correspondence between normal subgroups and homomorphic images
This leads to the following question: is there a correspondence between normal subgroups of and homomorphic images of ? We will try to answer this question, but before that, let us introduce some notion.
Definition. Let be a group. Pair is called a homomorphic image of iff is a group and is a surjective group homomorphism. We will say that two homomorphic images and of are isomorphic (or equivalent), if there exists a group isomorphism such that .
It is easy to see, that this isomorphism relation is actually an equivalence relation and thus we may speak about isomorphism classes of homomorphic images (which will be denoted by for homomorphic image ). Furthermore, if is a normal subgroup, then is a homomorphic image, where is a projection, i.e. . Let
Proof. First, we will show, that is onto. Let be a homomorphic image of . Let . Then (due to the first isomorphism theorem), there exists a group isomorphism defined by . This shows, that
and thus is isomorphic to . Therefore
which completes this part.
Now assume, that for some normal subgroups . This means, that and are isomorphic, i.e. there exists a group isomorphism such that . Let and denote by the neutral element. Then, we have
and (since is an isomorphism) this is if and only if . Thus, we’ve shown that . Analogously (after considering ) we have that . Therefore , which shows, that is injective. This completes the proof.
|Title||correspondence between normal subgroups and homomorphic images|
|Date of creation||2013-03-22 19:07:11|
|Last modified on||2013-03-22 19:07:11|
|Last modified by||joking (16130)|