7.9, Due April 1

Difficult:

I don't understand why we would factor permutations down. Why would we care to write the identity permutation as (12)(12)? Why is every transposition it's own inverse? Does that mean every element has order 2?

Reflective:

I think it's cool that we are cutting down on the notation for symmetric groups, I'm just worried that I will get confused when I try to follow the composition through.

7.8, Due March 30

Difficult:

I think I am still confused on cyclic groups and generators of groups. When they say does that mean the set of elements generated by e? Wouldn't that be the whole group? Or is it just e because when you add(mult) e, you just keep getting e over and over?

Reflective:

I thought it was interesting that instead of the kernel being the set of things that are mapped to zero, it's the set of things that are mapped to the identity. In an additive group like Z, it would be the same thing, but in say the multiplicative group of units of Z8 it would be 1.

7.7, Due March 28

Difficult:

I'm not sure I understand how thm 7.36 works in regards to the order of G/N. I think we established a normal subgroup of order 4 in D4 today. So D4/N would have order 2? How do I know what 2 elements are in there?

Reflective:

I thought it was cool how they related the quotient ring R/I to quotient groups and how R/I was really just the additive group of the quotient ring R/I.

7.6 Part 2, Due March 25

Difficult:

It was confusing when they said that you can conclude from Na=aN is that if na is in Na then na is also an element of aN, so that there is t in N such that an=at in aN. Does that mean that n=t?

Reflective:

I thought it was cool how they connected normal groups to abelian groups and centers of groups, and it was good that they made sure to mention that it doesn't imply that the normal subgroup N is commutative.

7.6 Part 1, Due March 23

Difficult:

I got very confused when they were talking about applying other definitions of congruence to groups. It was especially confusing when they were trying to explain/translate congruence in rings under addition with congruence in groups under multiplication.

Reflective:

I thought it was funny how they had to develop a whole new congruence for the left side, and go back and rename what we called "congruence" to be "right congruence".

Exam Questions, Due March 21

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

Obviously any theorems with names (1st Isomorphism Thm, Lagrange's Thm) are important. It seems that the basic ideas on this test will be ideals, kernals, groups, subgroups, and centers of groups.

What kinds of questions do you expect to see on the exam?

I kind of answered that above. I'd hope it would be like the last test, with some definitions, some questions asking for examples, a proof, and an application of some theorems to something we haven't worked specifically on.

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

I am not sure yet... I think I get confused with ideals and cosets and subgroups, and how they are different. I also find the order of an element confusing, and what it means for a group to be generated by two elements.

7.5 part 2, Due March 18

Difficult:

I had a hard time following the proof of thm 7.29 at first, when they were talking about how they knew ab=c. The other thing that I'm still confused about is that the theorems use "either or" language. Does that mean it doesn't matter which one we choose?

Reflective:

I think thm 7.28 would have been a lot more confusing if we hadn't talked about Lagrange's Theorem in class. I liked how Dr. Doud put it - if we have a group of order 24 and 6 distinct cosets, how many elements are in each coset? Obviously for prime numbers, it's either 1 or p.

7.5 part 1, Due March 16

Difficult:

I think the right coset idea is confusing for me. I know it came up before, but we never really addressed it so I haven't thought about it at all, but now it's in some of the theorems. I'm hoping we cover this in class so that I don't have to go searching through the book for it.

Reflective:

I thought LaGrange's theorem was an interesting idea but I'm not sure how knowing that will actually help me with anything. How is it significant that the order of a subgroup will divide the order of a group?

7.4, Due Pi Day

Difficult:

I still have a hard time understanding what an isomorphism is... I also think it might be hard to remember how the homomorphism part changes based on the 2 different operations that G and H could have.

Reflective:

For rings, there were 2 parts to a homomorphism, but for groups there is only 1. I suppose that's because there is only 1 operation on a group, it could be addition or multiplication (or something else that we define).

7.3, Due March 11

Difficult:

I got a bit confused after thm 7.13 when they started talking about cyclic groups. One thing they said was that every cyclic group is abelian, but I don't see why that would be the case.

Reflective:

Subgroups of groups follow along with the same idea as subrings of rings. It's a subgroup/subring if it is itself a group/ring under the same operations of the bigger group/ring. I also thought it was interesting that we only needed to check 2 of the 4 axioms because the identity comes naturally from those 2, and the associativity will be there.

7.2, Due March 9

Difficult:

I think the hardest part for me to follow was Thm 7.8 when they talked about order and finite order. I kept needing to refer back to the definition of "finite" order and try to make sense of statement 2. I'm not sure I understand why it happens the way it does.

Reflective:

I felt that most of the theorems presented in this section were easy to follow, and were just basic ideas that I was used to seeing, such as that a^m*a^n = a^(m+n). It was nice seeing some of those easy things again :)

7.1 Part 2, Due March 7

Difficult:

The previous homework, and the reading of the rest of this section didn't seem too hard to understand. The only thing that was frustrating was that they said to assume that unless said otherwise, we are thinking of the group operation as addition. Yet in the homework problems, they almost all dealt with multiplication.

Reflective:

I thought it was cool that we basically came up with Thm 7.1 in class. It was easy to see the relationship between groups and rings and try to figure out whether all rings are also groups, and so on.

7.1 Part 1, Due March 4

Difficult:

It seems strange to me to list the permutation the way they were doing it in the first example, but that may just be due to the fact that to me, it looks like matrices and would be confusing. I suppose It's basically set up like an x,y chart.

Reflective:

I'm so glad to be done with ring and field stuff. I'm hoping it's true what everyone has been saying, that group theory is much easier than ring theory. So far this section hasn't had any theorems or proofs, just definitions. There are a lot less axioms for a group than there was for a ring.