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.