# Abelian

From Example Problems

In mathematics, the term **abelian** is used in many different definitions:

## Contents

## 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.