Review, due on April 13

Which topics and theorems do you think are important out of those we have studied? - [All the ones about rings, groups, and ideals.]
What do you need to work on understanding better before the exam? - [Probably a little bit of everything]. 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 Wednesday. - [I would like to go through as many examples as possible listed in the section entitled, "Know lots of examples of all the things we talked about, especially:..."]
How do you think the things you learned in this course might be useful to you in the future? - [I am going to be taking Number Theory, so I figure this course would be very useful in understanding Number Theory. Other than that, this class has really taught me that I can understand difficult material and overcome difficult challenges.]

8.3, due on April 11

1. I want to know where they got the name Slyow! Also, in the example on page 8.3 says that S6 says there will be at least 60 subgroups of order 4. How are you supposed to figure that out? Is there a way to figure that out? We might have talked about how, but I am just not recalling it... Also, what is the relationship between abelian and normal? They seem pretty similar...

2. I don't know what else to say about this section, except that it is 8.3 which means we are almost done with the course which means I won't have to do these blog posts any more and I am so excited about that!!! :)

8.2, due on April 8

1. I feel like this section has a lot of interesting information to offer, but I didn't understand hardly any of it. Yep. There was a lot of it. I probably should have given it a little more time than I did to digest it all because there was so much of it.

2. I liked Theorem 8.12. It just made sense to me that if G and H were isomorphic to each other, then they would have the same elementary factors, or prime powers. The theorem was a bit to digest however, but it makes sense for the most part.

8.1, due on April 6

1. I think the only tricky part about this section was the proof of theorem 8.1. I guess that wasn't even too difficult because I am feeling like I am grasping how to prove isomorphisms a lot better now that we have done the First Isomorphism Theorem a few times.

2. This section was relatively simple. I especially thought Lemma 8.2 was cool. I did not see the significance or newness it brought until I realized we were talking about ab=ba where a and b are elements within DIFFERENT subgroups! That is crazy! Before we only talked about elements being normal within a single group.

7.10, due on April 4

1. I think I need to do my homework for section 7.9 because I don't think I am following too well this new notation for permutations. Like, I am pretty lost in this huge proof. I guess that is how most chapters end - hardest stuff at the end.

2. It was cool to see stuff about normality in this section, even if I didn't understand the permutation stuff. I feel like I understand normality pretty well.

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.