Abelian

Every subgroup of an abelian group is abelian.

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

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

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.

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

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

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.

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.