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^[1] 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 ${\displaystyle p}$, a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group ${\displaystyle G}$ is a maximal ${\displaystyle p}$-subgroup of ${\displaystyle G}$, i.e., a subgroup of ${\displaystyle G}$ that is a p-group (meaning its cardinality is a power of ${\displaystyle p;}$ or equivalently: For each group element, its order is some power of ${\displaystyle p}$) that is not a proper subgroup of any other ${\displaystyle p}$-subgroup of ${\displaystyle G}$. The set of all Sylow ${\displaystyle p}$-subgroups for a given prime ${\displaystyle p}$ is sometimes written ${\displaystyle {\text{Syl}}_{p}(G)}$.

The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group ${\displaystyle G}$ the order (number of elements) of every subgroup of ${\displaystyle G}$ divides the order of ${\displaystyle G}$. The Sylow theorems state that for every prime factor ${\displaystyle p}$ of the order of a finite group ${\displaystyle G}$, there exists a Sylow ${\displaystyle p}$-subgroup of ${\displaystyle G}$ of order ${\displaystyle p^{n}}$, the highest power of ${\displaystyle p}$ that divides the order of ${\displaystyle G}$. Moreover, every subgroup of order ${\displaystyle p^{n}}$ is a Sylow ${\displaystyle p}$-subgroup of ${\displaystyle G}$, and the Sylow ${\displaystyle p}$-subgroups of a group (for a given prime ${\displaystyle p}$) are conjugate to each other. Furthermore, the number of Sylow ${\displaystyle p}$-subgroups of a group for a given prime ${\displaystyle p}$ is congruent to 1 (mod ${\displaystyle p}$).