I don't know what it was about this section that really got to me, but I had very little understanding of this section. After theorem 7.11, the rest of the section went over my head. I need to go through the whole section again in class with lots of examples and then reread the section myself. I really don't understand what I read today.
I thought it was interesting that the proof of theorem 7.10 was so short. It seemed like we could almost just assume that H is a group. As interesting as this was though, I was confused by the fact that we could assume (ii) to prove the theorem. I thought assuming the theorem was never considered a proof.
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