natural projection

PropositionPlanetmathPlanetmathPlanetmath.  If H is a normal subgroupMathworldPlanetmath of a group G, then the mapping


is a surjectivePlanetmathPlanetmath homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath whose kernel is H.

Proof.  Because every coset appears as image, the mapping φ is surjective.  It is also homomorphic, since for all elements x,y of G, one has


The identity elementMathworldPlanetmath of the factor group G/H is the coset  eH=H,  whence


The mapping φ in the proposition is called natural projectionMathworldPlanetmath or canonical homomorphism.

Title natural projection
Canonical name NaturalProjection
Date of creation 2013-03-22 19:10:16
Last modified on 2013-03-22 19:10:16
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 4
Author pahio (2872)
Entry type Definition
Classification msc 20A05
Synonym canonical homomorphism
Synonym natural homomorphismMathworldPlanetmath
Related topic QuotientGroup
Related topic KernelOfAGroupHomomorphismIsANormalSubgroup