I would honestly say that this section is probably one of the most difficult for me to understand. I was glad it was so short because I don't think I got very much out of it and if there had been more information that would have been more information that went over my head. I think I am having a difficult time understanding what An is. I don't know what a group like this really is. I also didn't understand why lemma 7.53 is true. Nor did I understand case 3 of the proof of theorem 7.52 or actually why we could boil down the proof to three cases in the first place.
It is interesting to me whenever there is a theorem that is true for all n (here just n not 4). It is especially interesting when these proofs aren't by induction and are relatively short in length. Most proofs of theorems with that broad of an assumption turn out to be more difficult that just three cases like the one for theorem 7.52.
I had an additional question not on this section but on the homework for section 7.8. I won't be able to go to office hours since I have a test to take so I was hoping if there is time you could address it in class or I will ask you after class. On problem 1d) I can't figure out how to interpret the function. I guess f in Sn looks like some cycle (12...n) or something. But then I thought (k) must be one number inside parenthesis since 1 less than or equal to k less than n. So is (k) just the identity because isn't that how we write the identity in cycle notation? (1)=(2)=(3)=(4)=e in Sn? So then what is f(k)? Wouldn't that always be just f? And is the function just plugging in different values of f (different elements of Sn) or also different (k) (which would all be the same thing if my above assumption is correct)? Please help.
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