7.10 Due March 30

This was definitely difficult reading. I got the two lemmas pretty well, but the proof for 7.52, which was the whole point of the section, was a beast. I look forward to going through the proof in class.

Since every finite group is isomorphic to a group of permutations, it seems like every group is going to have a normal subgroup isomorphic to An so the same theorems that we learned in this section would apply to all those subgroups.

7.9 Due March 27

The thing I mostly found difficult was seeing how any permutation can be written as the product of disjoint cycles, but then it made sense when I thought of how you could just see it as different move made on the identity permutation.

I think this is neat information and I wonder where it is applied the most. I wonder what kind of subgroups you could make somehow using the disjoint factorization of permutations.

Due March 25

Before the exam, I think I just need to work on understanding what I can do in manipulating a group or subgroup and what can't necessarily be done.

It would be great if you did another problem where you show what orders all the elements of a finite group must have.

7.7 Due March 20

I guess it figures that after talking about congruence in groups, it's only natural that we then find some analogy to quotient rings in group theory. I didn't find the concept hard to understand but the difficult part for me was working through the examples.

In pondering the material, the idea that comes to mind is that we aren't really going to be interested in finite quotient groups since they're just going to be isomorphic to some other finite, simpler group. Infinite quotient groups on the hand are probably a lot more interesting.

7.6 Due March 18

It was fairly difficult to understand at first why exactly we should care about this seemingly small part of groups, but after reading and thinking, I guess since we're trying to understand the structure of groups as much as possible, it's important.

I'm not sure how this applies to other things. I'm not sure whether we're going to be more interested in normal groups or in non normal groups, but I want to find out.

7.5 part 2 due March 16

The reading wasn't too difficult. It was already easy to tell that there would only be a finite number of groups and so classifying a few of them isn't that hard. I found it a little hard to verify that there was only two possible nonisomorphic groups of order 6, though.

I would be interested to look at the classification of finite fields. It seems like there would have to be a finite number of nonisomorphic fields of arbitrary order.

7.5 part 1 due March 13

The thing I had the hardest thing to understand for me was the concept of congruence in groups. I found it difficult because when we studied congruence in rings, both addition and multiplication were involved in the definition of congruence.

In thinking about the subject material, I think this chapter is important in the context of structure of groups which is the soul of group theory.

7.4 Due Wednesday March 11

Most of the chapter wasn't difficult. Since we've already extensively studied isomorphism of rings, it was easy to understand how two groups could be isomorphic. The last two theories, however were harder to wrap my head around.

In reflection, I wonder how interesting cyclic infinite groups. Since they're just isomorphic to Z, studying any infinite cyclic group should just be like studying Z.

7.2 Due March 6

The thing I found most difficult today was reading about the definition of a^(-1), because I've been using that notation on the homework for 7.1. So I was distressed that I'd been using it before showing uniqueness.

In thinking about the material, I think it's neat how by just knowing a few things basically constructs our group for us (commutativity, order of elements, etc)

7.1 Due March 2

Honestly, I don't think I found anything in the chapter to be hard, since I've been studying group theory independently for a while. The only thing that was new is that the definition of a group that I first learned did not include the unit axiom. I don't know what implications this has, but I'm interested to find out.

In thinking about finite groups and permutations, I figure there can only be a finite number of nonisomorphic groups with an arbitrary number of elements. I'd be interested to learn how to figure out the number of possible groups given a number of elements.