I have problem understanding something I read of Dummit & Foote. I'm not sure I got it correctly, so I need to be checked:
If is the natural projection homomorphism of on and is a homomorphism, then let's prove that , given by is a function and homomorphism if and only if .
If I understand correctly, the issue here is proving that is a function. If we know that is a function, then proving that
is easy: , , and because is homomorphism.
So, assume and let (we want to prove that this implies ). Then and are in the same coset of N and this coset is contained in one coset of , because for any . So and must be in the same coset of , so which means that .
And I'm so far. I would like to know if the above checks up ok and how to prove the reverse statement.
Thanks in advance....