Editing Sylow theorems (section)

{{short description|Theorems that help decompose a finite group based on prime factors of its order}}
{{Group theory sidebar |Finite}}
{{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 [[p-group|''p''-group]] is a group whose [[cardinality]] is a [[Power (mathematics)|power]] of <math>p;</math> or equivalently, the [[Order of a group element|order]] of each group element is some power of <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 and 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>).