In mathematics, specifically in the field of finite group theory, the **Sylow theorems** are a collection of theorems named after the Norwegian mathematician Ludwig Sylow (1872) 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*, a **Sylow p-subgroup** (sometimes

**) of a group**

*p*-Sylow subgroup*G*is a maximal

*p*-subgroup of

*G*, i.e., a subgroup of

*G*that is a

*p*-group (so that the order of any group element is a power of

*p*), and that is not a proper subgroup of any other

*p*-subgroup of

*G*. The set of all Sylow

*p*-subgroups for a given prime

*p*is sometimes written Syl

_{p}(

*G*).

The Sylow theorems assert a partial converse to Lagrange's theorem that for any finite group *G* the order (number of elements) of every subgroup of *G* divides the order of *G*. For any prime factor *p* of the order of a finite group *G*, there exists a Sylow *p*-subgroup of *G*. The order of a Sylow *p*-subgroup of a finite group *G* is *pn*, where *n* is the multiplicity of *p* in the order of *G*, and any subgroup of order *pn* is a Sylow *p*-subgroup of *G*. The Sylow *p*-subgroups of a group (for fixed prime *p*) are conjugate to each other. The number of Sylow *p*-subgroups of a group for fixed prime *p* is congruent to 1 mod *p*.

Read more about Sylow Theorems: Sylow Theorems, Examples, Proof of The Sylow Theorems, Algorithms

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