Final

I need to understand group theory much, much better. A couple of nights ago, I had a math dream where I was playing chess with my brother and each piece was labeled with an ordered pair. For some reason, each side had only fourteen pieces. For some reason, in my dream I knew that the ordered pairs were in Z_2 x Z_7 and that (1,1) would generate it since gcd(2,7)=1. It was really weird. A few nights later (so, like, two nights ago), I had another dream about group theory. It was crazy. In class, I say we go over the Sylow theorems and their implications.

Section 8.5

1. I understand the reasoning they used to prove certain properties and classify the groups, but that's merely an artifact of knowing how to apply a theorem. At this point, they're little more than memorized facts and not the concepts they should be (which is necessary for any useful application).

2. Quaternions, as you probably know, have a direct application to rotations in 3D space and avoid the problem known as "gimbal lock" when performing such transformations. They therefore have a direct application to 3D graphics, which I suspected groups would since they are often given geometric interpretations. If I'm not mistaken, quaternions do not entirely "live" in three spatial dimensions--they comprise four--like D_n does not live in two.

8.2

1. The Fundamental Theory of Finite Abelian Groups sounds important. Why do the people who write the textbooks always lay the groundwork out so that the proofs of the fundamental theorems is so anti-climactic? I can't say I followed Lemma 8.6, an "intricate" proof.

2. I'm swimming in deep water here. I knew Abstract Algebra would be a bit abstract, but I have no intuition with this kind of stuff.

8.1

1. I'm not sure I understand the implications of Theorem 8.1. In fact, this is all a bit hazy.

2. I realized at the end of a lot of problem sets, there is a grey box headed "Application" containing a reference to a real-world application of the topic at hand. Am I to assume if there is no such grey box, there is no real-world application? I'm sorry; to really have a better grasp on this topic, I'm going to need time to digest it.