Studying for the Final, Due April 13

I feel like the most important things we have covered in this semester are the big topics. We have focused a lot on rings, ideal, and groups. I feel that these will be the main topics covered on the final, but the final will probably have more specific questions associated with them. I feel like through out the semester, most of the things we covered revolved around these topics.

The most difficult thing for me this semester has been quotient groups, rings, fields, and anything basically involving cosets. I just feel like I panic when I see R/I or G/N. I also feel like I never completely understood normal groups. But mostly its quotient stuff that I am really uncertain about.  I would really like to see a problem involving quotient rings or groups. Something involving a ring modulo an ideal would be helpful. I just don't know what to do with these things when I see them in a problem.

In the future, I think this class will help me form more logical arguments. I have learned to not freak out when I see the word proof. Honestly it has given me more confidence in proving things, (especially since theory of analysis shot down any confidence I had from 290). For the most part, I think this class has helped me understand modular arithmetic really well. I didn't have much exposure to this before, so its is nice to feel confident in my ability to do modular arithmetic and actually understand what is going on.

8.3, Due April 11

Difficult: I had a hard time understanding what a Sylow p-group is. I don't understand how to find these sylow p-groups and why they are even important. First of all, I don't even know what sylow means. How are sylow p-groups different than regular ones. Even the example following the definition didn't help me understand this concept any better. I don't know how to use the definition to help me use and find the sylow p-groups.

Connections: I felt like the first sylow theorem sounded really familiar. It seemed to me like Lagrange's theorem had some kind of relation to it. This theorem made sense to me, and it seemed that logically it just followed from Lagrange's Theorem. I would understand how to apply this theorem since it sounds a lot like previous theorems we have used in this class.

8.2, Due April 8

Difficult: In the beginning of the section, there was notation G(p) for the set of elements in G whose order is some prime power of p. This notation was confusing to me, and I didn't really understand what it was talking about. Even the example immediately following was of no help to me. Then the following lemmas and theorems all included this notation so I had a hard time following what they were even stating. So following the proof was even more difficult. I felt like this whole section pretty much followed this idea of prime powers and and p-groups, so I felt confused about what this section was stating.

Connections: This section uses a lot of things which we have visited previously in this class. For example, the notion of the orders of elements in a group being prime powers. We have spent lots of time discussing the orders of the elements of a group, and also discussing primes. We have also recently talked about cyclic groups and this is very important in the statement of the Fundamental Theorem of Finite abelian groups. So there are a lot of things in this section that we have visited before, but I feel like the prime power order is the thing that is most confusing.

8.1, Due April 6

Difficult: I was doing pretty good with the first part of this section, and then I started getting confused by the theorems. For example, theorem 8.1. I did not understand what this means at all. Also, lemma 8.2 was confusing to me as well. I didn't understand how this holds true. I might still be confused by normal subgroups and how they apply in proofs. I am trying to figure our how to follow the proof of 8.1, but I just don't understand all the subscripts and notation. I also don't get how to find what MxN is isomorphic to like the 2nd example of the section.

Connections: I recognize seeing the cartesian products in other sections of this text. I understand what to do with them and how to find the cartesian products, so using them in proofs should be pretty easy for me, but like I said above some of the notation is getting a little confusing for me. I feel like the cartesian products are pretty computational and are easy to work with and to find the elements of a cartesian product.

7.10, Due April 4

Difficult: As I was reading in this section, I got a little lost in the proof of 7.52 where it says that N is a normal subgroup of An. What does this subgroup look like? I think that it is just a some of the elements of Sn. So is it just a set of different cycles of An? I think I am also having a hard time figuring out what An looks like. How many elements are there? Is there a way to find this out by having a certain length of a cycle or knowing what symmetric group you are working in? Also, the proof of 7.52 is so long that I just got way lost in all of the notation and didn’t really understand what was going on.

Connections: Now that I feel like I better understand the new notation with symmetric groups, I feel like I can understand the notation used in the proofs involving cycles. I like that the cycles are pretty easy to compute the “multiplication” of them. I think that knowing this information helps me be able to follow the composition of the cycles when it is done in the proofs. 

7.9, Due April 1

I decided to write this blog about unicorns today instead of Abstract Algebra.. Enjoy!!!

APRIL FOOLS!!!

For reals though, here is the real blog information

Difficult: I felt that they new notation for the symmetric groups was really confusing and I feel like it isn't useful at all. What is the point of trying to write it this way, I have to think about it way more than I did when we just wrote it all out. So cycles are also confusing to me since I don't understand the notation. I got really confused about the notation because I felt like I understood some of it, but the notation on the middle of pg 231 didn't make any sense to me. So for the rest of the section, I was so confused because I couldn't make sense of cycles and the notation. So transpositions and alternating groups was really confusing to me.

Connections: The only connections I can really make with this section is that we have seen the symmetric groups before in other sections of chapter 7. I like them because they are good examples of non-abelian groups with different orders. However, they were kind of tedious to write them all out, so maybe once I can understand the new notation I will like them better and be able to understand how to have more applications of them.

7.8, Due March 30

Difficult: So in the first part of this section, the ideas are pretty parallel to those of rings. I understand homomorphisms, but the hardest part for me is finding the map f that is a homomorphism. It is easier for me to prove, but is there some kind of way to know what the map is? In all of the examples in the book, they choose their map, but this is something that is hard for me to do. Also, the idea of simple groups makes sense to me, but I'm just not sure how to go about testing if a group is simple. Is there a general way to do it? Also, I kind of got lost in the explanation of why simple groups are important. I didn't understand the point of them, I think I might have just gotten lost in the notation.

Connections: The First Isomorphism Theorem for groups is very similar to that for rings. I feel very confident in my ability to prove this theorem for rings, and I feel that I understand it pretty well. As a result, I think that I would be able to do this proof for groups as well. It follows the same general idea, but some of the notation is changed and some definitions would look different. Again, I am left wondering how they came up with the map for phi? How did they know to pick phi(Ka)=f(a)?