Abelian

Abelian, in mathematics, is used in many different definitions, named after Norwegian mathematician Niels Henrik Abel:

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 abelian
*Abelianisation

In Galois theory:

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

In real analysis:

*Abelian and tauberian theorems, used in the summation of divergent series

In functional analysis

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

In topology and number theory

*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:

*Preabelian category, an additive category that has all kernels and cokernels
*Abelian category, a preabelian category in which every monomorphism is a kernel and every epimorphism is a cokernel