Abelian, in mathematics, is used in many different definitions:

In group theory:

• Abelian group, a group in which the binary operation is commutative
  • Category of abelian groups Ab has abelian groups as objects and group homomorphisms as morphisms
• Metabelian group is a group where the commutator subgroup is contained in the center
• Abelianisation

In Galois theory: In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. ... In mathematics, a metabelian group is a group where the commutator subgroup is abelian. ... In mathematics, the derived group (or commutator subgroup) of a group G is the subgroup G1 generated by all the commutators of elements of G; that is, G1 = <[g,h] : g,h in G>. Note that the set of all commutators of the group is, generally, not a group (in...

• Abelian extension, a field extension for which the associated Galois group is abelian

In real analysis: In abstract algebra, an abelian extension is a field extension for which the associated Galois group is abelian. ...

• Abelian theorems, used in the summation of divergent series

In functional analysis In mathematics, a large number of methods have been proposed for the summation of divergent series. ...

• Abelian von Neumann algebra, a von Neumann algebra of operators on a Hilbert space in which all elements commute

In topology and number theory In functional analysis, an Abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all its elements commute. ...

• Abelian variety, a complex torus that can be embedded into projective space
• 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

In category theory: In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ... For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i. ... In mathematics, an abelian integral in Riemann surface theory is a function related to the indefinite integral of a differential of the first kind. ...

• Preabelian category is an additive category that has all kernels and cokernels
• Abelian category, a preabelian category in which morphisms and objects can be added.