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.