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Abelian Group

Defines Abelian Group, abelian group, commutative group

An abelian group is a group (G,⋅,e)(G, \cdot, e) in which the operation is commutative: a⋅b=b⋅aa \cdot b = b \cdot a for all a,b∈Ga, b \in G. The name honors Niels Henrik Abel.

The integers under addition, the rationals under addition, and the nonzero rationals under multiplication are all abelian groups. The lattice operations meet and join are commutative, so any lattice that forms a group under either operation is abelian.

Abelian groups are the groups whose structure is most completely understood. Every finitely generated abelian group decomposes as a direct sum of cyclic groups — this is the fundamental theorem of finitely generated abelian groups.

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@misc{emsenn2026-abelian-group,
  author    = {emsenn},
  title     = {Abelian Group},
  year      = {2026},
  url       = {https://emsenn.net/library/math/domains/algebra/terms/abelian-group/},
  publisher = {emsenn.net},
  license   = {CC BY-SA 4.0}
}