7.9, Due April 1

I decided to write this blog about unicorns today instead of Abstract Algebra.. Enjoy!!!

APRIL FOOLS!!!

For reals though, here is the real blog information

Difficult: I felt that they new notation for the symmetric groups was really confusing and I feel like it isn't useful at all. What is the point of trying to write it this way, I have to think about it way more than I did when we just wrote it all out. So cycles are also confusing to me since I don't understand the notation. I got really confused about the notation because I felt like I understood some of it, but the notation on the middle of pg 231 didn't make any sense to me. So for the rest of the section, I was so confused because I couldn't make sense of cycles and the notation. So transpositions and alternating groups was really confusing to me.

Connections: The only connections I can really make with this section is that we have seen the symmetric groups before in other sections of chapter 7. I like them because they are good examples of non-abelian groups with different orders. However, they were kind of tedious to write them all out, so maybe once I can understand the new notation I will like them better and be able to understand how to have more applications of them.

7.8, Due March 30

Difficult: So in the first part of this section, the ideas are pretty parallel to those of rings. I understand homomorphisms, but the hardest part for me is finding the map f that is a homomorphism. It is easier for me to prove, but is there some kind of way to know what the map is? In all of the examples in the book, they choose their map, but this is something that is hard for me to do. Also, the idea of simple groups makes sense to me, but I'm just not sure how to go about testing if a group is simple. Is there a general way to do it? Also, I kind of got lost in the explanation of why simple groups are important. I didn't understand the point of them, I think I might have just gotten lost in the notation.

Connections: The First Isomorphism Theorem for groups is very similar to that for rings. I feel very confident in my ability to prove this theorem for rings, and I feel that I understand it pretty well. As a result, I think that I would be able to do this proof for groups as well. It follows the same general idea, but some of the notation is changed and some definitions would look different. Again, I am left wondering how they came up with the map for phi? How did they know to pick phi(Ka)=f(a)?

7.7, Due March 28

Difficult: Just like with quotient rings with ideals, I don't understand how to determine if something is a quotient ring, and how to find the elements of it. Also, at the end of the section it was talking about the structure of the quotient groups. I don't understand why this is important and when it would be useful. I think the hardest part of this section was just being able to find the quotient group and know why it was useful.

Connections: The theorems and ideas associated with quotient groups is pretty similar to those of quotient rings. I feel like these are a lot alike. However, as I said before, quotient rings were difficult for me to understand, so I also feel like quotient groups will be hard to understand. Also, I noticed that quotient groups were given a new name as well, factor groups. Why are they called that? Is there a property about them that has to do with factoring?

7.6 Part 2, Due March 25

So I forgot to post on the first part of section 7.6... but here is the second stuff
Difficult: For me, the most difficult thing about normal subgroups is just realizing that aN=Na, not that an=na. The It is hard for me to realize that the sets just have to be equal, not that every element has to commute. Another thing that I find difficult is just being able to find whether aNa^(-1) is a subset of N. I just have a hard time remembering how to find if something is a subset... its just something that I need to review.

Connections: Theorem 7.33 is similar to most of the congruence theorems that we have seen this semester about multiplying two things that are congruent modulo something. However, this one is a little different because the two things are congruent modulo a normal subgroup. This one seems to have a little bit more of a restriction. I also noticed that there wasn't a theorem for adding two things that are congruent modulo a normal subgroup. Is this not allowed?

Midterm #2 Post, Due March 21

For this section of the book, I think that Lagrange's Theorem, Cayley's Theorem, and the first Isomorphism theorem are pretty important. Usually when theorems are named they are a big deal. I also think that other theorems will be really important too just because it is useful information to know in helping prove other things. For example, I feel like the theorems that give easier or faster ways to determine if something is a sub ring, or subgroup are really helpful. Also, the theorems that just are simple things to know to be able to use in other proofs, for example inverses are unique, or cancellation holds, or the identity is unique.

I expect to see questions about proving if something is an ideal, or a group. But I could also see things like find a maximal ideal, or prove something about the kernel, or show that this group is cyclic. I think there will also be questions about finding the cosets of an ideal or group. Also, like the last exam, I think there will be questions asking for examples of groups, ideals, subgroups, etc.

Something that I would like to discuss before the exam is that I really want to go over the main differences of the main topics we have covered in this unit. Another thing that I really struggle with is maximal ideals, cosets, and cyclic-generated groups. So I would really like to see examples of these things so that maybe some of my confusion could be cleared up.

7.5 Part 2, Due March 18

Difficult: For me, I have found that finding and proving groups are isomorphic is rather difficult. I feel like the only way to do this is to create some map, f, and then write out the multiplication/operation tables to show that they are the isomorphic. However, I feel like this could be an insufficient way of proving this, but I have no clue how to show this any other way. So from this section, I still just don't know how to show something is isomorphic with out multiplication tables. And beyond this section, I realized that I am just super confused about cyclic groups and how to know if a group is cyclic and how to find the order of a group as well as its elements. This is probably something I should know, but as I was trying to do the homework for the last section, I found that these concepts were causing me grief.

Connections: I guess the only connections I can make with this section is that the integers modulo a number can be isomorphic to other things, ie rings, groups, etc. So this was something I felt like I understood, however, as I mentioned above, I still find it really hard to find the isomorphic map and how to find if that is indeed a homomorphism. However, I still feel like the definition of an isomorphism is something I have seen before, but I find that I have a harder time showing isomorphisms for groups.

Vitaly Bergelson, Make-Up blog

Difficult: In his talk, Vitaly Bergelson talked about Poincare's Recurrence Theorem which had something to do with volume preserving transformations. I had absolutely no idea of what he was talking about. I tried to understand it, but I think I might have just gotten lost in the notation. He asked if we had any questions, but I was so confused so I didn't even know what to ask. He also talked about upper density functions. I didn't even know to what this is and how to find them.

Connections: The only connections I could find with this talk was about finding volume. This reminded me of multivariable calculus since I found volume of the three dimensional figures. Also, I felt like in linear algebra we talked about n-dimensional stuff, but I'm not sure exactly what we did with it. I couldn't really find many connections with this since I felt so confused.

7.5 Part 1, Due on March 16

Difficult: I think that the definition of right cosets is kind of confusing. However, it reminds me of some things we did with ideals. I think that the notation is just confusing to me. Also, I don't really understand what makes it a right cosets... what is a left coset? Thus, I didn't really understand theorem 7.25 because I don't understand even what the union of all the cosets would be.

Connections: In this section, I think I understand what it means for two elements of a group to be congruent modulo a subgroup. It is pretty similar to how it has been defined previously with rings, and ideals. I also understand how congruence modulo a subgroup is also an equivalence relation just as congruence has been in previous sections. Like I said before, I think I understood congruence, but the right coset thing is still a little weird to me, and also I am left wondering why do we care about the right coset anyway? What is so important about the right coset??

7.3, Due March 11

Difficult: I was doing pretty well with this section until I got to the section on cyclic groups. I did not understand this section of the reading at all. The notation was also kind of confusing to me. I just didn't understand what <a> means. The set of all the powers of "a" confused me because I thought that one of the powers of a should eventually equal the identity. Is this wrong? Or is that only when there is finite order? I just got confused by what cyclic groups really are I guess.

Connections: I like the idea of subgroups though. They seem pretty simple to me, especially after dealing with subrings. The theorems that go with subgroups, at least in the beginning of the section, seem to be pretty simple. The theorems seem pretty straight forward to compute to show whether a subset of a group is a subgroup. These theorems were easy for me to follow so I should be able to use them in the proofs for this section.

7.2, Due on March 9

Difficult: I had a hard time understanding the order of an element, and how to find it. The more difficult part for me though, was to find out whether or not an element has distinct powers by looking at the elements in a group. The example on the top of page 177 was difficult for me. I understood what it was talking about, but I don't think I would know to how figure out the powers on the exponents on my own. I also thought it was difficult to find out if an element has finite or infinite order.

Connections: I like in this section that some of the properties of groups are properties that I have used in other parts of mathematics. For example, the inverse of ab is b inverse times a inverse. This is similar to concepts used in linear algebra. Also, some of the properties of exponents hold as well. Since I have been using these concepts for a while, I am more comfortable using them and applying them to new ideas such as groups.

7.1 Part 2, Due March 7

Difficult: One thing that I thought was difficult about this section was the dihedral groups. I felt like I understood them, and I understood why they were groups since you can think about the group as a composition of functions. However, the thing I thought that was the most difficult part was trying to figure out the symmetries of the figures. Especially with the equilateral triangle. If I was asked to find the symmetries of a dihedral group of degree n, I wouldn't know how many symmetries there are supposed to be. Is there some way to know how many symmetries there should be in a dihedral group of degree n?

Connections: I like that groups seem to be really similar to rings, but even more simple. I also like that we can look at familiar rings and notice whether or not they can be groups. It helps me get more used to groups since we are looking at rings and other things that we have already considered in this course. At the end of the section, they talked about different groups that can be formed from things that we already have worked with. I like that groups are pretty simple and easy to work with, just like rings again.

7.1 Part 1, Due March 4

Difficult: For me, this section was pretty easy to understand, especially after making it through Chapter 6. However, I still have some questions. I was wondering what binary operations are allowed for groups. I guess I just am confused how to know how this operation is defined, or for that matter, how to define the operation. It seems like groups are a lot like rings, but I'm not quite sure what makes them different from rings. Is it just that only one operation is defined?

Connection: I like that in this section we are going back to something that seems more computational, and therefore a little less abstract. I am probably wrong though. I did like in this section how it talked about the composition of permutations. I like that this is very easy for me to follow and understand because I have been doing composition of functions for quite some time. I am hoping that groups will be a little more straightforward and simple to understand.

6.3, Due March 2

Difficult: I had a hard time understanding what maximal ideals are. I think that maybe it is notation that is confusing me, but I just couldn't quite understand what it means for an ideal to be maximal. As I was reading the definition, I also realized that I'm not really sure what it means for two ideals to be equal. So I don't know how to go about proving something is maximal. Mostly because I'm not sure how to prove that two ideals are equal.

Connections: I like dealing with prime numbers, or concepts having to do with primes. I think that things that are prime have very nice properties and are really nice to work with. Some of the examples were pretty simple, so I think for the most part I understood them. However, sometimes I think I understand things and I try to do the homework and I'm not quite sure how to apply things. So hopefully applying the concepts of prime ideals will be something I will be able to work with, and I will be able to understand how to use information about ideals being prime help me out in some of the proofs.