Sylow theorems

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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...

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