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单词 Abelianization
释义

abelianization


The abelianizationMathworldPlanetmath of a group G is G/[G,G], the quotient (http://planetmath.org/QuotientGroup) of G by its derived subgroup.

The abelianization of G is the largest abelianMathworldPlanetmath quotient of G, in the sense that if N is a normal subgroupMathworldPlanetmath of G then G/N is abelian if and only if [G,G]N.In particular, every abelian quotient of G is a homomorphic imagePlanetmathPlanetmathPlanetmath of G/[G,G].

If A is an abelian group and ϕ:GA is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/GroupHomomorphism),then there is a unique homomorphism ψ:G/[G,G]A such thatψπ=ϕ, where π:GG/[G,G] is the canonical projection.

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更新时间:2025/5/5 0:57:43