8.5, Due April 11th 2016

1) One of the most confusing parts to me while reading was how they were trying to complete the table on page 280.  Also for me, the most difficult groups we have studied are the dihedral groups and I had a hard time following the last half of the proof of Theorem 8.32.
2)  We were basically exploring theorem 8.30 before right?  I also love the chart on 281... it is very helpful to break it down in order to understand basically what each group is isomorphic to and why.  I feel like it is a good way to conclude our exploration of groups.


For the test:

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

As far as MOST IMPORTANT, I think it is the ones that you asked us to know how to prove on the previous midterm exams.  And just knowing the characteristics of 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.

I need to work on this last stuff that we have been studying.  I definitely feel like it is the most difficult.  I feel like my understanding is about 2 feet deep in a 20 foot pool!

 

#23 in 8.2 in the second edition.  Unless you feel like this won't help us that much with understanding this latter material.  I am not really sure what problems will help me to gain a better understanding for the final.

• What have you learned in this course? How might these things be useful to you in the future?

I think this class has really taught me how to prove something logically (identify my end goal, use the info given in the assumptions to get there, create a strategy, and go).  I am so grateful for this, because I don't fear proofs like I used to.  It makes me feel a little more comfortable and ok with the idea of theory of analysis in the fall.  When we studied rings, I felt like I better understood basic characteristics of the different sets that we work in and the reasons why things work or don't work in algebra - I feel like that is definitely something that I can use as a teacher.

8.3 and 8.4, due April 8th

1)  The most difficult thing for me in these sections were the proofs in 8.4.  I am also worried that I will be able to memorize all of these definitions.  Also, what is the importance of conjugacy being an equivalence relation in G?  How does that help us?
2) It is just cool to see all of these different theorems and I'm glad that I understand them (for the most part).  I am also glad to see a theorem from Cauchy - I have been learning about him in History of Math.  He wanted to axiomatize/generalize everything in calc.  Looks like in the meantime he helped develop abstract algebra.

8.2, April 6th

1) On the example on pg. 253 in the second edition it says that G(2) is the set of elements having orders 2^0, 2^1, 2^2, etc.  And then it says that it is the subgroup {0,3,6,9} which seem to be powers of three, not two?  I don't understand how those are supposed to be the same.  How can we know or prove if something is of maximal order.  I know that it says that elements of maximal order always exist in finite p-groups.  Is that the only way to know?
2)  I would be curious to know how long it takes people to come up with fundamental theorems, or even just theorems in general.  The proof is pretty simple for the Fundamental Theorem of Finite Abelian groups, but to even come up with a conjecture is cool.  This is probably a silly question, but are people still trying to play with different constraints and situations in order to come up with different theorems in 371?

8.1, Due April 4th

1) Is there any difference between the table of sums on pg. 246 and the direct product MXN in Z6.  They look identical, except that one is separated by an addition sign while the other is separated by a comma.
2) I guess I can't really understand what is different about this section than what we have been doing before.  Hopefully it will become more clear in class.

7.10, Due April 1

1)  If Sn is really huge, is there a way you can find all of the even permutations in Sn easily?  Will half of Sn always be even permutations and the other half odd permutations since it is isomorphic to Z2?  And for the last corollary 7.55, we know that all of the even permutations in Sn are normal, so don't you just have to show that any odd permutations are not normal?  I also couldn't follow the cases on pg. 240 too well.
2) After studying for the test, I am happy to feel like I can follow this section much more easily than the past few sections that we have been studying.  Again, I am hoping that we could talk a little about the application of these things - if there is application.  I know that often times, math is just studied for math's sake.