7.4, due on November 7

The ideas in the section made a lot of sense to me because we already have some familiarity with isomorphisms and homomorphisms. However, I had a difficult time following the proofs of the section. Specifically, I didn't understand the reason why we chose the function f(i)=a^i for theorem 7.18. I also didn't understand part 2 of the proof of theorem 7.19. And then the entirety of Theorem 7.20 was completely confusing and then of course the proof of corollary 7.21 plays off of that proof so I didn't get that one either.

I thought the most interesting part of the section was how many isomorphisms we can guarantee for groups. We know there is always an isomorphism from a finite group to a subgroup of the symmetric group. And we know that every group is isomorphic to a group of permutations and cyclic groups are isomorphic to the integers or the integers mod n. It seems like it is relatively easier to find isomorphisms for groups than for rings but maybe that's not true.