Oh for heavens sake! It's absolutely relentless. I have just 'finished' GR3 from M336 but towards the end I lost the plot in trying to follow the proof of the theorem for the canonical decomposition of finitely generated Abelian groups and gave up. Up until page 31 things had been going well and I was coping with the exercises and the proofs, but then the exercises on page 32 went up a level and I started to feel like I was drowning in complexity. I particularly hate the exercises that involve producing bits of proof that eventually are used in some much bigger proof. Often with the terminology, the difficulty of the arguments and the not really knowing where a proof is leading, I don't know my a&se from my elbow and I don't even understand what the hell they want us to do, let alone have an idea how to achieve it. It isn't very rewarding but you just have to say bu&&er it and carry on.
It is difficult producing proofs at the best of times. My irritation stems from the fact that whoever produced the proof in the first place had probably spent a great deal of time thinking about the problem and had a definite idea of what they wanted to prove. They were probably also fully conversant with all the terminology and had all the techniques at their fingertips. For us poor students it is rush, rush, rush. No sooner have you grasped one bit of terminology, another is thrown at you and before you can master anything very much you are on to the next topic. So do they really expect us to be able to be clever and come up with these sort of proofs in this type of environment? I think it is all a bit of a waste of time, to be honest.
The hilarious thing is that often having just finished a book, I am so subsumed by all the logic that I can't even recall what the book was about when I finished it!
So, progress. I finished ML4 on formal systems. That was ok and didn't cause any tantrums. I had to skip GE2 because at the time I needed the geometry envelope from the OU and it was due in the second mailing of books. So I went on to GR3.
What is GR3 all about? Well, it is trying to find out what the structure of an Abelian group looks like if it is presented as a finite set of generators with a finite set of relations. Towards the end of the book it goes on to give the structure of an Abelian group with a finite number of generators but with an unlimited number of relations (the bit that made me pull my hair out). Ok, so how do you do this?
Basically, we discovered in IB4 that the relations are essentially the elements of the kernel K of a homomorphism from the free group F with the same number of generators to the group G we want. The 'free' bit means that F is free from such non-trivial relations. It turns out that the quotient group F/K is isomorphic to G. In this book we consider only Abelian generated groups A. The clever bit is that we impose structure on the free group by demanding that it is also Abelian. It follows that the structure of the free Abelian group with n generators is just ZxZx...xZ, i.e. the direct product of n copies of Z. In the simple case where the relations that generate A are of the form da=0 where d is a multiple of the generator a, then A is isomorphic to a product of Zd's i.e. Z modulo d. For the case where the m relations are linear combinations of the n generators, then we can from an mxn matrix of the coefficients of the generators in each relation, which can be reduced to a diagonal matrix. Then A is isomorphic to a product of the Zd's again. There is a way of uniquely writing such a generated Abelian group - this is its canonical decomposition. The d's that are greater than 1 in this decomposition are called torsion coefficients.
