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.

Section 7.8

Difficult: This was a hard section. I understood the first half pretty well, but once I got to the Third Isomorphism Theorem I didn't really understand much after that. It seems to me that while we can prove all these theorems, they seem so ambiguous that they will have very little use for me. I am curious to see how we will use them.

Reflective: It's interesting that the First Isomorphism Theorem and the other two theorems before it can be applied so easily from rings to groups. I am interested to find out how we will use all the theorems that we learned in this section.

Section 7.7

Difficult: I am having a hard time understanding Theorem 7.38 and what uses it will have. It seems that if we wanted to use this theorem to find out if G is abelian, it would be easier to just show directly that G is an abelian group.

Reflective: I think it's interesting that the structure of a group can be discovered if the structure N and G are known. I also think it's interesting that there are so many properties that seem to be parallel to the properties of quotient rings.

Section 7.6 Part 2

Difficult: I don't understand what the difference between parts (2)/(3) and (4)/(5) are in Theorem 7.34. I know that the book explained a bit after the theorem was given, but I don't understand what it means and why these statements are not saying the same thing.

Reflective: I think it's interesting that a subgroup has to be normal in order for Theorem 7.33 to work. I think that the concept of normal subgroups is interesting because it seems like it would be much more unlikely that many normal subgroups, except for the center, would exist in a large group.

Section 7.6 Part 1

Difficult: I am having a hard time understanding the notation for congruence classes, both left and right. I don't really understand what the different sets on page 210 is under that definition for a left congruence class mean.

Reflective: I think it's interesting that there are some subgroups, called normal subgroups, that commute with every a in G.

Focus on Math Talk March 13

Difficult: I didn't really find anything difficult about this presentation, because it wasn't really about any concepts in math, just about the history of numbers as an idea. It was a really interesting to think about how numbers and counting came about. The information was very well presented, and everything was explained well so I was able to understand it all.

Reflective: I thought it was really interesting that so many words that we use today that don't even necessarily have much to do with math now came from mathematics, such as the word "stockholder." It was interesting to see how the first form of written language was to record mathematics, and I loved how he related this to everything else that we learn about every day, such as language and business.

Section 7.5 Part 2

Difficult: I am having a little difficult time understanding how they got the operation table in the proof for Theorem 7.30. I don't really understand some of the claims that the author makes when showing this part of the proof.

Reflective: I think it's really interesting that groups with prime number orders are cyclic and isomorphic to Z mod p. I also think it's interesting that this doesn't seem to apply only to prime numbers, but also to groups of order up to 8.

Section 7.5

Difficult: I am having a difficult time understanding Theorem 7.25. I don't understand why G is the union of all cosets of K, and how there can be a bijective function from K to Ka. It seems like Ka would be a lot smaller than K, so how can there be a bijection?

Reflective: I think it's really interesting that subgroups of a group G can also have congruence classes. I think it seems more similar to congruence modulo an ideal as opposed to congruence modulo an integer, so I hope that I will be able to understand it better this time.

Section 7.4

Difficult: I am having a difficult time understanding why the image of a function is important, what it will be used for, and what exactly part 4 of theorem 7.19 means. Also, I don't understand Cayley's Theorem and the proof of it.

Reflective: I think that the idea of 2 groups with different operations being isomorphic to each other is really interesting. It seems that isomorphisms of groups are more versatile than isomorphisms in rings, and I think they will be interesting and perhaps a little easier to work with, now that I understand isomorphisms better.

Test 2 Review

I think that the most important topics we have studied have been rings and groups. Within rings, we focused on congruence modulo n, congruence classes, polynomial arithmetic, and ideals. These are the main topics that we have covered, and they are the most important. Theorems that are important are the theorems that involve ring properties, subrings, and fields.
On the exam, I expect to see questions that test my knowledge on rings and fields, basic group properties that we have learned, proving that a set is a ring, group, field, or integral domain, isomorphisms, giving examples of different rings and groups, and proving that a subset is actually a subring or a subgroup.
For the exam, I need to study more examples of all different types of rings, fields, and groups. I also need to make sure I know the definitions in the book and that were emphasized in class. Some concepts I have trouble with are how to prove that two rings or groups are isomorphic, what a cyclic group is, and ideals and cyclic groups generated by a finite number of elements.

Section 7.3

Difficult: I am having a hard time understanding how the generators of groups work, and what Theorem 7.17 means. Specifically, I don't understand what exactly the set <S> is and what it means by "all possible products, in every order."

Reflective: I think that cyclic groups and the center of a group are interesting concepts, and I wonder what the connection between the two types of groups are, if there is any. I also wonder what the center of a group is used for.