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.