6.2 (second half), due February 28

I'm having a little trouble because the new stuff about kernels and the connections with homomorphisms made since in class and I understood it then, but the things they have extended it into aren't making sense to me know. For example, with theorem 6.12 I have a hard time even understanding what it is saying although the logic behind the proof makes sense. I also don't think I'm seeing all of the implications of an injective homomorphism versus a surjective homomorphism. The explanation between Theorem 6.13 and it's proof seems like it's really important and a good connection to make, but I just can't seem to understand it. Another weird thing is that I understand the examples at the end of the section just not most of what leads up to them.