Editing Sylow theorems (section)

== 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>.}}