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.

7.6 part 2

1. Not much of this was difficult, especially since we went over some of it in class.  The hardest part was probably following the proof of the interconnecting parts of Theorem 7.34.

2. It's interesting to me that even though Na = aN does not imply that na = an, that doesn't end up mattering for quotient groups.  Having the entire subgroup commute ends up being enough.

7.5 - 7.6, due on November 14

1. I think the hardest part for me is still getting  used to the multiplicative notation for cosets.  I'm sure it will come more naturally after I've done some homework on it.

2. I think it's interesting that the lack of commutativity is what makes quotient groups difficult to define, but that we don't actually need commutativity for each pair of elements in the group to make them work, only for the left and right congruence classes.

7.5, due on November 11

1. The multiplicative notation for cosets will take a bit of getting used to, since cosets in rings and vector spaces use addition as the operation.  The fact that right cosets aren't necessarily left cosets is also new, although it makes sense given that our operation may not be commutative.

2. I thought Lagrange's Theorem was quite interesting.  It's the kind of result that isn't intuitive without looking at examples.  This theorem will definitely make it easier to classify finite groups, especially with Corollary 7.27.

7.9, due on November 9

1. Theorem 7.51 in the second edition was rather confusing since we haven't learned about anything it uses in the proof.  We also haven't learned about normal subgroups or group indices, so the statement of the theorem didn't make any sense do me.  I'm guessing that's just a consequence of the difference in editions;  basically what the theorem says to me is that An is a subgroup of Sn, and not much more.

2. This section was pretty interesting.  I have played around with rearrangements before and noticed that any permutation can be achieved by just swapping two elements at a time, and I've also noticed the even-odd swap pattern.  It's nice to see both of those as theorems in the section.

7.4, due on November 7

1. Cayley's Theorem was a bit difficult to understand at first, especially since it isn't obvious from the statement of the theorem how you would go about proving it. It also relies on an earlier theorem  (7.19) that itself isn't intuitive.

2. It was interesting to see group representations come up in this section, since I am signed up for Group Representation Theory for next semester.  (I hope that class can stay scheduled!)  It was also interesting to see Z and Z mod n show up again in an abstract setting, in the characterization of cyclic groups.

4.3, due on November 4

1.  Mostly the difficult part of this section was the number of new ideas it introduced.  Most of them are things we've never seen before, which makes it harder to keep track of them all.  I read the section this morning, and I had completely forgotten about the center of a group by the time I finished reading about cyclic groups and generators.

2. I think it's really interesting that we only need to check one axiom for finite subgroups.  Subrings and ideals (and subspaces of vector spaces)
require two, even in finite rings.

7.2, due on November 2

1. The most difficult part for me was getting used to the multiplicative notation, especially the exponents.  It made it a bit difficult to follow the proofs of Theorems 7.8 and Corollary 7.9.

2. It was interesting to see how modular arithmetic pops up all over the place.  Whenever there were any calculations in the book with the order of an element of the group, they was almost always involved congruence in Z mod something.