Section 7.10

Difficult: This section was really abstract. The proof of Theorem 7.52 was especially hard for me to follow, and it would be helpful for me to go over the proof in class. I think it would make more sense if we went through the proof a little bit slower. It was confusing to me that the authors used both letters and numbers without much explanation.

Reflective: The fact that all alternating groups except when n = 4 are simple is quite interesting. I would think that it would be extremely helpful in classifying all of the groups. Using this fact, how did mathematicians classify all the groups?

Section 7.9

Difficult: The most difficult part of this section for me was Lemma 7.49. The proof of Lemma 7.49 was hard for me to follow, and I didn't quite understand why the authors used the elements xy, xd, cy, and cd. It would be helpful for me to go over it in class.

Reflective: The concept of symmetric and alternating groups is interesting. Since the alternating group A(n) is the set of all even permutations in S(n), what function does the set of all odd permutations have? What makes even permutations special? Why does the fact that they are even make it an "alternating" group? I am interested to see where this discussion and these definitions take us.

3rd Midterm Exam Preparation

Important Topics and Theorems:
I think that the most important theorems, definitions and examples of the sections that we have covered are 1. the definition of a group, 2. the order of an element/group, 3. Theorem 7.8, 4. Theorem 7.10, 5. definition and characteristics of cyclic groups, 6. Theorem 7.18, 7. Theorem 7.19, 8. Theorem 7.25, 9. Lagrange's Theorem, 10. the definition of a normal subgroup, 11. Theorem 7.34, 12. Theorem 7.36, 13. Theorem 7.37, and 14. Theorems 7.42, 7.43, and 7.44. These theorems cover the main concepts of the chapter.

Better Understanding:
I think that I need help especially with the latter part of this chapter. I need help understanding what congruence actually means in groups. I find myself confused often and always have to go back and review the definition because I can't picture it in my head. I also think it would be helpful to go over quotient groups and what they actually represent. It would be nice to do homework problems that we didn't already do in sections 7.5 and 7.8.

Section 7.8

Difficult: I can see the correlation between rings and groups, but some of the theorems were hard to simply accept because the book did not provide a proof. I would like to see a proof for Theorem 7.41. The first and third isomorphism theorems were okay to understand, but a little difficult because it is easy to forget what each group represents. It would be helpful to go over this in class.

Relfective: It is interesting that simple groups are really the key to classifying all groups. I never would have thought that simple groups would be the defining characteristics of groups and their structure! It is also interesting that so many of the theorems regarding rings have some kind of parallel application to groups. Where does this all go, though? We classify all the groups, but how does that help us get somewhere?

Section 7.7

Difficult: I didn't have much difficulty with this section; in fact, quotient groups make more sense to me than quotient rings did. I'm not sure why, but it makes a lot of sense to me. However, it makes me wonder two things:
1. What about left cosets? Can groups be defined by G/aN?
2. What if a subgroup K is not normal? Can we say anything about G/K? Is G/K even a group?

Reflective: It is interesting that we can come up with the same results about groups that we did with rings. Groups seem simpler to me, though. I think we can extend a lot of these theorems and ideas to other algebraic systems as well. That must be why research in mathematics continues to deepen!

Section 7.6

Difficult: I think the most difficult part of the section was the last Theorem. It was fairly understandable, but I would like to go over the proof in class.

Reflective: This chapter was interesting--it finally talked about left cosets! I was wondering when we would get them. I also thought it was interesting that there are normal subgroups in which na does not necessarily equal an for the same n. It is amazing that it still all works out.

Section 7.5 part 2

Difficult: The most difficult part of this section for me was Theorem 7.30. I think I can follow the proof, but it was hard to convince myself that the proof was sufficient to show that a group of order 6 is isomorphic either to Z(6) or S(3). Although it was hard to follow, I hope to understand it better in class today. Going through the proof more carefully would help me.

Reflective: I think that it is interesting that we are starting to classify all finite groups. It is incredible to me that we can use isomorphisms to examine their structure. I wonder, though, that if we have all finite groups up to order 8 classified, do the theorems that we are reading and proving now extend well to finite groups of large orders?

Section 7.5 part 1

Difficult: The most difficult part of this section for me was probably Lagrange's Theorem. I can follow the proof, but it is hard for me to understand the concept. It was always hard for me to understand the concept of congruence in rings with ideals. I can't "picture" it, so I still don't understand what congruence actually means. I think more examples would help me to think about it.

Reflective: It is interesting that the same concepts (like congruence) even extend to groups. I wonder, though, if we will study the left cosets of H in a group G. We have talked all about right cosets, but not yet about left cosets. Do they have similar qualities?

Section 7.4

Difficult: The most difficult part of the section for me was Theorem 7.20, along with Corollary 7.21. They express interesting ideas, but I couldn't follow the proofs very well. I would like to go over the proofs with more explanation in class. The rest of the text was fairly easy to understand. I just don't understand why it is important to know and use Thm. 7.20.

Reflective: It is interesting that homomorphisms and isomorphisms exist in groups as well. I also find it interesting that homomorphisms exist between groups that aren't even under the same binary operation. Thm. 7.18 also makes a lot of sense. When I was doing the homework for 7.3, I started to notice that the cyclic groups of an element with order n also had order n; of course, Z(n) also has order n.

Section 7.3

Difficult: The most difficult parts of this section for me were the theorems regarding cyclic groups. I understand what a cyclic group is, but some of the proofs were a little hard for me to follow. I struggled understanding especially the proofs of Thm. 7.14, 7.15, and 7.16. I think it would also be helpful for me to see some examples of subgroups and cyclic groups. Although there were lots of examples in the book, the authors never really proved that the examples actually worked. I think it would be helpful for me to see a proof of an example (that it is a subgroup/cyclic group, etc.) in class.

Reflective: The most interesting part of this section for me was Theorem 7.14. Although the proof was a little difficult to follow, it was interesting to look into the subgroup generated by one element. I think that it is interesting to see that one element can generate a subgroup of a group.