8.1, due on April 1

Difficult: So this section wasn't too bad. I would just like to see some more examples of groups that are direct products of some normal subgroups and go throught the material again.

Reflective: This makes groups seem more like numbers since they have factors. It would be kind of like the number 12 is a direct product of 1,2,3,4,6,12 where 1,2,3,4,6,12 are all normal subgroups. But it is a little different.

7.10, due on March 30

Difficult: So how in the proof for thm 7.52 does it come up with the cases, why doesn't 4 work, and where did it use Lemma 7.53?And in Cor 7.55, how does that proof go? This is tricky stuff!

Reflective: So A(sub)n is usually simple. I guess that is important so it is isomorphic to the integers mod p.

7.9, due on March 27

Difficult: Maybe it's cause it's late and I'm tired, but this is tricky to me. I'm still trying to get how (243) (1243) = (1423).

Reflective: This acts a lot like functions in a lot of ways, but is totally unlike anything I've ever done in a lot of ways.

I need to work on:

1. computations in D4
2. Right congruence
3. Left congruence
4. Cosets and what they "look like"
5. Quotient groups and what they "look like"
6. Knowing content of Thm 7.8

Questions for class:

1. a sample problem of application (kind of like #5 on last test. You know how we have to apply a thm to something we've never seen before. I'm not asking for the actual problem that will be on the test, but maybe something similar)
2. examples of quotient groups
3. examples of cosets

7.8, due on March 23

Difficult: The proof for part 3 on Thm 7.44 is still a little confusing. Also, factors in the integers you multiply to get that number. Do you compose composition factors to get a group?

Reflective: So multiplication factors are to integers as composition factors are to groups.

7.7, due on March 20

Difficult: It's a little tough to see what every quotient group "looks like." I'd like to go over the first example, D/N = Nr and Nv, again.

Reflective: This is a bit easier than quotient rings, but maybe that's because it sort of a second time through.

7.6, due on March 18

Difficult: I would like to cover both left and right congruence again and cover the definition of normal with examples just so it all sinks in.

Reflective: This is definitely different than rings.

7.5 part II, due on March 16

Note: So I read all of 7.5 for last blog for some reason so my old blog still stands.

Difficult: Corollary 7.27 makes sense when you take the time to take it apart. But I don't feel like I have the overall intuitive idea of what is going on because I'm still trying to memorize Theorem 7.8 and 7.14 and 7.18.

Reflective: Congruence is definitely a little different in groups than in rings and integers.

Addition: I will add that while the last 2 theorems make sense, they are long proofs! So it's a little hard to remember what you were even proving by the time you get to the end of the proof.

7.5 part I, due on March 13

Difficult: Corollary 7.27 makes sense when you take the time to take it apart. But I don't feel like I have the overall intuitive idea of what is going on because I'm still trying to memorize Theorem 7.8 and 7.14 and 7.18.

Reflective: Congruence is definitely a little different in groups than in rings and integers.

7.4, due on March 11

Difficult: I had a hard time following the proof for Theorem 7.20. It would be good to do that in class mayber.

Reflective: I noticed that Theorem 7.19 is a lot like Theorem 3.12. That's cool because it has a lot of nice properties.

7.3, due on March 9

Difficult: This is sort of a lot of new information. A little hard to take it all in. I'd like to see lots of cyclic subgroups and generated subgroups examples.

Reflective: Again, lots of information. The generated subgroups remind me of rings generated by two elements, like the polynomials with even constant terms generated by (x, 2). The cyclic supgroups seem completely new to me.

7.2, due on March 6

Difficult: The proofs of theorems 7.8 and 7.9 are a bit tricky. Also, the theorems themselves make perfect sense now, but I can them getting tricky if they are generalized in future sections.

Reflective: (ab) = b-1a-1 reminds me of something I think from linear algebra, but I can't seem to remember what.

7.1 part II, due on March 4

Difficult: One question. Why is a nonzero group under multiplication never a group?

Reflective: This section is nice because it gives us a lot of nice properties in groups that we are used to in rings. The whole group of shapes thing is kinda wierd, but cool. It reminds be of a game we played in elementary where we had to figure out what a piece of paper looked liked afer being folded, or rotated, or had a hole cut out and then was unfolded. Never thought of it as a function before though.