7.9, due on April Fools!

1. I don't think I'd be able to prove Theorem 7.46 or 7.49. Also when they started doing the 2-cycle transposition, such as (1234) = (14)(13)(12), I can't follow it as easily as say, (243)(1243). I hope we'll go over transpositions in class.

2. THIS SECTION WAS SO MUCH EASIER than the past few! I love the new notation for permutation, it is much more economic and interesting - it makes you think a little more. I also think it is interesting how the First Isomorphism Theorem keeps on coming up in the sections.

7.8, due on March 30

1. Did we ever do a THIRD ISOMORPHISM THEOREM FOR IDEALS? Because I know we did a FIRST one for ideals, so 7.42 does not look very new - 7.43 does however. This section is full of pretty intense proofs that I am not at all positive I would know how to prove on my own. Most likely I will have to for the test though!

2. I liked Theorem 7.45. It deals with the integers(mod p) where p is a prime. I think primes have a lot of very unique and interesting applications that have shown up a lot over the course of this course.

7.7, due on March 28

1. I know I have mentioned this before, but I still struggle with cosets a little. How are you supposed to find the distinct cosets, especially whey you are working with a group such as D4? I get confused with the notation we did in class versus the notation in the book and what order you are supposed to do r^3d or dr^3.

2. I liked this section because it really did not seem like it was presenting any new information - it is based off the sections previous, the only thing different is that the groups are now normal. It is interesting to see the relationship between abelian and normal.

7.6 (part 2), due on March 24

1. As far as I could tell to check and see if a subgroup of a group is normal, you just take each element of the group and 'multiply' it on the right and left side of each element in the subgroup and see if you get the same thing. I think this is wrong though - if Na=aN, then this means na is an element of Na AND aN, not that na=an. I'll have to remember that...

2. I liked the proof of Theorem 7.33. I think the theorem will be useful in proving other things too, such as possibly with quotient rings in the next section. I think I am going to have to spend a little more time on Theorem 7.34 though, it was long but I am sure very powerful! :)

7.6 (part 1), due on March 23

1. I think I am getting confused between left cosets as presented in 6.1 denoted a+I, and left cosets as presented in this section denoted a+K. Is the only difference that I is a subring and K is a subgroup?

2. It appears that a normal subgroup N of G is similar to abelian (commutative) groups as talked about before, except now we are talking about the commutativity(?) of cosets. So Na=aN for all a in G.

Review, due on March 21

* Which topics and theorems do you think are the most important out of those we have studied? The three listed on the study guide that we will have to state and prove, as well as ones concerning rings, groups, and ideals.
* What kinds of questions do you expect to see on the exam? Some computational questions and some proof questions, similar to the last test I suppose.
* 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 on Monday. The second theorem we have to know and prove: If R is a commutative ring with identity and I is an ideal of R, then R/I is an integral domain if and only if I is a prime ideal.

7.5 (part 2), due on March 18

1. I think I am getting caught up on the meaning of cosets - we've talked about them before and I can do proofs with them in it, but the concept behind what they really are I am struggling with a bit. I think that is why the proof of Lagrange's Theorem and Corollary 7.27 does not to clear to me.

2. I thought it very clever how the table on page 204 was filled out - they did a case by case method until they got the right ones. I don't think I would have thought of this! The one on page 205 was a bit more crazy however...

7.5 (part 1), due on March 16

1. One thing that I didn't really like is that all of the proofs say, "Copy the proof of ... with the obvious notational changes" instead of giving the proof. I'd rather like to see the proof right there instead of having to look it up and adapt it.

2. I thought it was interesting that you can have cosets in groups - I guess I just never thought about it yet. That is good though because I am still trying to understand them better! It is also cool that you can have congruence in groups. It seems like you can have congruence in so many applications.

7.4, due on Pi Day!

1. I liked the first few pages of this section. But when it started into Cayley's Theorem I got a little lost - I don't really like having to dissect large proofs with weird notation :)

2. I have found that this textbook is loaded with material that is very complex. However, the more time I give myself to read each section, the more things I pick up on, and the more I learn. I think I just need to give myself more time for the sections. For example, I spent a good portion of 20 minutes on page 194 trying to understand the theorem and proof of Theorem 7.19, and I think I got it now. Also, the second example on page 93, I was confused with why they did not show that f(a)+f(b)=f(a+b) and then I realized that the operation shown in the book is not necessarily multiplication, it is the operation in G. Anyway, so time is good, I just don't know if I always can come by the necessary amount.

7.3, due on March 11

1. One question that I had was the example on page 184 - it says that the multiplicative group of units in the ring Z(15) is U(15)={1,2,4,7,8,11,13,14} by Theorem 2.11, but I don't see how this theorem would prove this. Then it says that 7 has order 4 in U(15) but they only checked 4 of them - or is that just part of the construction?

2. It seems like there are SO many more applications in groups than there where in rings! I hope I can remember them all, because there are a lot. One that I found interesting was the center of a group - it commutes with all other elements like how the identity element does.

7.2, due on March 9

1. I did not understand the proof of Theorem 7.8 on page 177 - maybe it is because I need to understand finite order better first. Hopefully we go over this more in class!

2. I am glad I took liear algebra before taking this class. There seems to be a lot of applications from it. For example, in Corrolary 7.6 there is an important distinguising feature between Abelian and non-Abelian groups: (ab)(-1) = b(-1)a(-1) and not always a(-1)b(-1). Like with matrix multiplication, (ab)(-1) = b(-1)a(-1) also.

7.1 (part 2), due March 7

1. I think I am getting lost in the notation - for example, Theorem 7.4 has a lot of notation going on that makes it a little more confusing to understand compared to something like Theorem 7.1 or 7.2. I think I am alright on everything else though.

2. When I read Theorem 7.2, I kind of thought of the set U of all units in R as a universal union - it seems like this concept comes up a lot in mathematics, especially in 290 and 341. It is interesting to see how properties from one mathematics transfers to other mathematics, which in turn unites them all!