A set of three theorems about finite groups describing the existence, conjugacy and number of maximal p-subgroups (Sylow subgroups) for a given prime p dividing the group's order. Proved by Ludwig Sylow, they are the single most used tool for analyzing the structure of finite groups.
Facts
StatementFor every prime factor p of the order of a finite group G, there exists a Sylow p-subgroup of G of order p^n, the highest power of p dividing the order of G. Every subgroup of order p^n is a Sylow p-subgroup, all Sylow p-subgroups of G for a given prime p are conjugate to each other, and their number is congruent to 1 modulo p. 1 Classification
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In Branch
Sources
1. Sylow Theorems (Wikipedia)
Wikimedia FoundationLead section, statement sentence
The Sylow theorems state that for every prime factor p of the order of a finite group G, there exists a Sylow p-subgroup of G of order p^n, the highest power of p that divides the order of G.
History note on Sylow's 1872 proof
The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in Mathematische Annalen.
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