups.]]

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.

[[File:Hexagon reflections.svg|thumb|left|In ''D''<sub>12</sub> reflections no longer correspond to Sylow 2-subgroups, and fall into two conjugacy classes.]]
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.

{{Clear}}
Another example are the Sylow p-subgroups of ''GL''<sub>2</sub>(''F''<sub>''q''</sub>), where ''p'' and ''q'' are primes&nbsp;≥&nbsp;3 and {{math|''p''&nbsp;≡&nbsp;1&nbsp;(mod&nbsp;''q'')}} , which are all [[Abelian group|abelian]]. The order of ''GL''<sub>2</sub>(''F''<sub>''q''</sub>) is {{math|1=(''q''<sup>2</sup>&nbsp;−&nbsp;1)(''q''<sup>2</sup>&nbsp;−&nbsp;''q'') = (''q'')(''q''&nbsp;+&nbsp;1)(''q''&nbsp;−&nbsp;1)<sup>2</sup>}}. Since {{math|1=''q''&nbsp;=&nbsp;''p''<sup>''n''</sup>''m''&nbsp;+&nbsp;1}}, the order of {{math|1=''GL''<sub>2</sub>(''F''<sub>''q''</sub>) =&nbsp;''p''<sup>2''n''</sup> ''m''&prime;}}. Thus by Theorem&nbsp;1, the order of the Sylow ''p''-subgroups is ''p''<sup>2''n''</sup>.

One such subgroup ''P'', is the set of diagonal matrices  <math>\begin{bmatrix}x^{im} & 0 \\0 & x^{jm} \end{bmatrix}</math>, ''x'' is any [[Primitive root modulo n|primitive root]] of  ''F''<sub>''q''</sub>. Since the order of ''F''<sub>''q''</sub> is {{math|1=''q''&nbsp;−&nbsp;1}}, its primitive roots have order ''q'' − 1, which implies that {{math|1=''x''<sup>(''q''&nbsp;−&nbsp;1)/''p''<sup>''n''</sup></sup>}} or ''x''<sup>''m''</sup> and all its powers have an order which is a power of&nbsp;''p''. So, ''P'' is a subgroup where all its elements have orders which are powers of&nbsp;''p''. There are ''p<sup>n</sup>'' choices for both ''a'' and ''b'', making {{math|1 = {{!}}''P''{{!}} =&nbsp;''p''<sup>2''n''</sup>}}. 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''<sub>2</sub>(''F''<sub>''q''</sub>) 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 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]]. 
;Example-1: Groups of order ''pq'', ''p'' and ''q'' primes with ''p''&nbsp;<&nbsp;''q''. 
;Example-2: Group of order 30, groups of order 20, groups of order ''p''<sup>2</sup>''q'', ''p'' and ''q'' distinct primes are some of the applications. 
;Example-3: (Groups of order 60): If the order |''G''|&nbsp;=&nbsp;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''<sub>3</sub> be the number of Sylow 3-subgroups. Then ''n''<sub>3</sub> <math>\mid</math> 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).

=== Small groups are not simple ===
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)}}.

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.

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.

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.

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.

=== Wilson's theorem ===
Part of [[Wilson's theorem]] states that

:<math>(p-1)! \equiv -1 \pmod p</math>

for every prime ''p''. One may easily prove this theorem by Sylow's third theorem. Indeed,  
observe that the number ''n<sub>p</sub>'' of Sylow's ''p''-subgroups  
in the symmetric group ''S<sub>p</sub>'' is {{sfrac|1|''p''&nbsp;&minus;&nbsp;1}} times the number of p-cycles in ''S<sub>p</sub>'', ie. {{math|(''p''&nbsp;&minus;&nbsp;2)!}}. On the other hand, {{math|''n''<sub>''p''</sub> ≡ 1 (mod&nbsp;''p'')}}. Hence, {{math|(''p''&nbsp;&minus;&nbsp;2)! ≡ 1 (mod&nbsp;''p'')}}. So, {{math|(''p''&nbsp;&minus;&nbsp;1)! ≡ &minus;1 (mod&nbsp;''p'')}}.

=== 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<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]].

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.

==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,{{sfn|Waterhouse|1980}} Scharlau,{{sfn|Scharlau|1988}} Casadio and Zappa,{{sfn|Casadio|Zappa|1990}} Gow,{{sfn|Gow|1994}} and to some extent Meo.{{sfn|Meo|2004}}

One proof of the Sylow theorems exploits the notion of [[Group action (mathematics)|group action]] in various creative ways. The group {{mvar|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.{{sfn|Wielandt|1959}} In the following, we use <math>a \mid b</math> as notation for "a divides b" and <math>a \nmid b</math> for the negation of this statement.

{{math theorem|note=1|1=A finite group {{var|G}} whose order <math>|G|</math> is divisible by a prime power ''p<sup>k</sup>'' has a subgroup of order ''p<sup>k</sup>''.}}

{{math proof|1=Let {{math|1={{abs|{{var|G}}}} = ''p<sup>k</sup>m'' = ''p''<sup>''k''+''r''</sup>''u''}} such that <math>p \nmid u</math>, and let Ω denote the set of subsets of {{mvar|G}} of size ''p<sup>k</sup>''. {{mvar|G}} [[Group action (mathematics)|acts]] on Ω by left multiplication: for {{math|1={{var|g}} ∈ {{var|G}}}} and {{math|1={{var|ω}} ∈ Ω}}, {{math|1={{var|g}}⋅{{var|ω}} = {{mset| {{var|g}}{{var|x}} {{pipe}} {{var|x}} ∈ {{var|ω}} }}}}. For a given set {{math|1={{var|ω}} ∈ Ω}}, write {{var|G}}<sub>{{var|ω}}</sub> for its [[Group action (mathematics)#Orbits and stabilizers|stabilizer subgroup]] {{math|1={{mset| {{var|g}} ∈ {{var|G}} {{pipe}} {{var|g}}⋅{{var|ω}} {{=}} {{var|ω}} }}}} and {{var|G}}{{var|ω}} for its [[Group action (mathematics)#Orbits and stabilizers|orbit]] {{math|1={{mset| {{var|g}}⋅{{var|ω}} {{pipe}} {{var|g}} ∈ {{var|G}} }}}} in Ω.

The proof will show the existence of some {{math|1={{var|ω}} ∈ Ω}} for which {{mvar|G}}<sub>{{mvar|ω}}</sub> has ''p<sup>k</sup>'' elements, providing the desired subgroup. This is the maximal possible size of a stabilizer subgroup {{mvar|G}}<sub>{{mvar|ω}}</sub>, since for any fixed element {{math|1={{var|α}} ∈ {{var|ω}} ⊆ {{var|G}}}}, the right coset {{mvar|G}}<sub>{{mvar|ω}}</sub>{{mvar|α}} is contained in {{mvar|ω}}; therefore, {{math|1={{abs|{{var|G}}<sub>{{var|ω}}</sub>}} = {{abs|{{var|G}}<sub>{{var|ω}}</sub>{{mvar|α}}}} ≤ {{abs|{{var|ω}}}} = {{var|p}}<sup>{{var|k}}</sup>}}.

By the [[Group action (mathematics)#Orbits and stabilizers|orbit-stabilizer theorem]] we have {{math|1={{abs|{{var|G}}<sub>{{var|ω}}</sub>}} {{abs|{{var|G}}{{var|ω}}}} = {{abs|{{var|G}}}}}} for each {{math|1={{var|ω}} ∈ Ω}}, and therefore using the [[additive p-adic valuation]] ''ν<sub>p</sub>'', which counts the number of factors ''p'', one has {{math|1=''ν<sub>p</sub>''({{abs|{{var|G}}<sub>{{var|ω}}</sub>}}) + ''ν<sub>p</sub>''({{abs|{{var|G}}{{var|ω}}}}) = ''ν<sub>p</sub>''({{abs|{{var|G}}}}) = ''k'' + ''r''}}. This means that for those {{mvar|ω}} with {{math|1={{abs|{{var|G}}<sub>{{var|ω}}</sub>}} = ''p<sup>k</sup>''}}, the ones we are looking for, one has {{math|1=''ν<sub>p</sub>''({{abs|{{var|G}}{{var|ω}}}}) = ''r''}}, while for any other {{mvar|ω}} one has {{math|1=''ν<sub>p</sub>''({{abs|{{var|G}}{{var|ω}}}}) > ''r''}} (as {{math|1=0 < {{abs|{{var|G}}<sub>{{var|ω}}</sub>}} < ''p<sup>k</sup>''}} implies {{math|1=''ν<sub>p</sub>''({{abs|{{var|G}}<sub>{{var|ω}}</sub>}}) < ''k'')}}. Since {{math|1={{abs|Ω}}}} is the sum of {{math|1={{abs|{{var|G}}{{var|ω}}}}}} over all distinct orbits {{mvar|G}}{{mvar|ω}}, one can show the existence of ω of the former type by showing that {{math|1=''ν<sub>p</sub>''({{abs|Ω}}) = ''r''}} (if none existed, that valuation would exceed ''r''). This is an instance of [[Kummer's theorem]] (since in base ''p'' notation the number {{math|1={{abs|{{var|G}}}}}} ends with precisely ''k'' + ''r'' digits zero, subtracting ''p<sup>k</sup>'' from it involves a carry in ''r'' places), and can also be shown by a simple computation:

:<math>|\Omega | ={p^km \choose p^k} = \prod_{j=0}^{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>

and no power of ''p'' remains in any of the factors inside the product on the right. Hence {{math|1=''ν<sub>p</sub>''({{abs|Ω}}) = ''ν<sub>p</sub>''(''m'') = ''r''}}, completing the proof.

It may be noted that conversely every subgroup ''H'' of order ''p<sup>k</sup>'' gives rise to sets {{math|1={{var|ω}} ∈ Ω}} for which {{var|G}}<sub>{{var|ω}}</sub> = ''H'', namely any one of the ''m'' distinct cosets ''Hg''.}}

{{math theorem|name=Lemma|1=Let {{mvar|H}} be a finite {{mvar|p}}-group, let Ω be a finite set acted on by {{mvar|H}}, and let Ω<sub>0</sub> denote the set of points of Ω that are fixed under the action of {{mvar|H}}.  Then {{math|1={{abs|Ω}} ≡ {{abs|Ω<sub>0</sub>}} (mod&nbsp;{{var|p}})}}.}}

{{math proof|1=Any element {{math|1={{var|x}} ∈ Ω}} not fixed by {{mvar|H}} will lie in an orbit of order {{math|1={{abs|{{var|H}}}}/{{abs|''H<sub>x</sub>''}}}} (where ''H<sub>x</sub>'' denotes the [[Group action (mathematics)#Orbits and stabilizers|stabilizer]]), which is a multiple of {{mvar|p}} by assumption.  The result follows immediately by writing {{math|1={{abs|Ω}}}} as the sum of {{math|1={{abs|{{var|H}}{{var|x}}}}}} over all distinct orbits {{mvar|H}}{{mvar|x}} and reducing mod {{mvar|p}}.}}

{{math theorem|note=2|1=If ''H'' is a ''p''-subgroup of {{mvar|G}} and ''P'' is a Sylow ''p''-subgroup of {{mvar|G}}, then there exists an element ''g'' in {{mvar|G}} such that ''g''<sup>−1</sup>''Hg'' ≤ ''P''. In particular, all Sylow ''p''-subgroups of {{mvar|G}} are [[conjugacy class|conjugate]] to each other (and therefore [[isomorphism|isomorphic]]), that is, if ''H'' and ''K'' are Sylow ''p''-subgroups of {{mvar|G}}, then there exists an element ''g'' in {{mvar|G}} with ''g''<sup>−1</sup>''Hg'' = ''K''.}}

{{math proof|1=Let Ω be the set of left [[coset]]s of ''P'' in {{mvar|G}} and let ''H'' act on Ω by left multiplication. Applying the Lemma to ''H'' on Ω, we see that {{math|1={{abs|Ω<sub>0</sub>}} ≡ {{abs|Ω}} = [{{var|G}} : ''P''] (mod&nbsp;''p'')}}.  Now <math>p \nmid [G:P]</math> by definition so <math>p \nmid |\Omega_0|</math>, hence in particular {{math|1={{abs|Ω<sub>0</sub>}} ≠ 0}} so there exists some {{math|1=''gP'' ∈ Ω<sub>0</sub>}}. With this ''gP'', we have ''hgP'' = ''gP'' for all {{math|1={{var|h}} ∈ {{var|H}}}}, so ''g''<sup>−1</sup>''HgP'' = ''P'' and therefore ''g''<sup>−1</sup>''Hg'' ≤ ''P''. Furthermore, if ''H'' is a Sylow ''p''-subgroup, then {{math|1={{abs|''g''<sup>−1</sup>''Hg''}} = {{abs|''H''}} = {{abs|''P''}}}} so that ''g''<sup>−1</sup>''Hg'' = ''P''.}}

{{math theorem|note=3|1=Let ''q'' denote the order of any Sylow ''p''-subgroup ''P'' of a finite group {{var|G}}. Let ''n<sub>p</sub>'' denote the number of Sylow ''p''-subgroups of {{var|G}}.  Then (a) {{math|1=''n<sub>p</sub>'' = [{{var|G}} : ''N<sub>G</sub>''(''P'')]}} (where ''N<sub>G</sub>''(''P'') is the [[normalizer]] of ''P''), (b) {{math|1=''n<sub>p</sub>''}} divides {{math|1={{abs|{{var|G}}}}/''q''}}, and (c) {{math|1=''n<sub>p</sub>'' ≡ 1 (mod&nbsp;''p'')}}.}}

{{math proof|1=Let Ω be the set of all Sylow ''p''-subgroups of {{mvar|G}} and let {{mvar|G}} act on Ω by conjugation. Let {{math|1=''P'' ∈ Ω}} be a Sylow ''p''-subgroup. By Theorem 2, the orbit of ''P'' has size ''n<sub>p</sub>'', so by the orbit-stabilizer theorem {{math|1=''n<sub>p</sub>'' = [{{var|G}} : {{var|G}}<sub>''P''</sub>]}}. For this group action, the stabilizer {{mvar|G}}<sub>''P''</sub> is given by {{math|1={{mset|1= {{var|g}} ∈ {{var|G}} {{pipe}} ''gPg''<sup>−1</sup> = ''P'' }}  = ''N''<sub>G</sub>(''P'')}}, the normalizer of ''P'' in {{var|G}}. Thus, {{math|1=''n<sub>p</sub>'' = [{{var|G}} : ''N<sub>G</sub>''(''P'')]}}, and it follows that this number is a divisor of {{math|1=[{{var|G}} : ''P''] = {{abs|{{var|G}}}}/''q''}}.

Now let ''P'' act on Ω by conjugation, and again let Ω<sub>0</sub> denote the set of fixed points of this action. Let {{math|1=''Q'' ∈ Ω<sub>0</sub>}} and observe that then {{math|1=''Q'' = ''xQx''<sup>−1</sup>}} for all {{math|1=''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, {{math|1={{abs|Ω}} ≡ {{abs|Ω<sub>0</sub>}} = 1 (mod&nbsp;''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<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 Butler,{{sfn|Butler|1991|loc=Chapter 16}} including the algorithm described in Cannon.{{sfn|Cannon|1971}} These versions are still used in the [[GAP computer algebra system]].

In [[permutation group]]s, it has been proven, in Kantor{{sfn|Kantor|1985a}}{{sfn|Kantor|1985b}}{{sfn|Kantor|1990}} and Kantor and Taylor,{{sfn|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 Seress,{{sfn|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 ==
{{Div col}}
* [[Frattini's argument]]
* [[Hall subgroup]]
* [[Maximal subgroup]]
* [[p-group]]
{{Div col end}}

== Notes ==
{{reflist|refs=
<ref name=Sylow1872>{{Cite journal |last=Sylow |first=L. |author-link=Peter Ludwig Mejdell Sylow |date=1872 |title=Théorèmes sur les groupes de substitutions |journal=[[Mathematische Annalen|Math. Ann.]] |volume=5 |issue=4 |pages=584–594 |language=fr |doi=10.1007/BF01442913 |jfm=04.0056.02 |s2cid=121928336 |url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002242052 }}</ref>

<ref name=GraciaSaz_WebPDF>{{Cite web |last=Gracia–Saz |first=Alfonso |title=Classification of groups of order 60 |website=math.toronto.edu |url-status=live |url=http://www.math.toronto.edu/alfonso/347/Groups60.pdf |access-date=8 May 2021 |archive-date=28 October 2020 |archive-url=https://web.archive.org/web/20201028175748/http://www.math.toronto.edu/alfonso/347/Groups60.pdf}}</ref>

<ref name=Fraleigh_2004_322>{{cite book |last=Fraleigh |first=John B. |date=2004 |others=with contribution by Victor J. Katz |title=A First Course In Abstract Algebra |publisher=Pearson Education |page=322 |isbn=9788178089973}}</ref>
}}

==References==
=== Proofs ===
* {{Cite journal |last1=Casadio |first1=Giuseppina |last2=Zappa |first2=Guido |date=1990 |title=History of the Sylow theorem and its proofs |journal=<abbr Title="Bollettino di Storia delle Scienze Matematiche">Boll. Storia Sci. Mat.</abbr> |volume=10 |issue=1 |pages=29–75 |language=it |issn=0392-4432 |mr=1096350 |zbl=0721.01008
}}
* {{Cite journal |last=Gow |first=Rod |date=1994 |title=Sylow's proof of Sylow's theorem |journal=<abbr Title="Irish Mathematical Society Bulletin">Irish Math. Soc. Bull.</abbr> |volume=0033 |issue=33 |pages=55–63 |doi=10.33232/BIMS.0033.55.63 |issn=0791-5578 |mr=1313412 |zbl=0829.01011
|doi-access=free }}
* {{Cite journal |last1=Kammüller |first1=Florian |last2=Paulson |first2=Lawrence C. |date=1999 |title=A formal proof of Sylow's theorem. An experiment in abstract algebra with Isabelle HOL |journal=<abbr Title="Journal of Automated Reasoning">J. Automat. Reason.</abbr> |volume=23 |issue=3 |pages=235–264 |doi=10.1023/A:1006269330992 |issn=0168-7433 |mr=1721912 |zbl=0943.68149 |s2cid=1449341 |archive-date=2006-01-03 |url=http://www.cl.cam.ac.uk/users/lcp/papers/Kammueller/sylow.pdf |archive-url=https://web.archive.org/web/20060103223152/http://www.cl.cam.ac.uk/users/lcp/papers/Kammueller/sylow.pdf }}
* {{Cite journal |last=Meo |first=M. |date=2004 |title=The mathematical life of Cauchy's group theorem |journal=[[Historia Mathematica|Historia Math.]] |volume=31 |issue=2 |pages=196–221 |doi=10.1016/S0315-0860(03)00003-X |issn=0315-0860 |mr=2055642 |zbl=1065.01009 |doi-access=free 
}}
* {{Cite journal |last=Scharlau |first=Winfried |date=1988 |title=Die Entdeckung der Sylow-Sätze |journal=[[Historia Mathematica|Historia Math.]] |volume=15 |issue=1 |pages=40–52 |language=de |doi=10.1016/0315-0860(88)90048-1 |issn=0315-0860 |mr=931678 |zbl=0637.01006 |doi-access=free 
}}
* {{Cite journal |last=Waterhouse |first=William C. |author-link=William C. Waterhouse |date=1980 |title=The early proofs of Sylow's theorem |journal=<abbr Title="Archive for History of Exact Sciences">Arch. Hist. Exact Sci.</abbr> |volume=21 |issue=3 |pages=279–290 |doi=10.1007/BF00327877 |issn=0003-9519 |mr=575718 |zbl=0436.01006
|s2cid=123685226 }}
* {{Cite journal |last=Wielandt |first=Helmut |author-link=:de:Helmut Wielandt |date=1959 |title=Ein Beweis für die Existenz der Sylowgruppen |journal=<abbr Title="Archiv der Mathematik">Arch. Math.</abbr> |volume=10 |issue=1 |pages=401–402 |language=de |doi=10.1007/BF01240818 |issn=0003-9268 |mr=0147529 |zbl=0092.02403|s2cid=119816392 }}

=== Algorithms ===
* {{Cite book |last=Butler |first=G. |date=1991 |title=Fundamental Algorithms for Permutation Groups |series=[[Lecture Notes in Computer Science]] |volume=559 |location=Berlin, New York City |publisher=[[Springer-Verlag]] |doi=10.1007/3-540-54955-2 |isbn=9783540549550 |mr=1225579 |zbl=0785.20001
|s2cid=395110 }}
* {{Cite book |last=Cannon |first=John J. |date=1971 |chapter=Computing local structure of large finite groups |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 City, 1970) |series=<abbr title="SIAM-AMS Proceedings">SIAM-AMS Proc.</abbr> |volume=4 |location=Providence RI |publisher=[[American Mathematical Society|AMS]] |pages=161–176 |issn=0160-7634 |mr=0367027 |zbl=0253.20027
}}
* {{Cite journal |last=Kantor |first=William M. |date=1985a |title=Polynomial-time algorithms for finding elements of prime order and Sylow subgroups |journal=<abbr Title="Journal of Algorithms">J. Algorithms</abbr> |volume=6 |issue=4 |pages=478–514 |doi=10.1016/0196-6774(85)90029-X |issn=0196-6774 |mr=813589 |zbl=0604.20001 |citeseerx=10.1.1.74.3690 |url=http://uoregon.edu/~kantor/PAPERS/primeorder.pdf 
}}
* {{Cite journal |last=Kantor |first=William M. |author-link=William Kantor |date=1985b |title=Sylow's theorem in polynomial time |journal=<abbr Title="Journal of Computer and System Sciences">J. Comput. Syst. Sci.</abbr> |volume=30 |issue=3 |pages=359–394 |doi=10.1016/0022-0000(85)90052-2 |issn=1090-2724 |mr=805654 |zbl=0573.20022
|doi-access=free }}
* {{Cite journal |last1=Kantor |first1=William M. |last2=Taylor |first2=Donald E. |date=1988 |title=Polynomial-time versions of Sylow's theorem |journal=<abbr Title="Journal of Algorithms">J. Algorithms</abbr> |volume=9 |issue=1 |pages=1–17 |doi=10.1016/0196-6774(88)90002-8 |issn=0196-6774 |mr=925595 |zbl=0642.20019
}}
* {{Cite journal |last=Kantor |first=William M. |date=1990 |title=Finding Sylow normalizers in polynomial time |journal=<abbr Title="Journal of Algorithms">J. Algorithms</abbr> |volume=11 |issue=4 |pages=523–563 |doi=10.1016/0196-6774(90)90009-4 |issn=0196-6774 |mr=1079450 |zbl=0731.20005
}}
* {{Cite book |last=Seress |first=Ákos |date=2003 |title=Permutation Group Algorithms |series=Cambridge Tracts in Mathematics |volume=152 |publisher=[[Cambridge University Press]] |isbn=9780521661034 |mr=1970241 |zbl=1028.20002}}

== External links ==
* {{springer|title=Sylow theorems|id=p/s091560}}
* {{Wikibooks-inline|Abstract Algebra/Group Theory/The Sylow Theorems}}
* {{MathWorld |title=Sylow p-Subgroup |id=Sylowp-Subgroup}}
* {{MathWorld |title=Sylow Theorems |id=SylowTheorems}}

[[Category:Theorems about finite groups]]
[[Category:P-groups]]
[[Category:Articles containing proofs]]