Section 8.2, due on November 29

The most difficult part of this material for me was following Lemma 8.6. I got a little confused with the difference between a p-group and a cyclic group. Maybe it was just because it is a long theorem but there were some conclusions drawn that just didn't quite make sense to me.

The most interesting part of this material was the Fundamental Theorem of Finite Abelian Groups. When I think about all of the things that come from knowing a group is cyclic and has some characteristic associated with primes, it's very cool to think that every finite abelian group is the direct sum of cyclic groups and each of prime power order.

Section 8.1, due on November 23

The most difficult part of this material for me was understanding how you would write an element of a group that is a direct product of a bunch of different groups that don't have the same operation. Do you do it like a cross product like we had learned before?

The most interesting part of this material for me was the fact that just because G is a group that makes up some cross product, doesn't mean that G is a subgroup of that cross product.

Section 7.10, due on November 22

The most difficult part of this section was following all of the long proofs and trying to keep up with what was going on. I also don't really think I fully understand what the Alternating Group is and so some of the theorems made sense, but I didn't really understand the full implications behind them.

The most interesting part of this section was Lemma 7.53. I just thought it was interesting that even some An with n even has this property that every element is the product of 3-cycles. I also thought it was interesting to think about how all of these different theorems/lemmas of this section would apply to other areas and how they help.

Section 7.9, due on November 19

The most difficult part of this material for me was Theorem 7.5 and also just trying to keep the order of the cycles right. I don't think I quite understand why in Thrm. 7.5 they just remove the parantheses and put the element side by side because I thought the parentheses denoted cycles.

The most interesting part of this material for me was the Thrm. 7.47. I thought it was neat that every permutation in Sn can be written as the product of disjoing cycles. This seems like a really interesting concept and it just seems like a cool fact that has been proved.

Exam 3 review, due on November 17

Before the exam I need to better understand the concept of quotient groups and modding out by subgroups. I also need to go over again the various concepts of order and the theorems that apply to them. I feel like I also need to better understand some particular theorems such as the Second Isomorphism Theorem for groups.

I problem that I would like to see worked out in class would be one involving kernels/normal subgroups. I sometimes get confused in these types of proofs when transitioning between the different steps.

Section 7.8, due on November 15

The most difficult part of this material for me was understanding the Third Isomorphism Theorem and the theory that lead up to it. I still don't think I fully understand what subgroups of G/N look like and how they are related to G. I am also still a little confused about the section where they talk about how simple groups are the building blocks of all other groups and also the term composition factors.

The most interesting part of this section for me was theorem 7.45. I thought this was interesting considering that simple groups are the building blocks for all other groups and we know what the building blocks are isomorphic to, but when combined into a larger group we do not always know.

Section 7.7, due on November 12

The most difficult part of this material for me was thinking about what G/N actually does. The concept of modding out by a subgroup seems similar to modding out by ideals, but for some reason it still takes me a minute to really think about what is going on. Also some of their examples were a little confusing to follow.

The most interesting part of this material for me was noticing all of the similarities and differences to when we modded out by an ideal. I found it interesting that with groups we can talk about the order of G/N (thrm. 7.36). Why did the topic of order of a group just now come into play? Why didn't we talk about this when we were studying rings?

Section 7.6, due on November 10

The most difficult part of this material for me was following the all of the details that come along with Theorem 7.4. It was more difficult for me to understand in general why aN=N does not means that an=n for each n in N but rather that an=n1 for some n1 in N. The differences between these statements are a little confusing to me (ie. top of page 213).

The most interesting part of this material for me was the concept of a normal subgroup and how it is analagous to an ideal. It also made me wonder if it is at all related to a normal vector that we study in multivariable calculus and linear algebra. It is interesting, but still a little confusing, that normal subgroups aren't necessarily made up of elements that commute with all other elements of the group.

Section 7.5 (second part), due on November 8

The most difficult part of this material for me was completely following all of the logic and reasoning involved in Theorem 7.30 especially how they came up with the six elements from Ne and Nb that are elements of G.

The most interesting part of this material was that every group of order p (where p is a positive prime integer) is cyclic and isomorphic to Zp. I thought that this seemed to be a pretty powerful statement considering there are infinitely many primes. Considering the special behavior of Zp, it makes me wonder if a group of order p has special structure and behavior as well because it is isomorphic.

Section 7.5 (first part), due on November 5

The most difficult part of this material was understanding the concept of the index of H in G and also Lagrange's Theorem. The concept of congruence with subgroups doesn't quite make sense to me which is why the rest of these concepts aren't quite clicking. Also it talks about right cosets but not left cosets...are there left cosets and if there are do they have the same properties.

The most interesting part of this section for me was relating all of the similar properties to rings and integers. I also found it interesting that you can even define congruence when you mod out by a subgroup. This idea was really interesting to me, however it still doesn't quite make sense.

Section 7.4, due on November 3

The most difficult part of this material for me was trying to understand the proof of Cayley's Theorem. Also, I didn't really understand the second example of the section. I also don't remember what it means when they put two stars next to R, and whether that is just to denote something or if it is supposed to mean a certain something.

The most interesting part of this section for me was trying to figure out how isomorphisms of groups could be applied to the real world and if this is even a possibility. I was also wondering if certain things are preserved in a group isomorphism. Also, what is the difference between an isomorphism of rings and groups in a visual sense. Does it just sort of copy something more general whereas in the case of rings it is a copy of something more detailed? Or is the actual isomorphism different when dealing with rings?

Section 7.3, due on November 1

The hardest part of this material for me was understanding Theorem 7.17. I don't quite understand what is going on with the subgroup generated by S. I was also wondering in regards to the example under Theorem 7.13, why every unit mulitiplied by another unit must be a unit. It seems to make sense, I am just not connecting why this must be true.

The most interesting part of this section was the center. I found this interesting and was wondering if the center of a group is sort of like an ideal of a ring. They seem similar and so I was wondering if they are related in any way.