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.

7.6 Part 2, due on Friday, March 25

1. The hardest part of this reading for me is why in Theorem 7.34 - the 5 equivalent statements about normal subgroups - are statements #2 and #4 necessary as well as #3 and #5. Why do we need subsets and equal to for aNa-1, as well as with a-1Na. Wouldn't it be sufficient to just have statements 1-3 because 4 and 5 are included in 2 and 3? This is why I am confused and why this is the hardest part of the reading, because I do not see a purpose in this.

2. The most interesting part of this reading is that just because a subgroup N of group G is normal so Na = aN for every a in G does not mean that na=an for every n in N. This is fascinating to me. It is almost like a subgroup is normal is everything in G gets hit from both the left cosets and the right cosets but not necessarily that the things hitting are the same cosets from left and right. This is a really interesting idea to think about and this is what I liked learning the most in this section.

Extra Credit - Tuesday, March 22

Deanna Haunsperger - Carlton College - "Bright Lights on the Horizon"

1. The hardest part of this lecture was understanding exactly what the theme/purpose/topic of her lecture was. Every few minutes she started a new snippet about some interesting mathematics that someone is doing. But I did not know how each related to each other and what the point of talking about each of these people was other than it was really fascinating. I think she was just mentioning interesting articles that have been published in the journal she is in charge of, but it was still hard to follow and confusing to switch between all these different stories every two minutes.

2. The most interesting part of this lecture was EVERYTHING! She was so interesting to talk to and I actually understood what she was saying! The parts of the lecture I enjoyed the most were the parts where she would talk about mathematicians who did work that dealt with the arts. For example, mathematician Stephen D. Abbott studies "Turning Theorems into Plays." This is so cool because I am in love with plays and I love plays with math. She mentioned that the main playwright Abbott studies is Tom Stoppard. I actually saw one of his plays in London last summer, but I am ashamed to say that I did not know that Stoppard wrote other plays about math. She mentioned the play "Arcadia." I am now inspired to go read it! I also loved her talking about the topic "The Eccentricies of Actors." This deals with the Bacon Number and if Kevin Bacon is the center of the unverise and how many connections it takes for an actor to be connected to Kevin Bacon. Then she related it to Erdos numbers in mathematics. I absolutely loved this lecture. I learned things, and I was inspired by things she said, and it was just so interesting!!

7.6 Part 1, due on Wednesday, March 23

1. The most difficult part of this reading is keeping track of the left and right cosets. I already struggle with finding cosets, so now having to understand right and left cosets and be able to find them and compare them to see if they are different so that you can figure out if a subgroup is normal or do the right and left congruent principle is really confusing. I am worried that this is just going to make cosets even harder for me to understand.

2. The most interesting part of this section was that subgroups can have elements that are left congruent or right congruent mod the subgroup. This is so interesting. I have never thought about a subgroup or anything we have discussed having two different congruences - a right one and a left one. This is a nice layer added onto to the information we have already learned.

Midterm #2 Questions, Monday, March 21

1. Which topics and theorems do you think are the most important out of those we have studied?
Groups, subgroups, ideals, rings, and subrings. Knowing the properties and examples of these are going to be the most important.

2. What kinds of questions do you expect to see on the exam?
I expect to see questions about examples of different groups and ideals and such that we have talked about. I also expect to see a couple of proofs, maybe one or two we have done in class, and then a couple that we haven't done exactly but are similar -- like proving if something is a group, subgroup, finding ideals, kernels, etc.

3. 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 Monday.
I would like to review ideals. I have forgotten a lot of chapter 6, and that would be the best thing because other earlier principles build on newer stuff, but ideals would be something good to go over because I am still confused. I would like to see a problem of listing off all the big examples of groups, subgroups, etc.

7.5 Part 2, due on Friday, March 18

1. The hardest part of this reading was understanding the new theorems about how every group of 4 is isomorphic to either Z4 or Z2 x Z2 and the theorem about every group of order 6. I guess I do not understand why these theorems are important, and the proofs do not convince me that it is true for every group of order 4 and order 6.

2. The most interesting part of this reading was bringing back the principles about primes for theorem 7.28. It was cool to apply primes to groups.

7.5 Part 1, due on Wednesday, March 16

1. A lot of this reading was not difficult because it was just applying congruence stuff and cosets to groups now. So the hardest part was just the new terminology and symbols for the index of H in G is denoted [G:H]. This is kind of confusing for me just because it is something that is actually new in this section. So then using that and putting it into Lagrange's theorem is kind of confusing because I am having a hard time wrapping my mind around what it means and how to use it properly.

2. The most interesting part of this reading is that that theorem 7.26 is named Lagrange's theorem. I am curious to know how this relates to like Lagrange multipliers that I have studied before (in, I think, linear algebra but I'm not sure). Is it just the same guy who discovered both of these? Or are they connected principles to each other? I don't know because I don't actually remember Lagrange multipliers other than the name is something I have studied before. So I would be interested in learning the connection between that and the Lagrange's theorem in the reading.

Extra Credit - Tuesday, March 15 - Vitaly Bergelson's Lecture

1. The most difficult part of this special guest lecture to understand was where exactly everything he was saying fits into mathematics, particularly to my learning of mathematics. While many of the terms and steps he did in the proofs he was showing were familiar to me, I did not understand what he was trying to do. Additionally, I have a really hard time understanding many people that lecture, including him. So when I can't understand what they are saying, I have a hard time following their thoughts and knowledge.

2. The most interesting part of this lecture was the fact that it was the first mathematics guest lecture I have attended. Sometimes I get stuck in a bubble of thinking that the math we are doing is not universal and that BYU has its own way of doing proofs and math and such. However, attending this lecture opens my mind up to the world that mathematics and the way of proving them are universal. While I could not really follow what was going on, I could see him write down symbols and steps of the proof that I did understand and that I have used before and that was really cool. I am glad I attended.

7.4, due on Monday, March 14

1. The hardest part of the reading is probably understanding the new principle of an automorphisms. Everything else in this reading we have done, just with rings so it is pretty easy to understand. But an automorphism is something that is new, and it is used in the homework problem number 19, and I do not know how to do it. Like what is the point of an automorphism and an inner automorphism of G induced by c. It is so confusing.

2. The most interesting part of this reading was that isomorphisms and homomorphisms have come back and can be applied to groups. This is a really cool connection. I like learning things that I already know. It makes me feel smart, and it makes me feel like I can actually do the mathematics that is in this course.

7.3, due on Friday, March 11

1. The most difficult part of this reading was realizing that because I am still confused on an order of an element, I cannot understand subgroups, especially cyclic subgroups. And the biggest problem is is that I am not sure what it is about a, a^2, a^3,... elements and groups that I do not understand. But I cannot do any of the homework do tomorrow so clearly I do not understand something, and thus, I cannot understand cyclic subgroups and how they work and how to use them, so this is the hardest part of the reading.

2. The most interesting part of this reading was that once again a sub-"something" with only a couple of conditions was introduced. We have subsets, subrings, subfields, etc. and now we have subgroups! It is so interesting how everything in this class connects and parallels to each other. For example, eventually in this section we get to the only conditions for something to be a subgroup is to be a nonempty, finite subset and to be closed under the operation of the group that it is a subset under. Then it is a subgroup! Fascinating!

7.2, due on Wednesday, March 9

1. The hardest part about this reading was understanding the purpose and proof from Theorem 7.8. It kind of makes sense, but I do not really understand the purpose in knowing about an element in a group and the order of that element and the order of that element raised to an exponent, etc. So it makes it really hard to comprehend because I do not know where the book is going with the information in this theorem.

2. The most interesting part of this reading is tightly connected to the hardest part of this reading -- the order of an element of a group. This is so interesting. I would not have expected us to look at the order of an individual element of a group. It is so cool how everything in this course expands itself!

7.1 Part 1, due on Friday, March 4

1. The most difficult part of this reading is going to be understanding how to keep the rules for what is in a group G and what the rules for the binary operation of G are. After I read the section, I skimmed over the homework problems, and with problems like 4 and others, I found myself getting confused with the rules for what goes in the group as an element and then what the binary operation * is. That will be the most difficult part with this new idea of groups.

2. The most interesting part of this section is that I have seen all this before - I just cannot remember where! Maybe we did stuff like this in linear algebra... or calculus... or statistics... or math 300. I actually cannot remember, but as I was reading, I felt like I already knew how to do it all. So I know this first look at groups connects with math I have learned before, I just cannot remember where. But I do think it is cool because then it makes it feel like learning about groups will be knowledge that will be very attainable, which I really think is cool.

6.3, due on Wednesday, March 2

1. The most difficult part of this reading was understanding how the term prime can be applied to ideals. I do not understand how an ideal is prime when if b*c is in it then either b is in it or c is in it. Why does that make it prime? Then there is a theorem that says that something is a maximal ideal iff the quotient ring is a field, and yet on our homework it asks for the maximal ideal of rings that are not fields, so how does that work? I am so confused!

2. This reading connects to ideals to primes, which is really cool (once I am able to understand it!). I have only seen primes as associated with one number, but now we are calling an entire ideal prime! It is weird and confusing but still cool at the same time. I wonder if we will continue as the sections go on to add prime properties to things such as ideals and other topics in this abstract algebra course.

6.2 Part 2, due on Monday, February 26

1. The most difficult part of this section of the reading is keeping all the theorems and ideas straight with the kernels and quotient rings and regular rings and ideals and etc. There is so much to remember to then apply it to problems that are on the homework. I am so confused about which one of those terms when paired with another makes a function isomorphic or homomorphic (like theorems 6.12 and 6.13). It is all so unclear and the proofs do not help me understand them so I just keep getting more confused.

2. The most interesting part of this section is how kernels are being used in theorems to connect our rings back to homomorphic and isomorphic. I would have never guessed that the kernels we saw in linear algebra would come back and be applied to something we did at the beginning of the semester in Abstract Algebra. I love how connected mathematics is!