8.5, due on December 7

I felt like this section was easier than some of the others recently (it helped that a page and a half of the section was describing dihedral groups that we have already learned about). I didn't understand the first proof (theorem 8.27). I didn't know how they obtained the equation they call the class equation and I was unsure if the Ci's were just divisors of G or what they were. I also didn't really understand the proof of theorem 8.34. It was really long and I wasn't exactly sure how they used all these assumptions to construct the multiplication table. I would be helpful to go through that proof again.

I had one other question which was about the group Q, the quaternion group. I missed one or two days and I'm wondering if we talked about it on one of those days. I don't know what it is or why it exists and has no similar groups in other orders except of order 12 (is the reason there is a T group similar to the reason there is a Q group? It was a little disheartening that the chapter ends by telling us that there are 267 different groups of order 64 and that there was no way to know the number of distinct groups of order n. How do mathematicians find the maximum number of distinct groups for very large groups?

Looking over the study exam for the final made me feel a little overwhelmed. I feel like it is all important and that means I have a lot of stuff to memorize. I really need to review the stuff on rings because it has been a while since we covered that material. I also don't fully feel comfortable using the theorems from this chapter to prove things. I want to go over the practice problems from the study guide especially numbers 8, 9, 12, 17, and 18.

I have learned so much from this class. I don't know what will be specifically important to me in my future because I don't know what I want to do with my math degree. But I think that the improvement that I have made in being able to prove things will be really helpful in future classes, especially theory of analysis which freaked me out because I couldn't remember how to do proofs at all.

8.3/8.4, due on December 5

By far these were the most confusing sections of the class. I don't understand what is going on with the Sylow second theorem. I don't even know what they mean by P=x^-1Kx because I thought that doing x^-1Kx was used to determine if K is normal in G. What would it look like to have two sylow p-subgroups that were equal in this way? But my level of confusion really grew in section 8.4. I can read theorems and think I understand them, but then reading the proofs really showed my that I did not. I was so confused by the use of the conjugacy classes as used in the proofs. I also was getting mixed up and confused by the many uses of Lagrange's theorem. I need to reread these sections again and I need to go through the proofs in class really slowly.

I thought that the most interesting part of the section was talking about conjugacy classes and the centralizer. Throughout the semester as I have looked up proofs and information for clarification, conjugacy and centralizer have come up really frequently. People are always using them as the easiest way to prove things that we learned earlier on (which is obviously unhelpful at the time because we hadn't learned it). I am happy to finally be talking about them because (once I funny understand them) I think it will make a lot of aspects of group theory more clear - or at least I'll be able to understand some more of the online discussions and proofs of group theory.

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.