abelian group

English

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abelian group

Wikipedia

Alternative forms

• Abelian group

Etymology

Named in honour of Niels Henrik Abel (1802–1829), a Norwegian mathematician.

Pronunciation

• (US) IPA^(key): /əˈbi.li.ən ɡɹup/, /əˈbil.jən ɡɹup/

Noun

abelian group (plural abelian groups)

1. (algebra) A group in which the group operation is commutative.
   • 1986, Partially Ordered Abelian Groups with Interpolation, American Mathematical Society, 2010 softcover reprint, page 12,

     Let G and H be partially ordered abelian groups. A positive homomorphism from G to H is any abelian group homomorphism ${\displaystyle f:G\rightarrow H}$ that maps positive elements to positive elements, that is, ${\displaystyle f(G^{+})\subseteq H^{+}}$.

   • 2000, David Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, Springer, Softcover reprint of 1st edition, page 74,

     Chapter 2 is a brief introduction to some fundamental techniques for countable torsion-free abelian groups.

   • 2013, Karl H. Hofmann, Sidney A. Morris, The Structure of Compact Groups: A Primer for the Student: A Handbook for the Expert, Walter de Gruyter, page 299:

     By the end of Chapter 2 we had the full power of the Pontryagin Duality Theorem for compact abelian groups and for discrete abelian groups. Locally compact abelian groups are much closer to compact abelian groups than is apparent at first sight.

Synonyms

• commutative group

Hypernyms

• group
• solvable group

Hyponyms

• cyclic group
• free abelian group
• torsion subgroup

Derived terms

• free abelian group

Translations

Further reading

• abelian group, Abelian group at Google Ngram Viewer