Section 7.6 due Nov 10

Does left/right congruence affect other things? (For example, if there were a "divides" then things might get messy.) Can you further explain the definition of "normal" because they try to make a distinction at the top of p. 212 but I don't understand what they are getting at.

I was trying to decide if I like groups or rings better. I think I like groups better. You can define the group operation however you want as long as it follows the guidelines of a group.

Section 7.5 (part 1) due November 5

Can we go over Corollary 7.24 and review what right cosets are? And can you explain Lagrange's Theorem and the notation they use in the theorem?

The beginning of this is mostly review, so we have learned most of it before, applying it to rings and integers.

Section 7.4 due November 3

I don't understand the inner automorphism of G or Theorem 7.20.

It seems natural that this section is in here since there was a section on isomorphisms for rings.

Section 7.3 due Nov 1

I didn't understand the proof of Theorem 7.11. Many of the other theorems are obvious.

As in rings, it is helpful to know that G is a group so that you don't have to prove as many things, like associativity in subgroups.

Section 7.2 due 10/29

It says a^n=aaa...a (with n factors) but (ab)^n does not necessarily imply (a^n)(b^n). So does (ab)^n=(ab)(ab)(ab)(ab)...(ab) (with n factors of ab)? They said that we will use standard multiplicative notation, so shouldn't that mean that this is commutative?

The proof of part one of 7.5 was at first confusing but, upon further review, made a lot of sense and seemed extraordinarily creative. Then again, when are proofs not creative?

Section 7.1 (part 2) due 10/27

The section says that a nonzero ring is never a group because of the zero ring. Since multiplication is such a straightforward operation, why do they not make an exception in the group to say that for everything else, the group will work?

I thought the reflections were interesting because I haven't thought about reflections of shapes for a long time. I also think the theorems were somewhat interesting because they find some groups for us.

Section 7.1 (part 1) due Oct 25

With rings they say that if ab=ba then the ring is commutative. On p 163 they say that a group is abelian if it satisfies the commutativity axiom. Why is there a special word for it? Do they do it since multiplication commutativity is much more simple than what some operations can be? Can all groups' permuations be represented by an array?

Function composition is associative, but how often are composed functions commutative? Do they have to be simple, linear functions? That would be something interesting to explore a little bit. (Of course, they must not be too rare because there is a special name just for this type of group.)