abelian group

[/əˈbeɪliən ɡruːp/]
nounpl: abelian groups
grupo abeliano
1. In abstract algebra, a group in which the group operation is commutative, meaning that for any two elements a and b in the group, a * b = b * a. Named after the Norwegian mathematician Niels Henrik Abel.
The set of integers under addition is an abelian group because the order of addition does not affect the result.
O conjunto dos números inteiros sob adição é um grupo abeliano porque a ordem da adição não afeta o resultado.
2. A mathematical structure satisfying the group axioms with the additional requirement that the binary operation is commutative.
Every abelian group can be expressed as a direct sum of cyclic groups of prime power order.
Todo grupo abeliano pode ser expresso como uma soma direta de grupos cíclicos de ordem potência prima.
This is a specialized mathematical term used primarily in academic and educational contexts. It is not subject to regional variation in meaning, though teaching methods and emphasis may differ between Brazilian and Portuguese universities. The term honors Norwegian mathematician Niels Henrik Abel and is universally recognized in the mathematical community.
Synonyms / Sinônimos
commutative groupcommutative algebraic structure
Antonyms / Antônimos
non-abelian groupnon-commutative group

Regional Variations

General Brazilian
grupo abeliano
Standard mathematical terminology used in universities and academic contexts
Portugal
grupo abeliano
Same as Brazilian Portuguese; universal mathematical term
International Academic
grupo abeliano / grupo commutativo
Both terms are accepted, though 'grupo abeliano' is more common

Related Words

group theorycommutative propertycyclic groupabstract algebragroup operationNiels Henrik Abel

Related Idioms & Phrases

abelian group structure
properties of abelian groups
fundamental theorem of finitely generated abelian groups
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