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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

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

Abelian theorems, used in the summation of divergent series

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

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.