Section 8.3 Due June 14th

1. Wow there are so many proofs to remember. I hope that I can use then sufficiently on the test. I feel like the middle sections of chapter 7 and this section are the very hardest out of everything. I liked the cycles the best.
2. The properties of the order of groups are so interesting. Who is this Sylow person?

Sections 8.1-8.2 Due June 8th

1. I didn't understand how on the bottom of page 245 proving that the elements of Z_6 can be written as a sum of elements of M and N in one and only way proved that they were normal subgroups. The proofs in section 8.2 were very long and complicated, so I wonder whether I will be able to use them sufficiently in homework problems. I'm not understanding how we go about finding the elementary and invariant divisors.
2. It was interesting how prime power orders have such an effect on the groups and subgroups.

Sections 7.9-7.10 Due May 6

1. I don't understand how the product of the cycle groups works.I didn't understand the proof of lemma 7.54, especially when all the sudden they had 12k. I didn't see where that came from.
2. I really liked section 7.9 and thought it was kind of fun, but it seems a little pointless and like something that someone just made up and gave a name. I see no relevance in all of the connections that were made between cycles and the group S_n. They were interesting theorems, but seemed useless.

Sections 7.6-7.8 Due June 4

1. Wow the whole section on quotient groups and homomorphisms was confusing. I feel like we just keep building and building upon the original group idea and it's getting so complex that if I'm confused about a little thing in one section, then it's really detrimental in the next. Especially since we're moving so fast.
2. It's interesting that every subgroup of an abelian group is normal. I really liked quotient rings, but quotient groups are a lot more abstract.
   

Section 7.5 Due June 2

1. When it changes from {b in G such that ba^-1 in K}, to {b in G such that b = ka} it makes it hard to grasp what the set actually is talking about. The new notation H + 2 for <3> doesn't make sense. Theorem 7.25 was hard to follow.
2. I thought that Thm. 7.29 about how every group of order 4 is isomorphic to Z_4 or Z_2 X Z_2 was interesting. I don't like how instead of giving full proofs they say just to copy the proofs in previous chapters "with obvious notation changes".

Section 7.3-7.4 Due May 28

1. I'm having a hard time understanding cyclics. And I can't figure out what an automorphism would be with groups like S_3 and D_4. It started getting hazy when it was talking about the image and e_G and e_H, so if we could go over that a lot in class that'd be great.
2. It's interesting that every single infinite cyclic group or order n is isomorphic to Z_n. It was interesting how similar the properties of proofs for subgroups were to subrings.

Sections 7.1-7.2 Due May 26

1. Does part 2 of theorem 7.5 imply that G is commutative, and hence an abelian group? When it started talking about order of an element, I didn't understand what e was. When it says "if a has a finite order n" is n the number of times you have to multiply a to get e? and therefore is n not unique?
2. Groups seem to follow traditional rules with exponentials, and overall still seem fairly simple. It's interesting that we didn't talk about order in rings, but only in groups. The corollary on the top of page 178 was interesting because I wouldn't have thought that such a correlation between the orders of elements would exist.