8. 4 due April 8

This reading was neat. I was bugged at first that the book didn't offer proofs for the Sylow theorems but I didn't realize that they had a whole section for it. The hard part was actually reading through the proofs, but the weren't super difficult.

I'm interested in the idea of conjugacy equivalence. I wonder if groups can be constructed from conjugacy classes and if they would be any interesting.

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.