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 that ψπ=ϕ, where π:GG/[G,G] is the canonical projection.

Title abelianization
Canonical name Abelianization
Date of creation 2013-03-22 14:52:57
Last modified on 2013-03-22 14:52:57
Owner yark (2760)
Last modified by yark (2760)
Numerical id 7
Author yark (2760)
Entry type Definition
Classification msc 20F14
Synonym abelianisation
Related topic DerivedSubgroup