Section 9.4

Difficult: It was hard for me to understand Theorem 9.31. I don't get what this theorem is saying or how it will be important. The proof doesn't really make much sense to me either, and I don't understand how it proves the theorem.

Reflective: I think it's interesting that so much can be done with integral domains, and I think the concept of a field of quotients is really interesting. It makes sense to think of the rationals as a field of quotients of the integers.

Section 8.4-8.5

Difficult: The most difficult thing for me to understand in these sections was the proofs of the Sylow Theorems, especially the proof of the second Sylow theorem. I don't understand some of the steps they take in the proofs.

Reflective: I think the idea of conjugacy is really interesting, especially conjugacy classes. It's interesting how they fit into the classification of groups problem, and the proofs of the Sylow Theorems.

Section 8.3

Difficult: I am having a hard time understanding the second example on page 264. I am having a hard time following their steps to get to the point where they applied Corollary 8.16.

Reflective: I think it's really interesting that there are so many properties that can be found from finite groups. This section was really interesting to me because I think that the Sylow theorems are interesting, and I look forward to seeing how we will use them.

Section 8.2

Difficult: This was a hard section! I think I kind of understand most of it, but I feel pretty shaky on what exactly a the group G(p) is and what a  p-group is. I also don't really understand the proof of Lemma 8.6. Also, I wish I knew why they kept switching between additive notation and multiplicative notation.

Reflective: I think this section is really interesting because it determines how we can classify all abelian groups. I also think it's interesting that most abelian groups are additive.

Section 8.1

Difficult: I am having a bit of a hard time understanding what exactly Theorem 8.1 means. I understand the example before it, and I think that the theorem is just generalizing this example, but I don't understand how we will be using it.

Reflective: I think it's interesting that some finite groups can be classified as isomorphic to groups of ordered pairs, or direct products. I also think that it's interesting that if M and N are disjoint subgroups in G except for e, then the elements of M commute with the elements of N.

Section 7.10

Difficult: This was a hard section for me to understand. Specifically, I don't really understand why alternating groups are so significant and why it makes such a difference that a subgroup is classified as alternating.

Reflective: I think it's interesting that there are so many properties that can be found from simple groups and alternating groups, and I think it's interesting that every element of and alternating group is the product of 3 cycles.

Section 7.9

Difficult: I am having a difficult time understanding how they got all the different factorizations for permutations on page 233. I also don't really understand the proof of Lemma 7.49.

Reflective: I think it's interesting that this new cycle notation for permutations can show us so many more properties of permutations that we couldn't see before.