8.5 and final review, due on December 7

Reading:
1. The proof of Theorem 8.33 was a bit confusing since it mentioned so many results that weren't from this section.

2. I really liked seeing the table of groups extended to 15.  It's interesting to see T show up in there; it's a group we have never heard of before.


Exam preparation:
I think that being familiar with lots of examples will help more than anything else for the test.  I expect to see some of the group classification theorems on the final, since we haven't been tested on those yet.  I need to go over those a few times to make sure I know what they say, especially since some of them aren't intuitive.

One thing that I've learned already is that group theory makes recognizing patterns much easier.  If we have an unknown group, then knowing so much about the groups we have classified makes it easier to identify what the unknown group is isomorphic to, which gives us a lot of information about its structure quickly.  Abstract algebra will also help with my future math classes, since I will be doing grad school in math.

8.2, due on December 2

1. It was difficult to follow the proof of Lemma 8.6.  Could we go over that proof in class?

2. The main result of this chapter was pretty interesting.  It definitely makes working with abelian groups a lot simpler; we can classify them very quickly instead of starting from scratch with each.

8.1, due on November 30

1. The hardest part for me to understand is distinguishing between the first form of the direct product and the standard Cartesian product.  Is there any difference between the two?

2. I've seen direct sums before in vector spaces, and this seems to be a similar concept. We're decomposing G into normal subgroups (which are analogous to vector subspaces) that can be added together with no overlap to create G.

Test review, due on November 28

I think the most important theorems are the ones that allow us to manipulate groups more easily, such as Lagrange's Theorem, the Sudoku Theorem, and the fact that subgroups of a cyclic group are cyclic.

I expect to see questions on the exam that are similar to the homework problems but are things we haven't seen before.  I feel like the best way to test our knowledge is to have proofs on the test that are unfamiliar but follow easily from the theorems we have learned.

I need to practice proving the theorems on the practice exam.  I learned the proof of the First Isomorphism Theorem for the last test, but I haven't yet tried to reconstruct it with group notation instead of ideals, and I need to practice the others.

7.10, due on November 22

1. I had a hard time following all the permutation notation in the section.  When reading it's much easier to assume they're doing the steps correctly than it is to check every detail.

2.  The main theorem in this section was quite interesting, especially given the commentary on group classification in the previous section.  This section gives us an infinite set of easily-described finite simple groups, which will work well for examples in the future.

7.8, due on November 21

1. The hardest part of this section was Theorems 7.43 and 7.44.  Everything earlier in the section is similar to things we did with rings and ideals (and vector spaces), so it's not difficult to apply it to groups.  Once we start talking about quotient groups of quotient groups, though, it gets much harder to tell what's going on.

2. I thought the section on simple groups was really interesting.  It was a cool insight into some of the work there still is to do in group theory. (Also, it was much easier to understand than the rest of the section.)

7.7, due on November 18

1. The hardest part for me was keeping the elements of the group distinct in my mind from the elements of the quotient group.  It's mostly a lot of notation to keep track of.  Also, I don't really understand the example about Q/Z.  It's hard for me to figure out just what those cosets look like.

2.  It's interesting to me to see just how quickly we are learning about groups compared to rings.  I guess it makes sense because it's later in the semester, but it feels like there is a lot more packed into each section than there was in the first few chapters.