Difficult- In the first isomorphism theorem I do not know what canonical means, but I don't know if that is important or not. In all of the theorems I don't follow what they mean exactly. But I do like the figures/diagrams because they are so much easier to understand!
Reflective- I think the 34.4 Lemma! The join term is cool and I think I understand it so gotta love it. The examples are so helpful in understanding what these theorems mean.
Tuesday, November 30, 2010
Sunday, November 28, 2010
#35 due Nov 28
Difficult- There are two million and one symbols in this section! And so I am really confused what the exact definition of a factor group is and what 14.1 theorem means. The 14.2 and 14.3 examples help me understand what a factor group is a little bit more. 14.10 figure is so helpful! I was getting really lost with all the symbols and which meant what. I think the lecture will help me better understand these.
Reflective- I am glad we keep using homomorphisms and isomorphisms since we have used them and know a lot about them. 14.13 theorem we used last section so I know that one, yay!
Reflective- I am glad we keep using homomorphisms and isomorphisms since we have used them and know a lot about them. 14.13 theorem we used last section so I know that one, yay!
Sunday, November 21, 2010
#34 due Nov 21
So yeah I don't really know what I am supposed to be blogging about/if I'm supposed to blog? But I am going to do the exam questions.
Which topics and theorems do you think are important out of those we have studied?
Geesh there are lots. 7.8 Theorem is a big one because because it has 4 parts to it and they are very useful in a lot of proofs and problems we do. LaGrange's Theorem and its corollaries are also very important and are used a lot. I like 6.10 Theorem.
What do you need to work on understanding better before the exam?
I am working on understanding what we did in class Friday. I also am working on proofs of Theorems because I always get lost in them. Just trying to go over everything again before the test!
Which topics and theorems do you think are important out of those we have studied?
Geesh there are lots. 7.8 Theorem is a big one because because it has 4 parts to it and they are very useful in a lot of proofs and problems we do. LaGrange's Theorem and its corollaries are also very important and are used a lot. I like 6.10 Theorem.
What do you need to work on understanding better before the exam?
I am working on understanding what we did in class Friday. I also am working on proofs of Theorems because I always get lost in them. Just trying to go over everything again before the test!
Thursday, November 18, 2010
#33 due Nov 18
Difficult- The evaluation homomorphism confuses me, but if I think of it loud and not use the symbols it makes sense to me. The phi symbol (at least I think that is what it is) with the numbers next to it like the trivial homomorphism, identity map, etc is confusing to me. I follow a lot of the examples but they are somewhat confusing.
Reflective- I like how the division algrorithm comes back into play again! I like 13.12 Theorem because it makes sense and is what you would assume. Also the historical note is interesting as it is neat to read about people who have discovered things, such as cosets being the same.
Reflective- I like how the division algrorithm comes back into play again! I like 13.12 Theorem because it makes sense and is what you would assume. Also the historical note is interesting as it is neat to read about people who have discovered things, such as cosets being the same.
Tuesday, November 16, 2010
#32 due Nov 16
Difficult- Omgsh I do not like this notation! I am trying to make sense of 11.2 Theorem and I think I get it but it is a little bit confusing. Understanding the definitions and theorems are taking me a while to follow because I keep having to stop and make sense of the notation. When I see the examples I understand more what the definitions and theorems are saying.
Reflective- I think that 11.5 Theorem is very interesting! It is neat how Zm x Zn is cyclic and isomorphic to Zmn if m and n are relatively prime. As always the historical note is interesting. In general I just like the abelian concept so I like that part of the section.
Reflective- I think that 11.5 Theorem is very interesting! It is neat how Zm x Zn is cyclic and isomorphic to Zmn if m and n are relatively prime. As always the historical note is interesting. In general I just like the abelian concept so I like that part of the section.
Sunday, November 14, 2010
#32 due Nov 14
Difficult- Well notation is always confusing to me but I think I understand the relation statements in 10.1 Theorem. However in 10.7 Example I do not follow what is going on! I think I don't follow the index definition and why it is the number of just the left cosets. Maybe because the right has the same number so they just picked one? I don't know if that's even important.
Reflective- Well we have talked about left and right cosets previously so I remember those and they make sense. The Theorem of LaGrange is logical and makes sense!
Reflective- Well we have talked about left and right cosets previously so I remember those and they make sense. The Theorem of LaGrange is logical and makes sense!
Thursday, November 11, 2010
#31 due Nov 11
Difficult- Okay I think I get these and I like them but I am lost in 9.10 example and how they are coming up with the permutations so I guess I need to figure out how they are multiplying cycles if they are not disjoint. I also do not understand transpositions, which I feel like should be easy. But I don't get 9.13 and 9.14 examples.
Reflective- It is interesting that permutation multiplication is not commutative but but multiplication of disjoint cycles is. It is interesting that 9.15 Theorem has two very different proofs and different methods.
Reflective- It is interesting that permutation multiplication is not commutative but but multiplication of disjoint cycles is. It is interesting that 9.15 Theorem has two very different proofs and different methods.
Tuesday, November 9, 2010
#30 due Nov 9
Difficult- The permutations seem pretty straight forward and easy to understand. I do not know the word lemma so that is new to me. Towards the end of the section, with Cayley's theorem all of the symbols and notation start to really confuse me. However I think I understand a little bit so hopefully this section will not be too bad!
Reflective- Of course I always like the historical notes! And this one is indeed very interesting about permutations in Islamic and Hebrew cultures and their alphabets. Also it is very cool that S3 is also the group of symmetries of an equilateral triangle.
Reflective- Of course I always like the historical notes! And this one is indeed very interesting about permutations in Islamic and Hebrew cultures and their alphabets. Also it is very cool that S3 is also the group of symmetries of an equilateral triangle.
Sunday, November 7, 2010
#29 due Nov 7
Difficult- Well as always the notation seems to be confusing!! Also I don't get 7.6 theorem in words I don't think but the proof seems to make sense and I can follow that so maybe I just need some explanation. At first I was confused by the example 7.7 of the Cayley graphs because they were two different shapes but then it says that doesn't matter so never mind about that but in figure 7.9 I guess I don't get the relationship between all of the vertices and how they are connected.
Reflective- The subgroup generated by ai of the intersection is like the same of the cyclic group generator so that is nice and consistent. The Cayley graphs are very interesting! I think I like them, well the ones that I get that is!
Reflective- The subgroup generated by ai of the intersection is like the same of the cyclic group generator so that is nice and consistent. The Cayley graphs are very interesting! I think I like them, well the ones that I get that is!
Thursday, November 4, 2010
#28 due November 4
Difficult- I was following along fine until I got to 6.10 Theorem. I think I understand it somewhat, they are proving it is an isomorphism and I get that. I don't understand what is meant by "describing cyclic groups UP TO an isomorphism" but then the prove the isomorphism. Example 6.17 seems confusing but the book says it is straightforward and I think I am making sense out of it now.
Reflective- It is interesting that the division algorithm, the greatest common divisors and relatively prime come back into play with cyclic groups; a lot of things repeat themselves in this class, this is good. I like how the section ends on page 65, because I was just thinking that that example looked complicated.
Reflective- It is interesting that the division algorithm, the greatest common divisors and relatively prime come back into play with cyclic groups; a lot of things repeat themselves in this class, this is good. I like how the section ends on page 65, because I was just thinking that that example looked complicated.
#27 due October 30th
Difficult- I think I am struggling with the identity in groups. On the top of page 52 there is a subgroup diagram but it does not make sense to me. I don't know if there is a difference between a^(-1) and e or if they are the same inverse for a group or subgroup.
Reflective- The cyclic subgroups can relate back to principal ideals because they were generated by one element, as these cyclic subgroups are which is neat.
Reflective- The cyclic subgroups can relate back to principal ideals because they were generated by one element, as these cyclic subgroups are which is neat.
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