1. Which topics and theorems do you think are important out of those we have studied?
Um... I think that everything we've talked about since the last exam will be pretty important... It seems like the base for everything we have done in this class is knowing the definitions of groups and rings and using those definitions to prove different types of results. So, I'd say that knowing those definitions along with the theorems with names will be pretty darn important.
2. What do you need to work on understanding better before the exam?
Cauchy's Theorem. OMG, I have reviewed that thing four or five times since that day in class, and it still gives me nightmares and makes me want to cry.
3. How do you think the things you learned in this course might be useful to you in the future?
The things I have learned from this class, cheesy but true, are the importance of doing readings and homework before class because it really helps with understanding concepts explained in class. As I mentioned before when I did the reading I was able to listen more closely to things that seemed confusing during the reading, rather than trying to learn something during class and having more questions during homework. I also learned about proof techniques. While I still don't feel completely comfortable writing them on my own, I definitely got in the hang of using "If" statements to prove "then" statements and carefully examining definitions to help with the proofs.
Tuesday, April 12, 2011
Sunday, April 10, 2011
8.3
1. What was the most difficult part of the reading for you?
I think the hardest part of the reading for me is similar to the problems I've encountered at the end of the previous two unit's. It seems that the closer we get to tests the more all of the theorems kind of blend together. This is especially true as we are approaching the end of the semester where we have so many 'prime,' 'order,' 'isomorphic,' etc. proofs that things are kind of bleeding together. It certainly didn't help that they didn't really provide proofs for these theorems, but rather just said, "Here are a bunch of theorems you aren't really ready to prove, so we'll do that later." If I wasn't sitting here with my book in front of me, I doubt I'd be able to recount to you what any of the proofs actually said.
2. On a more positive note...
I know I just sounded like a big baby about all of the theorems bleeding together, but on a more positive note it is helpful that we have covered so many theorems that are so similar to each other because it has also made it more 'natural' to prove things, because it always feel like we've proven something similar before. So, hopefully I won't fail the final or the class, because as educational as this semester has been, I'd rather not do it again. :)
I think the hardest part of the reading for me is similar to the problems I've encountered at the end of the previous two unit's. It seems that the closer we get to tests the more all of the theorems kind of blend together. This is especially true as we are approaching the end of the semester where we have so many 'prime,' 'order,' 'isomorphic,' etc. proofs that things are kind of bleeding together. It certainly didn't help that they didn't really provide proofs for these theorems, but rather just said, "Here are a bunch of theorems you aren't really ready to prove, so we'll do that later." If I wasn't sitting here with my book in front of me, I doubt I'd be able to recount to you what any of the proofs actually said.
2. On a more positive note...
I know I just sounded like a big baby about all of the theorems bleeding together, but on a more positive note it is helpful that we have covered so many theorems that are so similar to each other because it has also made it more 'natural' to prove things, because it always feel like we've proven something similar before. So, hopefully I won't fail the final or the class, because as educational as this semester has been, I'd rather not do it again. :)
Tuesday, April 5, 2011
8.1
1. What was the hardest part of the reading for you?
Um... I'm kind of confused by Theorem 8.1, which sucks since it's the first theorem of the chapter. I am just not sure that I know exactly what this theorem is saying or what the proof is showing (other than proving the theorem).
2. May I say something about the final?
I'm not sure how else to say this, but I thought this was a pretty good place... For the love of everything holy, can we please not prove that theorem you proved yesterday in class on the test? Seriously, I was so confused and lost and with all of the new vocabulary and procedures, I wanted to cry and/or die. Or, if you're going to put it on the test, can you please review it or something? I mean, I wrote the whole thing down and as I was re-reading it after class it was more confusing than ever... Please, show some humanity. :)
Um... I'm kind of confused by Theorem 8.1, which sucks since it's the first theorem of the chapter. I am just not sure that I know exactly what this theorem is saying or what the proof is showing (other than proving the theorem).
2. May I say something about the final?
I'm not sure how else to say this, but I thought this was a pretty good place... For the love of everything holy, can we please not prove that theorem you proved yesterday in class on the test? Seriously, I was so confused and lost and with all of the new vocabulary and procedures, I wanted to cry and/or die. Or, if you're going to put it on the test, can you please review it or something? I mean, I wrote the whole thing down and as I was re-reading it after class it was more confusing than ever... Please, show some humanity. :)
Sunday, April 3, 2011
7.10
1. What was the hardest part of the reading for you?
Honestly, and I am embarrassed, I struggled with the whole thing. I'm just not exactly sure what this chapter is saying. It's obvious that it's all aimed at proving theorem 7.52, but I'm not exactly sure what that theorem is saying (because I can read) but I really struggled to follow the proofs of the lemmas and the meaning behind the theorem.
2. Figure out what Theorem 7.52 is saying
Okay, so I'm going to break this down and looks stuff up. The theorem states "For each n =/ 4, the alternating group An is a simple group."
Alright, I know the chapter addresses this, but I'm still not sure about n not equaling 4... but let's move on from there since it's just the assumption. Well, let's review "Alternating Groups". Chapter 7.9 says that an alternating group is the set of all even permutations (even permutations can be written as the product of an even number of transpositions [2-cycles]) in Sn. The set of these permutations is written An... And a simple group is a group whose only normal subgroups (or subgroups that have left and right cosets equal to each other) are <e> and itself (G). So, what the theorem is saying, as far as I understand it, is if the set of all permutations (not with 4 elements... or unique mappings containing the numbers 1-4, since A4 really has 24 elements...) that has permutations that are even, or can be written as an even number of 2-cycles, is a subgroup that only has the normal subgroups of <e> and itself (the alternating group of elements of Sn that has all even permutations.)
Honestly, and I am embarrassed, I struggled with the whole thing. I'm just not exactly sure what this chapter is saying. It's obvious that it's all aimed at proving theorem 7.52, but I'm not exactly sure what that theorem is saying (because I can read) but I really struggled to follow the proofs of the lemmas and the meaning behind the theorem.
2. Figure out what Theorem 7.52 is saying
Okay, so I'm going to break this down and looks stuff up. The theorem states "For each n =/ 4, the alternating group An is a simple group."
Alright, I know the chapter addresses this, but I'm still not sure about n not equaling 4... but let's move on from there since it's just the assumption. Well, let's review "Alternating Groups". Chapter 7.9 says that an alternating group is the set of all even permutations (even permutations can be written as the product of an even number of transpositions [2-cycles]) in Sn. The set of these permutations is written An... And a simple group is a group whose only normal subgroups (or subgroups that have left and right cosets equal to each other) are <e> and itself (G). So, what the theorem is saying, as far as I understand it, is if the set of all permutations (not with 4 elements... or unique mappings containing the numbers 1-4, since A4 really has 24 elements...) that has permutations that are even, or can be written as an even number of 2-cycles, is a subgroup that only has the normal subgroups of <e> and itself (the alternating group of elements of Sn that has all even permutations.)
Thursday, March 31, 2011
7.9
1. What was the hardest part of the reading for you?
I was fine with the reading (though this new notation feels confusing and silly to me so far) until the part about transpositions. I DO NOT UNDERSTAND THESE!!!!! I don't understand, with all of the establishing of cycles and stuff, where these transpositions come from or what they even mean. Because of that, I really felt stuck on the alternating groups section.
2. New Notation
I kind of understand the reasoning behind this new notation, and as annoying as it can be to write symmetric groups with the older notation (with the two lines) I would much rather write that because it is something you can track.... Like, I feel like I'm just going to get so lost when it comes to function composition if I'm just looking at a string of numbers (especially if it isn't separated by a comma! Oh Commas, how I love you.)
I was fine with the reading (though this new notation feels confusing and silly to me so far) until the part about transpositions. I DO NOT UNDERSTAND THESE!!!!! I don't understand, with all of the establishing of cycles and stuff, where these transpositions come from or what they even mean. Because of that, I really felt stuck on the alternating groups section.
2. New Notation
I kind of understand the reasoning behind this new notation, and as annoying as it can be to write symmetric groups with the older notation (with the two lines) I would much rather write that because it is something you can track.... Like, I feel like I'm just going to get so lost when it comes to function composition if I'm just looking at a string of numbers (especially if it isn't separated by a comma! Oh Commas, how I love you.)
Tuesday, March 29, 2011
7.8
1. What was the most difficult part of the reading for you?
I think that I understand the idea of simple groups, that doesn't really bother me. However I really struggled to follow the reading after the first paragraph of that section. I mostly think that the problem is that there aren't really very many examples given (other than Zp) but I think I really just need a class with examples to really grasp the concept and the following theorems and stuff.
2. Quotient Groups and Homomorphisms
As I mentioned in the last reading, this was the part where I really really got confused with rings and such. I think it's a mix of having so many theorems in this chapter and the notations make me dizzy and confused. For example, in the third isomorphism theorem for groups when it uses the notation (G/N)/(K/N), like... really? Am I reading this right? Is this "(G mod N) mod (K mod N)"?!? What is that? That is so crazy!
I think that I understand the idea of simple groups, that doesn't really bother me. However I really struggled to follow the reading after the first paragraph of that section. I mostly think that the problem is that there aren't really very many examples given (other than Zp) but I think I really just need a class with examples to really grasp the concept and the following theorems and stuff.
2. Quotient Groups and Homomorphisms
As I mentioned in the last reading, this was the part where I really really got confused with rings and such. I think it's a mix of having so many theorems in this chapter and the notations make me dizzy and confused. For example, in the third isomorphism theorem for groups when it uses the notation (G/N)/(K/N), like... really? Am I reading this right? Is this "(G mod N) mod (K mod N)"?!? What is that? That is so crazy!
Sunday, March 27, 2011
7.7
1. What was the hardest part of the reading for you?
Oh man, I hate to be a Negative Nancy here, but this is where I started to get lost and feel frustrated when talking about rings and quotient rings and yadda yadda. I feel nervous about going into this chapter and what it'll do for the rest of the semester. The part that was difficult for me was the part that talked about the structure of the groups. For example, Theorem 3.37 kind of lost me, along with 7.38. I don't like that they changed notation of the center of the group during the proof -- who just does that?!? and I really just didn't follow that proof very well.
2. Why the confusion?
This was the question I asked myself at the very beginning of this section. The first thing this chapter says is, "Let N be a normal subgroup of a group G. Then G/N denotes the set of all right cosets of N in G." Okay, my question is this: Why state that N is a NORMAL subgroup then make a notation for the right cosets? Why can't G/N just be notation for the normal cosets? Is there going to be a different notation for left cosets? I just don't really understand this part of it. I know that if N is normal to G then the right and left cosets of N will be the same, but why make the distinction here when talking about a normal set? That is what I would like to know.
Oh man, I hate to be a Negative Nancy here, but this is where I started to get lost and feel frustrated when talking about rings and quotient rings and yadda yadda. I feel nervous about going into this chapter and what it'll do for the rest of the semester. The part that was difficult for me was the part that talked about the structure of the groups. For example, Theorem 3.37 kind of lost me, along with 7.38. I don't like that they changed notation of the center of the group during the proof -- who just does that?!? and I really just didn't follow that proof very well.
2. Why the confusion?
This was the question I asked myself at the very beginning of this section. The first thing this chapter says is, "Let N be a normal subgroup of a group G. Then G/N denotes the set of all right cosets of N in G." Okay, my question is this: Why state that N is a NORMAL subgroup then make a notation for the right cosets? Why can't G/N just be notation for the normal cosets? Is there going to be a different notation for left cosets? I just don't really understand this part of it. I know that if N is normal to G then the right and left cosets of N will be the same, but why make the distinction here when talking about a normal set? That is what I would like to know.
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