1. Theorem 7.51 in the second edition was rather confusing since we haven't learned about anything it uses in the proof. We also haven't learned about normal subgroups or group indices, so the statement of the theorem didn't make any sense do me. I'm guessing that's just a consequence of the difference in editions; basically what the theorem says to me is that An is a subgroup of Sn, and not much more.
2. This section was pretty interesting. I have played around with rearrangements before and noticed that any permutation can be achieved by just swapping two elements at a time, and I've also noticed the even-odd swap pattern. It's nice to see both of those as theorems in the section.
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