7.9, due April 1

This section was pretty cool.  I found the new notation for permutations kinda fascinating.  It was really easy for me to understand.  The alternating groups were a littel tricky.  I kept wanting to do the operation from left to right but I'm pretty sure I'm supposed to do them right to left because it's still function composition.  I thought it was especially cool that you could write a single permutation as the product of two cycles.  That was really intriguing.

7.8, due March 30

Does everything in groups have a connection to something in rings?  I get so confused when they have the same notation or even names (kernel!).  I had a little bit of a hard time wrapping my head around all this in rings and now it's like I have to unwrap it and wrap it up again but this time in groups.  Another thing I noticed was that there is no Second Isomorphic Theorem For Groups.  I got a little caught up wondering why it wasn't included.  It must exist if there is a third one but was it just not important enough to make the book?  As far as the actual math goes, it all seemed straight forward until I got to the composite factors of G.  I don't even know where to start with those.

7.7, due March 28

Quotient groups, eh?  This section brought back old memories of the complexities of quotient rings.  It felt a little bit like the same song but the second verse.  When I read the first theorem, I thought to myself "isn't this something we already know?" and was able to follow the proof of it very easily.  The other theorems however were a little less obvious so me but they did make sense.  Their proofs weren't as straightforward to me but I thought I was able to get the gist of them.

7.6 (part 2), due March 25

Dr. Doud kinda spoiled the surprise for the rest of this section in class on Wednesday.  The thought that I struggled with throughout class was that just because Na=aN does not mean that na=an for some n in N.  When we went through the section on Wednesday, everything seemed to make sense.  One thing that sparked my curiosity was the theorem with five parts.  The way that I understood them is that all together they seemed to be saying that aNa^-1 = a^-1Na.  Again I got caught up thinking that this was true for all n in N.

7.6 (part 1), due March 23

The title of this section is "Normal Subgroups".  I thought these subgroups were kind of weird.  Does left congruent and left coset mean that you perform the group operation aka "multiply" on the left?  The theorems and properties of these subgroups entered my mind as the same properties of the other not normal subgroups we talked about earlier in the chapter.  I found myself wondering if these properties were the same or if there was a difference that I was somehow missing?

EXAM 2, due March 21

Oh boy! As I have started getting ready for this exam, I have started to get a little worried.  I get worried about a lot of things though, so I shouldn't be surprised.  I think the thing that has me worried the most are the functions composed of a ring modulo some ideal.  I had a little bit of a hard time understanding that when we were studying that chapter in class.  I hope that the format of this test is very similar to the format of the first exam.  I expect to see questions about groups and questions about rings of functions and ideals. 

7.5 (part 2), due March 18

The rest of this section was pretty straightforward.  I thought it was very interesting that we could tell by the order of a group which group it was isomorphic to.  I got a little lost in the proof of the theorem that said that a group of order 6 was isomorphic either to Z_6 or S_3.  The origination of ba=(a^2)b and b^2=e was not really clear, but I got really confused when compared the correspondence to S_3. I didn't really follow that.  How did they know what each element of the group mapped to in S_3?

7.5 (part 1), due March 16

This section didn't make much sense to me.  I understood the idea and the concept that I was supposed to be understanding but I just couldn't make sense of it.  I understood the definition of congruence in groups  but the section about Lagrange's Theorem went over my head, just barely out of reach.  The theorem just before Lagrange's was kinda confusing and that confusion carried over a little bit into Lagrange's Theorem too.

7.4, due March 14

HAPPY PI DAY!!!  So the way I understand it, an isomorphism of groups is an automorphism but it has all the properties of a isomorphism of rings but instead of multiplication and addition, only the group operation has to be conserved.  A homomorphism of groups of permutations is called a representation.  SO far it all seems to be making sense.  The only hang up is the symbol for isomorphism/automorphism.  The only way to tell if it is representing an isomorphism or an automorphism.  I guess that that will have to be specified in the context of the theorem or problem which makes sense.

Oh and have a great time in Italy!

7.3, due on March 11

Subgroups, centers, and cycles!  Oh my!  I feel like this section was more difficult because there was so much information in it.  The individual parts weren't too bad but all together it seemed like there was too much to chew.  The part on generators was a little hard to follow because I felt like we had already talked about the elements that generate a group, but we had actually only discussed the elements that generate a cyclic group.  That first little taste of confusion lasted through the whole theorem, so I got a little lost.  Most of the theorems seemed to be straightforward maybe because they answered some of the questions that came to my own mind.  They're proofs however weren't so straightforward.  I found myself getting hung up on notation again and trying to read the symbols instead of just interpreting their proper meanings.  I've got to stop doing that!

7.2, due March 9

This section felt very straightforward and seemed to be almost logical.  I did get caught up in the finite order of an element of a group.  I read through the first few examples thinking that they were talking about the whole group, which wasn't making much sense.   Then I went back and read them more carefully and noticed that they were talking about specific elements.  This lead me to wonder if only the groups with finite order had element that had finite order as well.  Maybe this was answered in the reading and I just missed it somehow.

7.1 (part 2), due March 7

The rest of 7.1 wasn't too bad.  I recognized a few examples from our class discussion on Friday.  I thought the dihedral groups were pretty neat.  I thought it was really interesting how abstract algebra could apply to geometry.  Some of the examples towards the end of the chapter were hard to follow the first time around but after looking at them again, I was able to get the jist of them.  I'm still a little fuzzy on the general linear groups but that will probably be ironed out in class.

7.1 (part 1), due March 4

Tis a gift to be simple! I really enjoyed this section.  It was easy to understand and it was fairly straight forward.  The only question that crossed my mind as I read was "Does * always denote the composition of two functions or is that just one interpretation of that symbol?"  From the text it seemed to just be one interpretation of the symbol which then lead me to wonder what some other interpretations might be.  I was only a little confused about waht exactly a symmetric group was.  The definition was buried in the example and I wasn't sure what properties made the group symmetric.

6.3, due March 2

Ok, so this section wasn't all that bad.   I was able to follow the bit about primes fairly well.  The part about maximals is still a little rough though.  I understand the definition...I think.  It was the proof that kinda threw me.  I started getting mixed up in the notation and then I had hard time keeping all the rings and stuff straight in my mind.  I feel like there should be a road map or a picture or something that helps us visual learners like myself keep it all organized in our minds.