abelian

1. not comparablemathematicsOf an algebraic structure (usually a group or algebra), having a commutative defining operation.First known use Mid 19ᵗʰ century.

   The interesting aspect here is that U₃ is irreducible, even though all irreps over the complexes are one-dimensional because ℤ₄ is abelian.

   1996, Barry Simon, Representations of Finite and Compact Groups, page 50

2. not comparablemathematicsOf a binary function, commutative.

1. datednot comparablegroup theoryAlternative letter-case form of abelian.

   All irreducible Abelian equations of the tenth order belong to the cyclical group of order ten,[…].

   1914, Charles Gustave Paul Kuschke, The Abelian Equations of the Tenth Degree Irreducible in a Given Domain of Rationality, University of California Press, page 118

   2 more examples

   Ex. 2. Show by §§ 187, 188 that there can be no transitive Abelian group of prime degree other than the cyclic group, and that there is no irreducible Abelian equation of prime degree other than the cyclic equation.

   1904, Florian Cajori, An Introduction to the Modern Theory of Equations, The MacMillan Company, page 214

   In this chapter "group" means torsion free Abelian group. The "bible" of Abelian group theory is the two volume work of Fuchs [11].

   2004, Friedrich Kasch, Adolf Mader, Rings, Modules, and the Total, Springer (Birkhäuser), page 101

plural Abelians

1. historicalChristianityA 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.

   If we include heretical sects there is perhaps some similarity to be traced between the syneisaktism of certain Catholic enthusiasts and the customs of the Abelians or Abelonians, who took their name from Abel.

   1951, Reginald Reynolds, Beds: with many noteworthy instances of lying on, under, or about them, Doubleday, page 256

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   1826, unnamed translator, Pierre Bayle, An Historical and Critical Dictionary [1697, Dictionnaire Historique et Critique; 1709, Historical and Critical Dictionary], Selected and Abridged, Volume 1, page 50, The Abelians were only a moderate sort of Encratites and Novatians, who absolutely condemned matrimony, while the Abelians approved of and retained it.

   A difference of opinion existed as to the moral effects of matrimony among the early Christians; the Abelians, for instance, suffered no member of their sect to remain single, while the Novatians condemned marriage under every circumstance.

   1833, James Flamank, A Treatise on Happiness, page 222

   Synonyms: Abelite, Abelonian, Abelonite, Abeloite, Abenonite