Final Exam Questions, due on Wednesday, April 13

1. Which topics and theorems do you think are important out of those we have studied?
I think the following items are important: the Cauchy theorem, cyclic groups, normal subgroups, quotient groups, other types of groups: symmetric, alternating, simple, abelian, finite abelian, etc.

2. What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out in class on Wednesday.
I am still confused about things that deal with quotient groups, cyclic groups, and direct products. These are the three things that have not sunk in yet. I still do not know how to create cyclic groups, and I do not know how to apply quotient groups, and I do not know how to figure out direct products. Any problem that deals with one of these three questions would be nice to see on Wednesday. Like an example of writing a direct sum/product of certain groups would be nice to see on Wednesday.

3. How do you think the things you learned in this course might be useful to you in the future?
I do not think I am going to take any of the theorems or properties we have learned in this class in my future for teaching mathematics to middle school students. However, I do think I can take a more non-literal approach with what I have learned in this class and apply it in my future. I have learned how to work really hard, and I have developed my ability to analyze and think about problems and create my own proof. This class has taught me to really think about mathematics and construct mathematical ideas on my own. And I will be able to use these things that I have learned in my future with harder classes that might come my way and for when I am trying to get my future math students to construct their own proofs and properties for mathematics.

8.3, due on Monday, April 11

1. The hardest part of this reading to understand is theorem 8.15, which is the second sylow theorem. I do not know how if P and K are both Sylow's how P = (x^-1)Kx for some x in G. I do know see how that would work and it does not make much sense. Can't P and K just be differnt prime number-subgroups, so then they would not be related in that way? I'm confused.

2. The most interesting part of this reading was that there was such a classification as a Sylow p-subgroup. I like when new properties and terminologies are introduced to us. It makes me feel like people have really studied abstract algebra before us and have created the most important things you can do with groups so that I know these things are important and are not just random theorems and properties to random groups of numbers. So I like this and I like new things like Sylow p-subgroups and that is what I find so interesting about this section.

8.2, due on Friday, April 8

1. Although there were lots of proofs and lemmas in the section to go through and understand, the hardest one to grasp was theorem 8.7 - the fundamental theorem of finite abelian groups. It makes me think that this theorem is most important because its name is the "fundamental theorem." However, I think it is really confusing that it is so important and fundamental to know about finite abelian groups that they are direct sums of cyclic groups, each of prime power order. Why is this so important? And why do we care? I do not understand.

2. The most interesting part of this section was that there was sooo many different proofs and properties for finite abelian groups. I would have never thought they were so important or unique in the sense that they have all these special theorems that people have discovered, studied, and now teach to anyone in abstract algebra. It is pretty cool.

8.1, due on Wednesday, April 6

1. The hardest part of this reading is understanding some of what theorem 8.1 is saying. I do not understanding G's elements being of the form a1a2a3... with each one pairing up with N1, N2, etc. What is an example of these types of elements in G and their corresponding normal subgroups in actual numbers and groups and not just letters. I cannot follow anything with letters if I do not have an example of what it is saying in numbers. It is just very confusing.

2. The most interesting part of this reading was that you can write groups as a sum of two or more of its subgroups. I think that is really cool and interesting. Also, you can know how many elements are in a group made of up crossing between other group because if those other groups are finite, you just multiple each's order to get the order of the cross of them all. That is really cool too. I like doing fun stuff like that.

7.10, due on Monday, April 4

1. The hardest part of this section to understand is the three cases in the main proof of theorem 7.52. Once we start having elements that are products of cycles written as (123...) and lots of them together, I start getting really confused and I cannot follow the work of the proof. So this is the hardest part to understand.

2. It is interesting that this section is basically just one theorem and its proof (with a couple of lemmas and their proofs included). This tells me that this theorem is so important that an entire section and day in class needs to be dedicated to it. So, I am determined to understand this theorem well enough to appreciate its importance in our study of abstract algebra, specifically in our understanding of the classification of finite groups.

7.8, due on Wednesday, March 30

1. The hardest part of this reading is understanding the importance in some of the theorems discussed and proved in this section. Some of them may make sense, but I wonder why they are important and why we have to learn them. Like, for example, why is it important to know theorem 7.45: G is a simple abelian group iff G is isomorphic to the additive group Zp for some prime p. Also, why is it important to know that if you have a situation like theorem 7.44 part 3. It just seems like these things are not very pertinent to our study of mathematics, thus it becomes very difficult for me to learn these things and understand them when I think they are somewhat not very useful.

2. The best part of this reading is being about to create homomorphisms and isomorphisms with/to quotient groups. I love coming back to concepts I actually understand and I actually understand homomorphisms and isomorphisms! This makes the new math for the day better because I feel like I have a strong handle on some of the new stuff because it is old stuff that I really understand still.

7.7, due on Monday, March 28

1. The hardest part of this reading is keeping straight all the different applications to quotient groups such as cyclic, abelian, normal, orders, and cosets. It is a lot of different properties of groups to be pushed into just one little section. Quotient rings are hard enough, thus making quotient groups hard, thus adding all the properties of groups to quotient rings is very confusing and hard to commit to memory and understanding.

2. It was really interesting that quotients came back into the class. We had studied quotient rings but now we are studying quotient groups. I like that this connection exists and that quotient things are coming back. Although, I was never very good with understanding quotient rings, so I probably will have to work hard to understand quotient groups.