# Abelian

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Abelian

Abelian, in mathematics, is used in many different definitions, named after Norwegian mathematician Niels Henrik Abel:

In group theory:

*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

In Galois theory:

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

In real analysis:

*Abelian and tauberian theorems, used in the summation of divergent series

In functional analysis

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

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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### Look at other dictionaries:

• Abelian — A*bel i*an, Abelite A bel*ite, Abelonian A bel*o ni*an, n. (Eccl. Hist.) One of a sect in Africa (4th century), mentioned by St. Augustine, who states that they married, but lived in continence, after the manner, as they pretended, of Abel. [1913 …   The Collaborative International Dictionary of English

• abelian — adjective Having a commutative defining operation. See Also: abelian algebra, abelian group, abelian operation, abelian ring …   Wiktionary

• abelian — adjective Usage: often capitalized Etymology: Niels Abel died 1829 Norwegian mathematician Date: 1847 commutative 2 < abelian group > < abelian ring > …   New Collegiate Dictionary

• abelian — ə¦bēlyən, lēən adjective Usage: often capitalized Etymology: Niels Abel died 1829 Norwegian mathematician + English ian : commutative 2 abelian ring the real numbers under addition comprise an abelian group …   Useful english dictionary

• Abelian — /euh bee lee euhn, euh beel yeuhn/, adj. Math. 1. of or pertaining to Niels Henrik Abel or his theorems. 2. pertaining to an algebraic system in which an operation is commutative: an Abelian group. 3. pertaining to such an operation. [1905 10;… …   Universalium

• Abelian — noun A member of a sect in fourth century Africa mentioned by St. Augustine, who states that they married but lived in continence after the manner, as they claimed, of Abel. Syn: Abelite, Abelonian …   Wiktionary

• abelián — adj. m. (sil. li an), pl. abeliéni; f. sg. abeliánã, pl. abeliéne …   Romanian orthography

• abelian — [ə bi:lɪən] adjective Mathematics (of a group) having members related by a commutative operation (i.e. a*b = b*a). Origin C19: from the name of the Norwegian mathematician Niels Henrik Abel + ian …   English new terms dictionary

• abelian — abe·lian …   English syllables

• Abelian — A•be•li•an [[t]əˈbi li ən, əˈbil yən[/t]] adj. math. of or pertaining to an algebraic system in which an operation is commutative • Etymology: 1905–10; after Niels Henrik Abel (1802–29), Norwegian mathematician …   From formal English to slang