proof of third isomorphism theorem

We’ll give a proof of the third isomorphism theorem using the Fundamental homomorphism theoremMathworldPlanetmath.

Let G be a group, and let KH be normal subgroupsMathworldPlanetmath of G. Define p,q to be the natural homomorphismsMathworldPlanetmathPlanetmath from G to G/H, G/K respectively:


K is a subset of ker(p), so there exists a unique homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath φ:G/KG/H so that φq=p.

p is surjectivePlanetmathPlanetmath, so φ is surjective as well; hence imφ=G/H. The kernel of φ is ker(p)/K=H/K. So by the first isomorphism theoremPlanetmathPlanetmath we have

Title proof of third isomorphism theorem
Canonical name ProofOfThirdIsomorphismTheorem
Date of creation 2013-03-22 15:35:09
Last modified on 2013-03-22 15:35:09
Owner Thomas Heye (1234)
Last modified by Thomas Heye (1234)
Numerical id 5
Author Thomas Heye (1234)
Entry type Proof
Classification msc 20A05