Group theory

• Abelian group
  Abelian group
  In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
  
  , a group in which the binary operation is commutative
  • Category of abelian groups
    Category of abelian groups
    In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category....
    
     Ab has abelian groups as objects and group homomorphisms as morphisms
• Metabelian group
  Metabelian group
  In mathematics, a metabelian group is a group whose commutator subgroup is abelian. Equivalently, a group G is metabelian if and only if there is an abelian normal subgroup A such that the quotient group G/A is abelian....
  
   is a group where the commutator subgroup is abelian
• Abelianisation

Galois theory

• Abelian extension
  Abelian extension
  In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable....
  
  , a field extension for which the associated Galois group is abelian

Functional analysis

• Abelian von Neumann algebra
  Abelian von Neumann algebra
  In functional analysis, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute.The prototypical example of an abelian von Neumann algebra is...
  
  , a von Neumann algebra of operators on a Hilbert space in which all elements commute

Topology and number theory

• Abelian variety
  Abelian variety
  In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
  
  , a complex torus that can be embedded into projective space
• Abelian surface
  Abelian surface
  In mathematics, an abelian surface is 2-dimensional abelian variety.One dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic...
  
  , a two-dimensional abelian variety
• Abelian function, a meromorphic function on an abelian variety
• Abelian integral, a function related to the indefinite integral of a differential of the first kind

• Pre-abelian category
  Pre-Abelian category
  In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels.Spelled out in more detail, this means that a category C is pre-abelian if:...
  
  , an additive category that has all kernels and cokernels
• Abelian category
  Abelian category
  In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
  
  , a preabelian category in which every monomorphism is a kernel and every epimorphism is a cokernel

• A gauge theory
  Gauge theory
  In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
  
   is abelian or non-abelian depending on whether its symmetry group is commutative or non-commutative, respectively

Other uses

• Abelians
  Abelians
  Abelians were a Christian sect that sprang up in the 4th century near Hippo Regius in north Africa during the reign of Arcadius. They lived in continence as they affirmed Abel did. They were required to be married but were forbidden to consummate the marriage. Each couple was required to adopt two...
  
  , 4th-century Christian sect
• Hovhannes Abelian
  Hovhannes Abelian
  Hovhannes Harutyuni Abelian was an Armenian actor, People's Artist of Armenian SSR .- Biography :Since 1882, he worked in Armenian and Russian theatres of Baku and Tiflis...
  
  (1865–1936), Armenian actor