Abelian Abelian Boyar In Mathematics, the term abelian is used in many different definitions:

In Group theory

An Abelian group is a group in which the Binary operation is commutative.
The Category of abelian groups Ab has abelian groups as objects and group homomorphisms as morphisms.
A Metabelian group is a group where the commutator subgroup is contained in the center.
Any group is "made abelian" by its abelianisation.

In Galois theory

An Abelian extension is a Field extension for which the associated Galois group is abelian.

In Real analysis

Abelian theorems are used in the summation of Divergent series.

In Functional analysis

An Abelian von Neumann algebra is a Von Neumann algebra of operators on a Hilbert space in which all elements commute.

In Topology and Number theory

An Abelian variety is a complex Torus that can be embedded into Projective space. This includes the elliptic curves and the abelian varieties of CM-type.
An abelian function is a Meromorphic function on an abelian variety.
An Abelian integral is a function related to the indefinite integral of a Differential of the first kind. This includes the elliptic integrals.

In Category theory

A preabelian category is an Additive category that has all kernels and cokernels.
An Abelian category is a preabelian category in which morphisms and objects can be added.