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 ${\displaystyle p}$, a p-group is a group whose cardinality is a power of ${\displaystyle p;}$ or equivalently, the order of each group element is some power of ${\displaystyle p}$. A Sylow p-subgroup (sometimes p-Sylow subgroup) of a finite group ${\displaystyle G}$ is a maximal ${\displaystyle p}$-subgroup of ${\displaystyle G}$, i.e., a subgroup of ${\displaystyle G}$ that is a p-group and is not a proper subgroup of any other ${\displaystyle p}$-subgroup of ${\displaystyle G}$. The set of all Sylow ${\displaystyle p}$-subgroups for a given prime ${\displaystyle p}$ is sometimes written ${\displaystyle {\text{Syl}}_{p}(G)}$.

The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group ${\displaystyle G}$ the order (number of elements) of every subgroup of ${\displaystyle G}$ divides the order of ${\displaystyle G}$. The Sylow theorems state that for every prime factor ${\displaystyle p}$ of the order of a finite group ${\displaystyle G}$, there exists a Sylow ${\displaystyle p}$-subgroup of ${\displaystyle G}$ of order ${\displaystyle p^{n}}$, the highest power of ${\displaystyle p}$ that divides the order of ${\displaystyle G}$. Moreover, every subgroup of order ${\displaystyle p^{n}}$ is a Sylow ${\displaystyle p}$-subgroup of ${\displaystyle G}$, and the Sylow ${\displaystyle p}$-subgroups of a group (for a given prime ${\displaystyle p}$) are conjugate to each other. Furthermore, the number of Sylow ${\displaystyle p}$-subgroups of a group for a given prime ${\displaystyle p}$ is congruent to 1 (mod ${\displaystyle p}$).

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 ${\displaystyle G}$ 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. ${\displaystyle |G|=60}$.

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 ${\displaystyle \operatorname {Syl} _{p}(G)}$, all members are actually isomorphic to each other and have the largest possible order: if ${\displaystyle |G|=p^{n}m}$ with ${\displaystyle n>0}$ where p does not divide m, then every Sylow p-subgroup P has order ${\displaystyle |P|=p^{n}}$. That is, P is a p-group and ${\displaystyle {\text{gcd}}(|G:P|,p)=1}$. These properties can be exploited to further analyze the structure of G.

The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in Mathematische Annalen.

Theorem (1)—For every prime factor p with multiplicity n of the order of a finite group G, there exists a Sylow p-subgroup of G, of order ${\displaystyle p^{n}}$.

The following weaker version of theorem 1 was first proved by Augustin-Louis Cauchy, and is known as Cauchy's theorem.