8.5, due April 1

Difficult: For theorem 8.26 why is true for all n except for 4.

Interesting: A_n is the product of 3 cycles.

Midterm 3, Due March 30

Most important: First Isomorphism Theorem, knowing how to prove something is a group and subgroup.

Questions to expect: Proving that something is a normal subgroup, finding kernels, definitions, examples.

Need work understanding: I struggle with coming up with examples for things and being able to explain why.  I also struggle with knowing where to start with proofs.

8.4, Due March 28

Difficult: Why a third isomorphism theorem? What is the point?

Interesting:  Definitions are all pretty much the same just different notations so that you can recognize that you are talking about a group.

Carl Simon: Complex Systems

Difficult: He was talking a lot about like sicknesses and stuff so it made things kind of hard to follow just because I don't know a whole lot about that stuff.  Also how can you measure/ write an equation for how contagious something is?

Interesting: Mathematics is an essential part of solving just about any problem. Everything is a system and you use calculus for just about anything things just need to be kept simple.

8.3, Due March 25

Difficult: I was noticing that |G|/|N|=x then G/H is isomorphic to Z_x.  Is this always true.

Interesting: That (Na)(Nb)=Nab

8.2, Due March 23

Diffucult: I am confused by the note where it says that a^(-1)Na=N does not mean that a^(-1)na=n for each n in N, all it means is that a^(-1)na=n_1 for some n_1 in N.

Interesting: It was interesting to see it mentioned that just because Na=aN doesn't mean that a commutes. That is something I was noticing in the homework.

8.1&8.2, Due March 21

Difficult: So every finite group is isomorphic to exactly one group?

Interesting:  It is interesting that you can take one subgroup from a group and it not be normal and then you can take another and it is normal. Also interesting that just because Na=aN doesn't make the group abelian, but it works the other way around.

8.1, Due March 18

Difficult: How you can have something modulo a subgroup.

Interesting:  The parallel of congruence with integers and rings to groups.

7.5, Due March 16

Difficult: I don't understand how this idea of cycles connect to what we have been doing.  Like why is it important?

Interesting: Every transposition is it's own inverse.

7.3, Due March 11

Difficult: There is so much notation that was used one way in earlier math classes and I feel like in this class we are using it in a completely different way.

Interesting: If something is a subset of a group then it is a subgroup if that group.

7.2, Due March 9

Difficult: So just to be clear ab doesn't mean multiplication we are just being lazy and not writing the *? So is a^2=aa, would that be multiplication or is that still what ever operation we are performing on the group? Also this is from the previous section but is order the number of element in a group or is it something else?

Interesting: It's interesting that an element can have order.

7.1, Due March 7

Difficult: In the third edition book what is the point of having a 7.1 and a 7.1 A.  They were essentially the same except 7.1 had theorems and 7.1 A went into more depth on the examples.  So some of the examples in 7.1 were hard to understand but looking at 7.1 A cleared things up.

Interesting: It's interesting that if you take the full ring of Z, R, Q, etc that it is not a group under multiplication but is under addition.  It's just interesting that when you change the operation it is no longer a group.

7.1, Due March 4

Difficult: I'm not quite understanding how example 5 is a representation of a group.  Is it just each rotation represents an element of a group? The *, can that be any operation of your choosing or is it always the composition?

Interesting: It is nice that a group is defined by one operation.

Midterm 2, Due March 2

Most important: Ideal, irreducible/reducible polynomials, showing that polynomials are fields, integral domains, or rings, First Isomorphism Theorem for rings.

Questions expected: Definitions, examples, state and prove a theorem.

Need to work on:  I am very unfamiliar with all of the theorems. There is so much to know and it's not easy stuff to just memorize.  I am not good at proving things.  I just don't know where everything comes from.