7.8

1. Composition factors look interesting and deep. I wouldn't mind another explanation of them.

2. I think the real insight here is in the definitions of kernel, the homomorphic mapping from one group to another, the standardization of these theorems and definitions. These are given freely and then the proofs are relegated to previous results in analogous ring constructs. I suppose this is a good thing, since, while the definitions and homomorphisms seem trivial, it is easy to see these things after the fact.

7.7

1. The most difficult part was... I don't know. I don't know what I don't know.

2. Quotient groups seem like the natural thing to construct after all that we saw with quotient rings.

7.5

1. Whoa. Who saw this coming? Congruence for groups? Considering we have (supposedly) been exposed to the proofs twice before, this is sitting pretty okay with me. Lagrange's Theorem seems important.

2. Okay, I know I always ask this question (mostly rhetorically, and never accusatory): what can this be used for? I still see applications with computer graphics and digital interfaces, but that's my bent. I suppose the von Neumann computer model is one avenue where the abstract meets the actual, but I'm blind to other applications. I guess I'll give you a rundown of what I'm thinking since this is the "making connections" portion of the blog: I read a question on Slashdot a few weeks ago that was essentially, "How big is the intersection between philosophy and computer science?" and the general answer was, "Not very." This struck me as odd, because it seems to me that problem definition/specification/nature borders on the philosophical if it isn't outright. In my experience, I see a lot of what I'd call poorly coded applications that imposed arbitrary limitations which I conjecture is because the developer could not or would not prove to his/her satisfaction that certain conditions could not occur without some sweeping restriction. One naive approach to writing anything that models things geometrically might impose arbitrary limits on the types of transformations one could do one things as "tangible" as 3D models or as abstract (conceptually) as graphical user interfaces (which, I think, must exist in spatial dimensions, by definition...?). As a sort of contrived example, I can imagine one ensuring that an entire group of discrete transformations were applied iteratively (since computers do that well) by using a generator and then removing a lot of "logic" code that ensures each transformation was applied with the efficient and elegant approach that, they must have been applied since it's been proven such and such is a generator for the group. Of course, this is the approach Donald Knuth took with his software, in that he said, "Beware of bugs in the above code; I have only proved it correct, not tried it."

7.3

1. The most difficult part of the section was the proof that a cyclic subgroup is the smallest subgroup containing elements of a given set. I take it since there is notation for such a group, it will come up more.

2. I think that a bunch of groups exist in Pascal's triangle with numbers in the proximity of one another. I think all elements a in Z sub n where (a,n)=1 form a group under multiplication. I think the set of all physical manipulations can be broken down to primitives which form a group.

7.1 Part 2

1. Nothing too difficult here. I either understand it really well or not well at all. Sorry I can't be of more help here.

2. I'm going to assume that I really don't understand this well at all (even though I think I do). I tutor kids in math, and I hear a lot of kids say, "This is the most useless stuff. I'm never going to use this. Why am I learning this?" Of course, I can see the applicability of what they're learning. They could too, if they really understood the story problems. But there aren't story problems in this book. I can think of using these, uh, constructs as tools toward proving that certain systems will have certain behavior, that could help in the design of, well, a lot of things, I think.

7.1

1. I hate to be not helpful, but I had Math 190 last semester, so groups are pretty fresh in my mind. No problems here.

2. I get the impression that transformations are associated a lot with groups. If one were designing/creating a computer program that had an interface that mimicked that natural, tangible, physical interface we have, I bet it would be useful to be able to prove that any transformation was the composition of perhaps simpler transformations, proving that transformations were closed under composition and that any manipulation the user might make could be predicted, or at least its result could be. This is probably a very rudimentary application, if it even is one.