Sylow theorems

Algebraic structure → Group theory Group theory
Basic notions Subgroup Normal subgroup Quotient group (Semi-)direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable action Glossary of group theory List of group theory topics
Finite groups Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Classification of finite simple groups cyclic alternating Lie type sporadic
Discrete groups Lattices Integers (${\displaystyle \mathbb {Z} }$) Free group Modular groups PSL(2, ${\displaystyle \mathbb {Z} }$) SL(2, ${\displaystyle \mathbb {Z} }$) Arithmetic group Lattice Hyperbolic group
Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve
v t e

This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (November 2018) (Learn how and when to remove this message)

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^[1] 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 Sylow p-subgroup (sometimes p-Sylow subgroup) of a group ${\displaystyle G}$ is a maximal ${\displaystyle p}$-subgroup of ${\displaystyle G}$, i.e., a subgroup of ${\displaystyle G}$ that is a p-group (meaning its cardinality is a power of ${\displaystyle p,}$ or equivalently, the order of every group element is a power of ${\displaystyle p}$) that 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 ${\displaystyle p}$).