Sylow theorems

In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed 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 p {\displaystyle p} , a p-group is a group in which the order of every element is a power of p {\displaystyle p} ; for finite groups, this is equivalent to the group's cardinality being a power of p {\displaystyle p} . A Sylow p-subgroup of a group G {\displaystyle G} is a maximal p {\displaystyle p} -subgroup—that is, a p-subgroup of G {\displaystyle G} not contained in any strictly larger p-subgroup. The set of all Sylow p {\displaystyle p} -subgroups of G {\displaystyle G} for a given prime p {\displaystyle p} is denoted Syl p ( G ) {\displaystyle {\text{Syl}}_{p}(G)} . The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group G {\displaystyle G} the order (number of elements) of every subgroup of G {\displaystyle G} divides the order of G {\displaystyle G} . The Sylow theorems state that for every prime factor p {\displaystyle p} of the order of a finite group G {\displaystyle G} , there exists a Sylow p {\displaystyle p} -subgroup of G {\displaystyle G} of order p n {\displaystyle p^{n}} , the highest power of p {\displaystyle p} that divides the order of G {\displaystyle G} . Moreover, every subgroup of order p n {\displaystyle p^{n}} is a Sylow p {\displaystyle p} -subgroup of G {\displaystyle G} , and the Sylow p {\displaystyle p} -subgroups...