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In mathematics, abelian refers to any of a number of different mathematical concepts named after Niels Henrik Abel:

Group theory[edit]

Galois theory[edit]

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

Real analysis[edit]

Functional analysis[edit]

Topology and number theory[edit]

Category theory[edit]

  • Pre-abelian 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


  • A gauge theory is abelian or non-abelian depending on whether its symmetry group is commutative or non-commutative, respectively

Other uses[edit]