Editing Sylow theorems (section)

=== 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.