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.
Thursday, March 24, 2011
Tuesday, March 22, 2011
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!!
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!!
Monday, March 21, 2011
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.
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.
Sunday, March 20, 2011
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.
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.
Thursday, March 17, 2011
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.
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.
Tuesday, March 15, 2011
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.
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.
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.
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