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.)

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!

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.

7.6 Part 2

1. What was the hardest part of the reading for you?
I know I'm naughty because I haven't blogged or done my homework in a couple of days. Call it my spring break. That being said, tonight's reading was pretty easy for me, mostly because we talked about the whole chapter in class last time... but the thing that I found the most 'interesting' or something that I think is kind of tricky is the stuff in bold letters on the top of page 212: "The condition aN=Na does NOT imply that na=an for every n in N."

2. Theorem 7.34
While proving this in class yesterday, the question was asked, "Why do we need all 5 of these pieces of the theorem? Aren't 2 and 3 kind of redundant?" I thought the answer to that question was kind of interesting, and that was "Though 4 and 5 are stronger statements than 2 and 3, it's easier to prove a weaker statement, and from the weaker statement it becomes easier to prove the stronger statement, and that was evident from the proof of 7.34.

7.5 Part 2

1. What was the hardest part of the reading for you?
I am just feeling a tad confused with the way that all of the theorems are coming together, mostly because I don't remember a lot of them and I'm struggling to keep them straight. It's evident that everything is coming to a head in this section. As I read through the proofs of these theorems, I know I could (and should) spend more time reviewing and looking at the different theorems that are used here, but I didn't... so that is a good part of my confusion.

2. I love sudoku...
The one thing that I did enjoy about these proofs was how similar they were to sudoku, like we have discussed. I like that you start with knowing that something is cyclic and looking at it's order and then filling in the table. The only problem is, it can be tricky to keep these theorems straight! Like I said already...

7.5

1. What was the hardest part of the reading for you?
Well, now I'm kind of embarrassed and confused. When looking at D4 (like in the example on page 200) under function composition, I realized that I am probably crazy, but h*d would have you look at h on the left side of a chart and over to d on the top of the chart. But, according to this, you look at d along the side then to h on the top. Maybe I've been writing my tables wrong, but that is nuts and confusing. The other thing that confused me is how h was congruent to d mod that subset of rotations. What?!? That made no sense to me. It made sense, I suppose, that congruence classes here are ab^-1 in K... but... what? I also hated that they left me confused then changed concepts.

2. Lagrange's Theorem
Um.... what is this malarky? It's more confusing notation and it's a bunch of confusing stuff, and I feel sad and confused.  Pretty much the only thing that I really took out of that section is with these distinct right cosets, where every element of a group belongs to one and only one coset, then of course by adding the number of elements of the right cosets will be the same as the number of elements of a group... But that was pretty much the only thing I took out of that section.

7.4

1. What was the most difficult part of the reading for you?
The thing that I'm struggling with is, if there are two groups who work under two different operations, I feel like I'm going to struggle how to use which operation. For example, I struggled to follow the second example on page 192 where were were looking at the additive group of real numbers and it's mapping to the multiplicative group of positive real numbers. This seems confusing and no fun to me.

2. Theorems
It's probably because I haven't done the homework for 7.3 yet, but the different theorems discussed and proven in 7.4 kind of through me for a loop. Again, it's probably because I haven't done the exercises for the things discussed in the theorem (like cyclic groups and such) but these theorems kind of make my head feel mushy.

7.3

1. What was the most difficult part of the reading for you?
I think the hardest part of the reading for me today was understanding "Generators of a Group." I'm not convinced that I'm convinced all the way yet about this concept. Well, I see how, for example {7,11} generate the group U15, like in the example.... but.... I'm not sure that I know where that comes from or why that matters. Well, it isn't even that I need to know why it matters, but I"m confused by the explanation of where this "generator" comes from.

2. Cyclic Subgroups Generated By "a"
I decided that I like this to be named "cyclic" because it's cyclical!!!!! Well, I guess if it's of infinite order it isn't cyclical, but I like the idea of it being of finite order and being cyclical. I know this is a lame thing to talk about, but that's how I feel. However, I would like to say that while I understand the change of notation for additive groups, I don't like it. I wish it would just be consistant notation for groups and that would make me happy.

7.2

1. What was the hardest part of the reading for you?
I'm honestly kind of confused by some notation here. Mostly with corollary 7.9 on the part that talks about how "that is, |a|=<|c|..." I'm confused because doesn't |a| mean the size of the group? Like.... shouldn't this be |G| not |a|? That is what I"m most confused about right now.

2. Embarrassed
Well, that was really the hardest thing for me, but as I was looking around for what I could write here, I thought "Find the earliest use of |a| and see if you can figure it out." Well, then I ran into the part where it talks about it and now I'm embarrassed that I didn't see it the first time through. It looks as though |a| is the order of a, which is fine, but I just get so darn confused by the crazy use of notation all over the place! Oh well, I guess I'm just a goober.

7.1 Part 2

1. What was the most difficult part of the reading for you?
I guess that the hardest part of the reading for me was the same thing that was difficult for me in the homework. With these complex operations and definitions of a group, it is difficult for me to figure out if each element has an inverse. I didn't really have a hard time finding the identities of the different groups in number 4, but figuring out the inverses and if they were in the group kind of tricked my head a little, and I think that is the difficult part for me.

2. Shape Symmetry
I really liked the examples they had where they reflected, rotated, etc. the different shapes. I think the reason this really sunk in with me is because I'm currently in Survey of Geometry where we are really looking at those symmetries and such, so having a visual between the shapes and "functions" moving the corners (or whatever) was very helpful for me to understand what is going on in groups (mostly how after you  look at the symmetries the you have to end up with the same shape, which really painted a picture for me about being closed under the operation). So.... the moral of the story is that it made me happy.  

7.1

(First of all I would like to point out how stupid I am because I totally read 6.3 the other night, and spaced blogging about it. Dumb dumb dumb....)

1. What was the most difficult part of the reading for you?
Okay, I think I am understanding this idea... kind of... but I am really struggling with something pretty basic and silly: What the heck are the elements of this group? Like, in the main example they were using of functions mapping 1,2,3 to 1,2,3, so are the elements 1,2,3? or are the elements the mappings? I understand that the one operation was the "of" function, but what are the elements? If I had to guess, I'd guess that the elements are the mappings, but if that is the case, I think I'm mostly confused about what groups (in general) are or look like.

2. An 'axiom' summary of groups:
i) Closure under operation
ii) Operationally associative
iii) Identity element
iv) Inverse element for EACH element (whatever those are) of your Group.