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.