#39 due Dec 7

Difficult- Section 37 seems to be pretty straight forward in theorems and examples. Sometimes I don't follow how the book comes up with certain things, but usually when I go back over it I can figure it out. 37.8 Lemma and its associated examples are the ones that cause the biggest challenge for me.


Reflective- I think 37.6 theorem is very interesting as well as 37.8 lemma.

#38 due Dec 5

Difficult- I have followed pretty much everything I think I am confused a little bit about how the conjugates work. I also think I need to review maximal subgroups since that is part of the definition of Sylow subgroups. But most of the rest seems to make sense it is just building on previous knowledge.

Reflective- Well I googled Cauchy to find out more about him and his theorem because I like mathematicians and he is cool. I also like this section a lot because it seems to have a large amount of repeated things like the normal subgroup and primes and order and everything.

#37 due Dec 2

Difficult- The Cartesian product of sets makes sense I am just really bad at remember exactly what the notation means. I am confused by 11.3 example because I don't see how that relates to anything new in the section because it seems like something we have already done. I do not understand 11.9 theorem very well. So I'm thinking I don't understand the section period. But then I see 11.10 example and that makes perfect sense. So I think it's notation and wording that is holding me up.


Reflective- Gotta love 11.5 Theorem because it goes back to isomorphisms and relatively prime. The historical note is interesting about how findings were finally turned into abstract theory.

#36 due Nov 30

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.

#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!

#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!

#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.

#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.

#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!

#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.

#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.

#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!

#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.

#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.

#26 due October 28th

Difficult- I think I am terrible at understanding this notation! It is kind of confusing. I don't think I understand the binary operation and what that means. I am definitely going to need to get this during the lecture! Also the group tables seem to be a little confusing. I think everything in this class confuses me.


Reflect- The historical note on page 38 is interesting because it relates the groups to other types of math like geometry and some of calc 2 or 3 I believe with LaGrange and permutations. I also like the historical note on page 39. They just add something cool so the text isn't as boring!

#24 due October 24th

Which topics and theorems do you think are important out of those we have studied?

We have done so much this section geesh! Well I feel like the division algorithm theorem is very important which is number 4.1.7, we used that one a lot. Then the factor theorem is also important. The extension field, K, theorem is important and also cool. And then we learn what an ideal is and we have used it a ton since then so that is important. In what we have done lately the isomorphism theorems have also been very important.

What do you need to work on understanding better before the exam?

I think I need to work on understanding notation, and how to understand what questions are asking me. I will need to review homework problems and examples to know how to better do that. I also need to work on how ideas and theorems relate to each other so when constructing proofs and such I can actually do them. Proofs are what kill me.

#23 due October 21st

I think I am confused on the blog schedule. It says for today to read section 26 but I think that's what we did in class yesterday! So I'm blogging on section 27.

Difficult- Okay I think I do not remember what a proper ideal is so I do not understand the definition of a maximal ideal of a ring. I will look that up though so I can understand this. I think I follow everything else. I am getting really worried for the test though. I also need to figure out what proper nontrivial ideals are.

Reflective- Abstract algebra is really cool the way that primes play into it. Prime ideals seem to be neat. If I can understand them correctly!

#22 due October 19t

Difficult- As you mention, the notation with all of the plus signs is confusing. It requires slowing down and thinking about it for it to make sense, which is fine it just isn't easy right away. I think I am still confused also on what the notation R/I means. Once the lecture turns into the isomorphism theorems with the quotient ring I get very confused. I don't know if maybe once I see the theorems worked out if I'll get it or not but it looks to be really hard.

Reflective- It is neat that the quotient rings were not intended to have anything to do with homomorphisms and isomorphisms yet they do. And it is also cool that we're bringing homomorphisms and isomorphisms back into lectures. The old stuff makes me feel like I at least know something  because this new stuff is confusing to me! I will definitely be in before the exam with questions :)

#20.5

(end of 20)

Difficult- I don't understand how you would do examples 6.1.13-6.1.15. I think I understand the paragraph about principle idea but then I do not get how you do problems with that. The rest of the lecture about congruence seems to make sense though. Cosets might confuse me a little but I think I can figure them out.


Reflective- I think this ideal stuff is cool because it ties together more stuff that we have done previously like subrings, congruence and mod stuff.

#20 due October 12th

Difficult- Okay the definition of the word ideal is confusing. Although I remember you starting to talk about it in class on Monday so it seems familiar and the examples make more sense. I think of all the symbols confuse me . Reflective- The idea of ideal is pretty cool and there are a lot of examples so I think it will be a big topic. This section doesn't seem too bad and I will probably understand it more after class.