Abelian

Abelian is an adjective used in mathematics to honor the work of the 19th‑century Norwegian mathematician Niels Henrik Abel and to denote commutativity or commutative structures in algebra and related areas. In modern use, many concepts carry the label Abelian because their defining operations are independent of order.

The most familiar Abelian object is the Abelian group. An Abelian group is a set G with a binary operation, usually written as addition, that combines any two elements to another element of G and satisfies closure, associativity, the existence of an identity element, and inverses, together with the commutativity a + b = b + a for all a, b in G. Examples include the integers under addition, real numbers under addition, and more generally any vector space under vector addition. Abelian groups are closed under subgroups, quotients, and homomorphisms, and play a foundational role in algebra. Finite Abelian groups are classified up to isomorphism by the fundamental theorem of finite Abelian groups.

Beyond groups, the term Abelian appears in category theory and algebraic geometry. An Abelian category is a category with a zero object, finite limits and colimits, and exactness properties that generalize module categories; it provides a natural setting for homological algebra. An Abelian variety is a complete algebraic variety equipped with a group law defined by regular maps; these objects generalize elliptic curves to higher dimensions and are central in number theory and algebraic geometry, with applications to rational points and moduli spaces.

The concept also extends to notions like the Abelianization of a group, which yields the largest Abelian quotient by factoring out the commutator subgroup, reflecting the passage from noncommutative to commutative structure.