abelian group

Noun

Meaning

a group that satisfies the commutative law

Synonyms

Hypernyms

Examples

Additive groups of rings The additive group of a ring is an abelian group, but not all abelian groups are additive groups of rings (with nontrivial multiplication). Source: Internet

Classification The fundamental theorem of finite abelian groups states that every finite abelian group G can be expressed as the direct sum of cyclic subgroups of prime -power order. Source: Internet

Conversely, given any ring, (R, +, · ), (R, +) is an abelian group. Source: Internet

Every subgroup and factor group of a finitely generated abelian group is again finitely generated abelian. Source: Internet

Examples * The integers are a finitely generated abelian group. Source: Internet

For example, an abelian group may have two different subgroups which are isomorphic, but they are never conjugate. Source: Internet

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