abelian group

See also: Abelian group

English

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 $f:G\rightarrow H$ that maps positive elements to positive elements, that is, $f(G^{+})\subseteq H^{+}$.

   • 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

group in which the group operation is commutative

• Chinese:

  Mandarin: 阿貝爾群／阿贝尔群 (zh) (Ābèi'ěr qún)

• Czech: abelovská grupa f
• Dutch: Abelse groep m, Abelse ring m
• Esperanto: abela grupo, komuta grupo
• Estonian: Abeli rühm (et)
• Finnish: Abelin ryhmä
• French: groupe abélien (fr) m, groupe commutatif (fr)
• German: abelsche Gruppe f, kommutative Gruppe f
• Greek: αβελιανή ομάδα (el) f (avelianí omáda)
• Hungarian: Abel-csoport (hu)
• Icelandic: Abel-grúpa f, víxlgrúpa f, abelsk grúpa f, víxlin grúpa f
• Italian: gruppo abeliano m
• Japanese: アーベル群 (Āberu-gun)
• Norwegian:

  Bokmål: abelsk gruppe m or f

• Polish: grupa przemienna f, grupa abelowa (pl) f
• Portuguese: grupo abeliano m
• Romanian: grup abelian n, grup comutativ n
• Russian: а́белева гру́ппа (ru) f (ábeleva grúppa)
• Serbo-Croatian: Abelova grupa f
• Spanish: grupo abeliano m
• Swedish: abelsk grupp c, kommutativ grupp c
• Telugu: ఎబిలియన్ సమూహం (ebiliyan samūhaṁ)

The translations below need to be checked and inserted above into the appropriate translation tables. See instructions at Wiktionary:Entry layout § Translations.

Translations to be checked

• Chinese:

  Mandarin: ^{(please verify)} 模 (zh) (mó)

Further reading

• abelian group, Abelian group at the Google Books Ngram Viewer.