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.