FINAL EXAM Reflections, due April 13

I think that the important stuff to know for the final is the material on rings and groups.  I know that sounds like something to which you could reply "DUH! That's the whole course" and you might be right.  But I mean it in a more specific way.  Yeah, modular arithmetic is cool and all but I think it's more important to know about the groups and rings that are formed by the set of integers modulo some n.  I would really like all of chapter 8 to be a nice little time filler, for-your-own-enjoyment type of thing and not be included on the final.  I feel that if we've had to stick with the other section for the entire semester thus far, we might as well endure to the end and not change things up now.  I mean if we were going to break away from them, I wish we would have done that before the second midterm.  This class has taught me a lot of really cool things that I found to be very fascinating.   It helped me embrace my inner nerd. :)  I don't know how much of it I will use as a teacher but I don't plan on just forgetting it all as soon as the semester ends.  It's been fun!

8.3, due April 11

I really liked how this section stated the Sylow Theorems and then showed how to apply them, putting their proofs off until later.  Some of the times when all that information was crammed into the same section it got really confusing.  I enjoyed the plethora of examples that the authors gave and the humor they sneaked into the footnotes :) As I read, I noticed that I was picking up on the gist of the theorems and the math the involved, but they weren't sinking in very deeply.  I was getting a little turned around in the general forms of the statements of the theorems themselves, but the examples helped set things right.

8.2, due April 8

Throughout this section I could tell that I was getting more and more confused.  I was getting so wrapped up in trying to understand the new stuff that even some of  the old stuff that mention was confusing.   I thought the G(p) subgroup was pretty cool and the maximal element reminded me of something that I learned in MATH 341 last semester with least upper bounds.  I also thought that Lemma 8.8 would be very handy in some of the proofs and homework problems we would be assigned.

8.1, due April 6

One thing that confused me in this section came when they defined the coordinatewise operation on the Cartesian product G1xG2x....xGn.  I expected it to be an ordered pair with n entries.  Instead it was like to cycles each with n elements composed together.  I thought the with M={0, 3}, N={0, 2, 4}, and Z_6 was EXTREMELY fascinating!  Things weren't as awesome after that (How could you top something that cool?) and I think there's a typo in the last example.  They claim that M={1, 11} has cyclic order 4 and N={1, 2, 4, 8} has cyclic order 2.  I'm pretty sure that's supposed to be the other way around.

7.10, due April 4

This section wasn't too bad, but the proof of that first theorem was REALLY long and at times was a little hard to follow.  It was hard to keep track of what each symbol/variable was representing and what they were being used for.  One thing I learned while reading this section was that normally when you see (123) you read it "one hundred twenty-three" but this is not how it is meant to be read in cycle notation.  In cycle notation these are individual integers.   I found myself instinctively reading it the way it is meant for cycle notation instead of the normal way I would usually think about it.

7.9, due April 1

This section was pretty cool.  I found the new notation for permutations kinda fascinating.  It was really easy for me to understand.  The alternating groups were a littel tricky.  I kept wanting to do the operation from left to right but I'm pretty sure I'm supposed to do them right to left because it's still function composition.  I thought it was especially cool that you could write a single permutation as the product of two cycles.  That was really intriguing.

7.8, due March 30

Does everything in groups have a connection to something in rings?  I get so confused when they have the same notation or even names (kernel!).  I had a little bit of a hard time wrapping my head around all this in rings and now it's like I have to unwrap it and wrap it up again but this time in groups.  Another thing I noticed was that there is no Second Isomorphic Theorem For Groups.  I got a little caught up wondering why it wasn't included.  It must exist if there is a third one but was it just not important enough to make the book?  As far as the actual math goes, it all seemed straight forward until I got to the composite factors of G.  I don't even know where to start with those.