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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 L. 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 psubgroup (sometimes pSylow subgroup) of a group G is a maximal psubgroup of G, i.e., a subgroup of G which is a pgroup (so that the order of any group element is a power of p), and which is not a proper subgroup of any other psubgroup of G. The set of all Sylow psubgroups 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 psubgroup of G. The order of a Sylow psubgroup of a finite group G is p^{n}, where n is the multiplicity of p in the order of G, and any subgroup of order p^{n} is a Sylow psubgroup of G. The Sylow psubgroups of a group (for fixed prime p) are conjugate to each other. The number of Sylow psubgroups of a group for fixed prime p is congruent to 1 mod p.
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