8.1, due November 30

My mind was overwhelmed with this section. It was interesting but I'm not sure what I read. I was really confused by the proof of theorem 8.1. In fact I had no idea what was going on to prove that f is a homomorphism. I would also like to go over the general idea on which the whole section is based which is writing the elements of a group as sums or products of elements of two subgroups. I don't really understand why that works.

I think it is interesting to look at the problem of what a group is isomorphic to by crossing some of its normal subgroups. How do we know which normal subgroups to use? Is it based on the size of the subgroup or is it trial and error?

Test Review, due on November 27

I think some of the most important theorems and topics we have covered are related to homomorphisms and isomorphisms of groups. I feel like it is easiest to interpret groups when we see what they are isomorphic to. Thus theorems that tell us about normal subgroups and give us the ability to use the first, second, or third isomorphism theorems are the most helpful to understanding groups. I could see this test being very similar to the last one which asks us true false questions about whether certain sets are groups, certain types of groups, whether groups have elements of certain order, etc. Then there will be some questions asking for examples of groups that are isomorphic to Zn or ZnxZm and examples of other types of groups. Then there will probably be some large proof problem asking for some use of the First Isomorphism Theorem for groups like the last test had for rings. I think that there will also be some problem asking for a proof about the order of elements in a group and proving what those orders are (so using Lagrange's theorem).

I need to work on a lot for this test. I think I understood a lot of the material but I didn't really internalize it yet. I also need to work on the last couple of sections that we covered. I have the most difficult time with the material regarding Sn groups (like talking about permutations being odd or even and talking about groups An). I would like to go over problems 7 and 8 on the practice exam because I have a difficult time finding subgroups of groups and I am wondering if there is an easier way to do it that to just check a lot of options.

7.10, due on November 22

I would honestly say that this section is probably one of the most difficult for me to understand. I was glad it was so short because I don't think I got very much out of it and if there had been more information that would have been more information that went over my head. I think I am having a difficult time understanding what An is. I don't know what a group like this really is. I also didn't understand why lemma 7.53 is true. Nor did I understand case 3 of the proof of theorem 7.52 or actually why we could boil down the proof to three cases in the first place.

It is interesting to me whenever there is a theorem that is true for all n (here just n not 4). It is especially interesting when these proofs aren't by induction and are relatively short in length. Most proofs of theorems with that broad of an assumption turn out to be more difficult that just three cases like the one for theorem 7.52.

I had an additional question not on this section but on the homework for section 7.8. I won't be able to go to office hours since I have a test to take so I was hoping if there is time you could address it in class or I will ask you after class. On problem 1d) I can't figure out how to interpret the function. I guess f in Sn looks like some cycle (12...n) or something. But then I thought (k) must be one number inside parenthesis since 1 less than or equal to k less than n. So is (k) just the identity because isn't that how we write the identity in cycle notation? (1)=(2)=(3)=(4)=e in Sn? So then what is f(k)? Wouldn't that always be just f? And is the function just plugging in different values of f (different elements of Sn) or also different (k) (which would all be the same thing if my above assumption is correct)? Please help.
                                                                         

7.8, due on Novermber 21

I feel similarly about this section as I did about the section on the First Isomorphism Theorem for rings. It was interesting, but the proofs were long and confusing. I didn't understand the proof of theorem 7.45 and since the proof of the First Isomophism Theorem for rings confused me the first time, I would love to run through the proof of the First Isomorphism Theorem for Groups in full instead of just saying "translate the proof of theorem 6.13".

I really liked the Third Isomorphism Theorem for Groups. I thought it was interesting that the book defined another isomorphism theorem in the section rather that just having us discover it in the practices (like in the section about rings). Is that because this theorem is more useful in groups than in rings?

7.7, due on November 18

The most difficult part of this section for me to understand was actually one of the examples. The second example on page 218 was confusing because they did the arithmetic for the example but didn't do any of the verification that G/N is cyclic and of order 4. I think that I am confused as to what a cyclic quotient group looks like and how to use a multiplication table to determine that. If I could get to the determination that it had an element of order four then I would know that it was cyclic and that it was isomorphic to Z4. But I can't figure out how to tell from the table that something has order 4.

I thought the most interesting part of the section was the last theorem where it is shown that if G/C is cyclic then G is abelian. Do we know in what cases G/C is cyclic? Does it happen often? It would be nice to look at examples where this theorem is true. I would also like to look at examples where G is abelian but G/C is not cyclic.

half of 7.6, due on November 16

I am really glad that we are going through the proof of theorem 7.34 tomorrow in class because I didn't quite catch what was going on. I also didn't really understand the proof of theorem 7.33. I would like to go through that again if possible.

I was wondering when we would get into quotient groups so I suppose it's good that we've got the background for those now. I don't think that I really grasp how normal subgroups have similar properties to ideals. I really want to take a class on the application of rings and groups because I find them interesting but sometimes too abstract to grasp without application.

half of 7.5 and half of 7.6, due on November 14

I thought the rest of section 7.5 was mostly understandable. What I read from section 7.6 was very confusing though. I'm sure missing class on Friday is a main reason for my confusion. I am having a really hard time conceptualizing what a congruence class in a group is. So talking about left congruence classes now is increasing my confusion. I think that going through examples in class will really be the only way for me to grasp this.

I really enjoy being able to classify finite groups as being isomorphic to some particular modulo of the integers. I feel like isomorphism in groups seems to be much more easy to identify than in rings and thus I feel like I know so much more about a finite group when I see it than I did about a ring. I actually enjoy rings more I think. Maybe that's just because I'm having a harder time understanding groups.

half of 7.5, due on November 11

The most difficult part of the reading to understand was corollary 7.27 and it's proof. Maybe I had been reading too long but it didn't make sense to me. I also didn't quite get the proof of theorem 7.25. All in all though this section was easier to understand than most.

I thought Lagrange's theorem was really interesting. The proof was relatively easy to understand so it makes me believe that it has been developed over time from an original much more complicated proof by Lagrange. He must've done much more work in group theory developing more than just that one theory or else I feel like someone else would have figured out this result sooner.

7.9, due on November 9

I didn't understand the whole second half of the section starting with transpositions. I guess what was the source of my confusion was starting with how (12)(12)=(1). I think I might be just not understanding the notation. After that, I couldn't follow the whole odd or even number of transpositions proofs and the last theorem 7.51 was the most confusing. I did not understand the definition of An and the proof that followed.

I thought the most interesting part of the section was that we developed the notation for symmetric groups to simplify problems. It seems like a really clever and simple solution to writing the long form of elements in a symmetric group. I also thought it was interesting that theorem 7.51 was proved using the First Isomorphism Theorem. I want to know how this theorem is proved when you learn about group theory before ring theory and don't have the First Isomorphism Theorem to use.

7.4, due on November 7

The ideas in the section made a lot of sense to me because we already have some familiarity with isomorphisms and homomorphisms. However, I had a difficult time following the proofs of the section. Specifically, I didn't understand the reason why we chose the function f(i)=a^i for theorem 7.18. I also didn't understand part 2 of the proof of theorem 7.19. And then the entirety of Theorem 7.20 was completely confusing and then of course the proof of corollary 7.21 plays off of that proof so I didn't get that one either.

I thought the most interesting part of the section was how many isomorphisms we can guarantee for groups. We know there is always an isomorphism from a finite group to a subgroup of the symmetric group. And we know that every group is isomorphic to a group of permutations and cyclic groups are isomorphic to the integers or the integers mod n. It seems like it is relatively easier to find isomorphisms for groups than for rings but maybe that's not true.

7.3, due on November 4

I don't know what it was about this section that really got to me, but I had very little understanding of this section. After theorem 7.11, the rest of the section went over my head. I need to go through the whole section again in class with lots of examples and then reread the section myself. I really don't understand what I read today.

I thought it was interesting that the proof of theorem 7.10 was so short. It seemed like we could almost just assume that H is a group. As interesting as this was though, I was confused by the fact that we could assume (ii) to prove the theorem. I thought assuming the theorem was never considered a proof.

7.2, due on November 2

For the most part I followed this section well. The most confusing parts were parts 3 and 4 of theorem 7.8 and their proofs, and the proof of corollary 7.9. I understood what corollary 7.9 was saying but I couldn't follow the proof (probably because I didn't the references of how to use theorem 7.8 part 4.

I was glad we had already spent a few minutes of class discussing the order of an element because I don't know that I would have understood the discussion without having already gone through some examples. I'm curious to know how order plays a role in groups. Like it was mentioned in class, we will be using it probably in conjunction with isomorphism between groups. Because of theorem 7.8 part 3 where a^i=a^j iff i is congruent to j mod n, I'm wondering if we'll be making congruence classes in groups and using those in different ways. I can see how this chapter could go from relatively easy to understand to completely abstract relatively quickly.