Last Blog Entry! Due April 11

Which topics and theorems do you think are important out of those we have studied?

 I think Legrange's Theorem, the remainder theorem, and the first Isomorphism theorem are the most important theorems we have studied. Some of the topics I think are important are the properties of groups and how the groups and rings are isomorphic to other groups and rings.

What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out in class.


 I think one of the hardest things to do is finding a function to create an isomorphism between two groups or rings. One example of this is problem 15 on the homework review. Another concept I am struggling with is proving that a group is cyclic, like problem 8, 14, and 18. I would really like to go over problem 1 in class though, with An being a normal subgroup, and also problem 12 would be good since I am don't have the best understanding of automorphisms.

How do you think the things you learned in this course might be useful to you in the future?

Throughout this semester, I have realized that there are things in our every day life that we may be able to consider a group, and since we know so many things about groups now, it seems like I could understand those everyday things better, like the rubiks cube, or the equivalence classes when painting the sides of a cube (i am in combinatorics and we discussed this idea).

Section 9.4, Due April 9

Part I: What was the most difficult part of the material for you?

I don't really understand why [a,b] + [c,d] = [ad+bc, bd] in this section because in my mind, I would think it should be [a,b] + [c,d] + [a+c, b+d] but it is not even close to that. So why not? 

Part II: Write something reflective about the reading.

I see how the field F and the integral domain R relate to the rationals and the reals, but I am wondering how any of this stuff came up. Why would someone want to find something that is isomorphic to the rationals and reals when we have them already? I know I probably can't get an answer unless I studied more math, but it is something I am thinking about. 

Section 8.4 & 8.5, Due April 6

Part I: What was the most difficult part of the material for you?


I am stuck on theorem 8.21 because I am having a hard time seeing how the number of distinct conjugates of a are related to [G:C(a)]. The proof really confused me because I don't really know why they are doing what they are doing and what it is accomplishing. The example kind of makes sense, but the proof is way to confusing for me to see that clear connection.



Part II: Write something reflective about the reading.

I thought it was interesting to learn about conjugates and conjugacy classes. At first, I didn't like thinking of another kind of class, but I can see how this class is similar to congruence classes with modular arithmetic, it is just a computation and a set of elements with that computation. At least, that is how I kind of think about it. 

Section 8.2, Due April 4

Part I: What was the most difficult part of the material for you?

I was most confused once Theorem 8.15 was introduced. I don't really understand the x^-1Kx and why that is needed for this section. I think it would be more helpful with an example because most of the other theorems have examples but theorem 8.15 doesn't have any, so I am confused. 

Part II: Write something reflective about the reading.

Most of this chapter I have been really confused, but the First Sylow Theorem actually made sense and seemed like an interesting extension idea with groups. So I am glad that this section is making a little more sense. 

Section 8.1, Due April 2

Part I: What was the most difficult part of the material for you?


There was a lot of stuff in this section, but one of the most confusing parts was the definition of invariant factors and elementary divisors of G. The book gave an example but I don't really understand it still and maybe it is because I don't really understand the previous theorems but I was very lost at this part.



Part II: Write something reflective about the reading. 


So I am really struggling with chapter 8, I just don't see what we are trying to do and where this is going. Chapter 7 made some sense because we were relating many of the things from rings to groups, but now it just seems like random ideas that don't make any sense to me and I don't understand the importance of it all, but maybe it will change as I learn more about this section.

Section 8.1, Due March 30

Part I: What was the most difficult part of the material for you?

I don't understand the definition of the Cartesian product of G1, G2, ..., Gk because the book says that it is (a1, a2, ..., an)(b1, b2, ...bn)=(a1b1, a2b2, a3b3, ...,anbn). That doesn't make sense to me at all, because I think of it as G1=a1,a2,a3,....,an or G2=b1,b2,....,bn). So I don't see where G3, G4,....,Gk came into the product.  

Part II: Write something reflective about the reading. 

The first example in the book is really interesting since the books shows us the connection that M and N make to MxN. I don't really understand why all of this may be useful to us working in groups, but it seems interesting. It feels like we are just finding ways that other things about groups are similar to simpler groups, so I guess I just wonder why we even study the complicated stuff when it is similar to the simpler ideas in group theory.

Section 7.10, Due March 28

Part I: What was the most difficult part of the material for you?


Because we haven't really discussed the alternate group yet, I think the whole proof of 7.52 was really confusing. I am still trying to understand multiplying a permutation with cyclic notation and I am struggling the proofs involving cyclic notation, so this section didn't really help with that since a lot of it was about that stuff.



Part II: Write something reflective about the reading.


I think Lemma 7.53 is really interesting because I understand it well and the proof makes a little more sense than the rest, but it seems like a really good way to understand the structure of alternate groups. Plus, it is crazy that this section was pretty much written so to prepare the reader for a proof so this lemma seems pretty important.