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.

Section 7.9, due March 26

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

I was very confused with the transposition part. I do not understand corollary 7.48 because I don't see at all how (1234)=(14)(13)(12). So after I read that part, the rest of the section was very confusing.

Part II: Write something reflective about the reading.

Even though I didn't understand the transposition idea, I understood the disjoint cycle notation because I am in Math 450 right now and we have discussed this subject before. It is really interesting taking these classes together because they have a lot of parts that are similar so it is easy for me to make connections.

Section 7.8, Due March 23

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

The third part of Theorem 7.44 confused me the most. I didn't really understand the proof and where T came from. I don't really see how that part of the theorem is even helpful, and so I guess I don't know what it is meaning in "English."

Part II: Write something reflective about the reading.


I am taking Math History right now and we are discussing set theory and group theory and how that started to develop, and reading this chapter made me think back to the different views on math in the 20th century. I always thought math has always been there and we discover it, but reading this chapter gets me thinking of the other view that math may have been something we created to solve our problems. I can see how group arithmetic is starting to be similar to normal arithmetic and it seems like it was sort of constructed so it would be like that. Crazy.

Section 7.7, Due March 21

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

 I think I am still struggling with a specific example of a quotient group, but I think after the lecture tomorrow and after the homework, it will make more sense to me because I will be able to actually work with it.

Part II: Write something reflective about the reading.  


One of the examples was with the integers and the rational numbers and how Q/Z is a quotient groups but no a quotient ring. That is crazy because they try and relate much of what we are learning with groups back to rings yet even though something may be group, I guess it is possible that it won't be a ring.

Rest of Section 7.6, Due March 19

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


I had to use Theorem 7.34 in the homework, but since I hadn't read about it at that point and we hadn't discussed it as a class, I was a little lost on that. I kind of make sense of the proof but it is a pretty big theorem with 5 parts all equivalent so I think I just need to have a better understanding of that proof.


Part II: Write something reflective about the reading.  

I really like how this section led up to theorem 7.33. I feel like what we did in class and what the book describes eventually gets you to the point that theorem 7.33 is true, and then they state it and it just fits perfectly.  

Section 7.6, Due March 16

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

 The very first example in the book is what started my confusion. It isn't a big confusion but I am still confused. I don't understand how r1*t^-1=r1*t. I really just don't understand what the inverse of t is and such. For the most part, this section was easy to understand, but I am a little lost on the group D4, so I think I just need to revisit that. I also was confused on how theorem 6.5 for groups would only work on Abelian groups, but I am sure we will discuss that more in class.

Part II: Write something reflective about the reading.  
 
 It is really interesting to connect the ideas of rings and ideal to groups now. It scares me a little bit because ring quotients were pretty confusing so I hope that connecting those ideas to groups isn't too confusing.
 

Rest of Section 7.5, Due March 14

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

 I am still struggling with theorem 7.28. I used it in the homework already but I think that confused me more because I don't really grasp what the theorem really means in "English."
 
Part II: Write something reflective about the reading.  


I think it is incredible that some of the groups of certain orders are isomorphic to a certain amount of groups. I remember when studying for the last test, the TA mentioned this idea and it blew my mind away. At first when learning about groups, it seems like there are so many possible ways, but now it seems like the list can be narrowed a little bit which is cool.

Section 7.5, Due March 12

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

 I am struggling with the proof of theorem 7.25. First, I am just confused on why it is important, I guess it is hard for me to understand in "English" and the proof for the first part I'm not quite understanding. I think what is the hardest for me to grasp is the meaning of a right coset.

Part II: Write something reflective about the reading.  


I loved the rules that they gave at the beginning of how multiplication and addition relate. It made sense and I understood why in groups, we have to distinguish the two yet we can see how they still both work.

Section 7.4, Due March 9

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

The proof for Cayley's Theorem had me really confused. I don't really understand what A(G) really is and it is hard to follow the proof without understanding that notation. It seems like such an abstract proof, so I hope I can understand it in class.

Part II: Write something reflective about the reading.  

From working on the review for the exam and looking at all the different groups and isomorphisms from rings, it doesn't surprise me that we can relate groups and rings with isomorphism and that groups can be linked to the permutations. I really don't know how anyone was genius enough to come up with all this stuff!  

Questions, Due March 7

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


I think some of the most important topics is understanding F[x]/(p(x)) and what that means. Also, the first isomorphism theorem is quite important along with the proof. Examples and non-examples I think will also help a lot.

What kinds of questions do you expect to see on the exam?


I expect to see questions that ask to list examples and non-examples and various things. I also expect to see questions that deal with F[x]/(p(x)) and possibly listing the possible congruence classes. Also, proofs with groups and if something is a group.

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.

I need to work on thinking of examples of maximal ideals, like with problem number 5 as a sample problem it says, Give an example of a maximal ideal in a ring that does not contain all proper ideals of the ring.  I also need more work on describing what a quotient ring means. So like problem 6 says to give an example of a prime ideal in ZxZ that is not maximal and describe the quotient ring ZxZ/I.

Section 7.3, Due March 3

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

The proof for theorem 7.15 was really confusing. It kept referencing back to previous theorems and I kept trying to make sense of the flow of the proof, but I didn't understand how they got that every element of G is a root of the polynomial x^m -1. I don't even understand what that means and why it is even mentioned in this theorem.

Part II: Write something reflective about the reading.  

At first the cyclic subgroup generated by a made sense and it seemed pretty easy to grasp. Then when they talked about a subgroup generated by S, it seemed a little bit difficult but then I saw the relation between the two and it made more sense.  

Section 7.2, Due March 1

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

Theorem 7.8 was a lot to take in, since it really is like 4 theorems. When reading the proof for 3 and 4, I was really confused and I think it will help to actually go over them in class. The proof for part 2 was a little confusing as well, but it makes sense why proving 2 would also prove part 1. 

Part II: Write something reflective about the reading.  

The finite order of an element a in a group is a really interesting idea. I first tried to think of examples to make sense of what this means, but in the integers and rational and real numbers, there is no number that has finite order. Then, when I read the examples in the book, it made sense of what kinds of elements would have finite order.  

Rest of Section 7.1, Due February 29

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

I can understand the part of how a ring is an abelian group under addition, but I think I am still confused on how it doesn't work with multiplication. And since a ring is an abelian group under addition, does that mean that the binary operation is addition and you only need that operation. I guess I am still confused on what a binary operation still means. 

Part II: Write something reflective about the reading. 
 
When they book started taking about a cut out square and rotating it and reflecting it, I thought of the example in class with the Rubik's cube. It is really cool to make that kind of connection and to see how both are groups.

Section 7.1 first half, Due February 27

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

The most difficult part of the material is that a group has a binary operation *. At first I didn't think much of what that was saying, but then when they went into the examples of compositions of functions, I realized that the composition was the binary operation, which seems weird. 

Part II: Write something reflective about the reading. 

I am in a combinatorics class right now and we are talking about permutations, so I could easily see the connection of why the order of S(n) is n! and that the notation for these functions. It was really cool to make the connections of this section to my combinatorics class.