Questions, Due April 11, 2012

Honestly, I feel that all the theorems that we need to know for the final are the most important. :) But written below is a theorem that I feel is important, but I am not exactly sure why. I can tell that if I understand this theorem better than a lot of other things will make much more sense to me, like quotient rings and groups for example. Anything dealing with those I feel is just very difficult for me for some reason, I do not know why.

• Theorem: If R is a commutative ring with identity and I is an ideal of R, then R/I is an integral domain if and only if I is a prime ideal. 

The general topics of the different kinds of rings and groups I think would be good topics to know. 

Any past homework question relating to A(n) is something that I need to understand how to do because I am not even sure what A(n) is. How do I even go about finding it? 

If there is one thing that this course has taught me I think it would be the importance of being exact, precise, and disciplined. There is great power that comes from exactness, and I think I am just starting to understand it, because it is a hard thing to learn how to always do. Also I have learned better how to problem solve and look at problem from the inside out, step by step, instead of the whole mountain at once. And of course this class has taught me patience. Patience in myself, my work and in those around me.  

Sections 8.4-8.5, Due April 5, 2012

Difficult:

As I read section 8.4, I had difficulty distinguishing between C and C(a). I know that C(a) is the centralizer of a. But then they used C in place of C(a) and then they used it as the representation of the conjugacy class and I got a little confused. Also, with how intricate all of the details are becoming about all the theorems and what we can and cannot say about any given group, it is difficult for me to really understand the why behind all of the theorems, especially those in 8.5, because to me many of them seem somewhat trivial. It seems random, even though I know it is not. I know that there are a lot of good reasons why someone discovered these theorems but, like I said, a couple of more examples would be quite help.

Reflective:

It is a sad thought, but I think I am just now starting to really appreciate abstract algebra better. It really is a fine, and complicated art. What is so funny about it is that there are times when the book seems to be making things so very simple and easy. And then all of a sudden it comes to a theorem and its proof and all simplicity gets thrown out the window at times. I just hope that I can study enough so that I can look at the final from the simplest view possible rather than trying to cram everything in and only make it unnecessarily complicated for myself.

8.3-8.4 Sections, Due April 3, 2012

Difficult:

I understand what a Sylow group is, the difficult part is understanding how it can be used and all of the different proofs about it. Any proof or theorem involving the group "K" for some reason to me is confusing. I do not see it as plainly as it seems like it is supposed to be seen from the theorems. Theorem 8.17 is particularly confusing to me.

Reflective:

I am really looking forward to the day in this class when we will stop moving forward. I know that that may sound foolish , but I honestly only just want to stop and give myself time to understand what we have already learned. Just when I think that I am about to understand things, we are introduced to a brand new kind of way to organize letters and groups. It can be very frustrating.

Section 8.2, Due April 1, 2012

Difficult:

I do not completely understand the concept of G(p), therefore it was difficult for me to understand the proof of Lemma 8.4. The paragraph at the top of page 255 is also confusing to me. I do not understand exactly what a p-group is or an element of masimal order. Also, even though the book said that I should be able to follow the logic of the proof of lemma 8.6, it was difficult for me to follow. I still do not understand it.

Reflective:

What we are now learning about groups has gotten to be very detailed that it is difficult for me to follow most ideas and concepts. I do like groups better than rings for some reason. I think it is because I think that they are more visual. A cyclic group, for example, is a group that I can visualize very well. So, hopefully as I do problems relating to this section I can better visualize what is happening.

Section 8.1, Due March 29, 2012

Difficult:

This whole section was a little difficult for me. I read through it seeming to somewhat understand what they were talking about but I really didn't. I do not really know what they mean by the product of two groups. I do not know what it means to multiply two groups together. The proof of theorem 8.1 was also difficult to follow.

Reflective:

I liked the example starting at the end of page 245. It helped to be able to look at the tables and visually see why the two groups were isomorphic to each other. I am not anticipating how this new section will complicate our homework. I think at this point I am ready to only focus on what we have learned to prepare for the final. Though I am glad that we did not have to learn about a brand new entity. I like working in groups. If we stay in groups then that will be a good thing.

Section 7.10, March 27, 2012

Difficult:

I do not really understand the concept of alternating groups. It is difficult for me to understand what they mean without doing a couple of examples. Reading through the proofs I think I understand what the theorems mean and why they are true. But I still do not think I have grasped the concept of alternating groups.

Reflective:

I think mathematics is this never ending way to group numbers together and name them something. Anything that has a pattern it seems like you can have a different name for it. I would even venture to say the there is an infinite number of patterns that you can find using numbers and therefore mathematics is a kind of infinite subject. :)

Section 7.9, Due March 25, 2012

Difficult:

I understood most about cycle notation and the product of permutations, etc. But it is a little difficult for me sometimes to see what the product between two permutations will look like. Especially when there are more than two permutations to combine into one. It was also difficult for me to follow the proof of Lemma 7.49. I am sure there is an easy way of explaining it to someone, but the proof was difficult to follow.

Reflective:

I like the idea of cycle notation. I know that it will make it easier for me to write permutations. But I also know that I can potentially get very confused and start mixing up all the numbers. This will call for some serious careful calculations when I am doing my homework. It is sections like these that make me see mathematics as more of a history of ideas; one day someone was thinking about simpler ways to write permutations and then BOOM! he thought of cycle notation.