[uh-bee-lee-uhn, uh-beel-yuhn]
Abelian, in mathematics, is used in many different definitions, named after Norwegian mathematician Niels Henrik Abel:

In group theory:

In Galois theory:

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

In real analysis:

In functional analysis

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

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