I’m just now getting around to writing this post, although I worked through these exercises last week. These exercises come from the chapter on cosets and Lagrange’s theorem.
My solution is as follows (included as an image so that I could more easily include well-formatted math):
The exercise that we couldn’t solve is this one:
Although it looks fairly straightforward, it really isn’t, at least using any methods that I know so far. We struggled to figure out what facts or theorems to take advantage of in order to prove the desired result. (I even tried looking up solutions online, and all of them rely on results that I haven’t proven yet.) This highlights how often times the most difficult part of solving an exercise isn’t executing a proof once you have a plan of attack, but coming up with the plan of attack in the first place.