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 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 , a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group is a maximal -subgroup of , i.e., a subgroup of that is a p-group (meaning its cardinality is a power of or equivalently, the order of every group element is a power of ) that is not a proper subgroup of any other -subgroup of . The set of all Sylow -subgroups for a given prime is sometimes written . The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group the order (number of elements) of every subgroup of divides the order of . The Sylow theorems state that for every prime factor of the order of a finite group , there exists a Sylow -subgroup of of order , the highest power of that divides the order of . Moreover, every subgroup of order is a Sylow -subgroup of , and the Sylow -subgroups of a group (for a given prime ) are conjugate to each other. Furthermore, the number of Sylow -subgroups of a group for a given prime is congruent to 1 (mod ). The Sylow theorems are a powerful statement about the structure of groups in general, but are also powerful in applications of finite group theory. This is because they give a method for using the prime decomposition of the cardinality of a finite group to give statements about the structure of its subgroups: essentially, it gives a technique to transport basic number-theoretic information about a group to its group structure. From this observation, classifying finite groups becomes a game of finding which combinations/constructions of groups of smaller order can be applied to construct a group.