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.