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 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 p-Sylow subgroup) of a group 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.