- #1
moo5003
- 207
- 0
Problem:
" Prove (Third Isomorphism THeorem) If M and N are normal subgroups of G and N < or = to M, that (G/N)/(M/N) is isomorphic to G/M."
Work done so far:
Using simply definitions I have simplified (G/N)/(M/N) to (GM/N). Now using the first Isomorphism theorem I want to show that a homomorphism Phi from GM to G/M exists. Such that the Kernal of Phi is N.
I constructed phi such that GM -> G/M
where it sends all x |----> xN.
My problem is as follows: How do I know xN is actually in the set G/M. It may just be that I'm going about the proof in a way that is very complicated then it should be. Any help would be greatly appreciated.
" Prove (Third Isomorphism THeorem) If M and N are normal subgroups of G and N < or = to M, that (G/N)/(M/N) is isomorphic to G/M."
Work done so far:
Using simply definitions I have simplified (G/N)/(M/N) to (GM/N). Now using the first Isomorphism theorem I want to show that a homomorphism Phi from GM to G/M exists. Such that the Kernal of Phi is N.
I constructed phi such that GM -> G/M
where it sends all x |----> xN.
My problem is as follows: How do I know xN is actually in the set G/M. It may just be that I'm going about the proof in a way that is very complicated then it should be. Any help would be greatly appreciated.