correspondence between normal subgroups and homomorphic images

Assume, that G and H are groups. If f:GH is a group homomorphismMathworldPlanetmath, then the first isomorphism theoremPlanetmathPlanetmath states, that the function F:G/ker(f)im(f) defined by F(gker(f))=f(g) is a well-defined group isomorphism. Note that ker(f) is always normal in G.

This leads to the following question: is there a correspondence between normal subgroupsMathworldPlanetmath of G and homomorphic imagesPlanetmathPlanetmathPlanetmath of G? We will try to answer this question, but before that, let us introduce some notion.

First of all, homomorphic image im(f) is not only a subgroupMathworldPlanetmathPlanetmath of H. Actually homomorphic image contains also some data about homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. This observation leads to the following definition:

Definition. Let G be a group. Pair (H,f) is called a homomorphic image of G iff H is a group and f:GH is a surjectivePlanetmathPlanetmath group homomorphism. We will say that two homomorphic images (H,f) and (H,f) of G are isomorphic (or equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath), if there exists a group isomorphism F:HH such that Ff=f.

It is easy to see, that this isomorphism relationMathworldPlanetmathPlanetmathPlanetmath is actually an equivalence relation and thus we may speak about isomorphism classes of homomorphic images (which will be denoted by [H,f] for homomorphic image (H,f)). Furthermore, if NG is a normal subgroup, then (G/N,πN) is a homomorphic image, where πN:GG/N is a projectionPlanetmathPlanetmath, i.e. πN(g)=gN. Let

norm(G)={NG|N is normal subgroup};{[H,f]|(H,f) is a homomorphic image of G}.

PropositionPlanetmathPlanetmath. Function T:norm(G) defined by T(N)=[G/N,πN] is a bijection.

Proof. First, we will show, that T is onto. Let (H,f) be a homomorphic image of G. Let N=ker(f). Then (due to the first isomorphism theorem), there exists a group isomorphism F:G/NH defined by F(gN)=f(g). This shows, that


and thus (G/N,πN) is isomorphic to (H,f). Therefore


which completesPlanetmathPlanetmathPlanetmathPlanetmath this part.

Now assume, that T(N)=T(N) for some normal subgroups N,Nnorm(G). This means, that (G/N,πN) and (G/N,πN) are isomorphic, i.e. there exists a group isomorphism F:G/NG/N such that FπN=πN. Let xN=ker(πN) and denote by eG/N the neutral element. Then, we have


and (since F is an isomorphism) this is if and only if xker(πN)=N. Thus, we’ve shown that NN. Analogously (after considering F-1) we have that NN. Therefore N=N, which shows, that T is injectivePlanetmathPlanetmath. This completes the proof.

Title correspondence between normal subgroups and homomorphic images
Canonical name CorrespondenceBetweenNormalSubgroupsAndHomomorphicImages
Date of creation 2013-03-22 19:07:11
Last modified on 2013-03-22 19:07:11
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Theorem
Classification msc 20A05
Classification msc 13A15
Related topic HomomorphicImageOfGroup