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 ).
Theorems
Motivation
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. For example, a typical application of these theorems is in the classification of finite groups of some fixed cardinality, e.g. .
Statement
Collections of subgroups that are each maximal in one sense or another are common in group theory. The surprising result here is that in the case of , all members are actually isomorphic to each other and have the largest possible order: if with where does not divide , then every Sylow -subgroup has order . That is, is a -group and . These properties can be exploited to further analyze the structure of .
The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in Mathematische Annalen.
The following weaker version of theorem 1 was first proved by Augustin-Louis Cauchy, and is known as Cauchy's theorem.
Consequences
The Sylow theorems imply that for a prime number every Sylow -subgroup is of the same order, . Conversely, if a subgroup has order , then it is a Sylow -subgroup, and so is conjugate to every other Sylow -subgroup. Due to the maximality condition, if is any -subgroup of , then is a subgroup of a -subgroup of order .
A very important consequence of Theorem 2 is that the condition is equivalent to saying that the Sylow -subgroup of is a normal subgroup. However, there are groups that have normal subgroups but no normal Sylow subgroups, such as .
Sylow theorems for infinite groups
There is an analogue of the Sylow theorems for infinite groups. One defines a Sylow -subgroup in an infinite group to be a p-subgroup (that is, every element in it has -power order) that is maximal for inclusion among all -subgroups in the group. Let denote the set of conjugates of a subgroup .
Examples
A simple illustration of Sylow subgroups and the Sylow theorems are the dihedral group of the n-gon, D2n. For n odd, 2 = 21 is the highest power of 2 dividing the order, and thus subgroups of order 2 are Sylow subgroups. These are the groups generated by a reflection, of which there are n, and they are all conjugate under rotations; geometrically the axes of symmetry pass through a vertex and a side.
By contrast, if n is even, then 4 divides the order of the group, and the subgroups of order 2 are no longer Sylow subgroups, and in fact they fall into two conjugacy classes, geometrically according to whether they pass through two vertices or two faces. These are related by an outer automorphism, which can be represented by rotation through π/n, half the minimal rotation in the dihedral group.