The Proof is in the Pudding

Doesn't that sound like an excuse that a grad student in mathematics might give his or her professor? "I don't have my homework today... the proof is in the pudding."

Finally, after a lot of effort, I proved (I think) that if integer matrices A and B satisfy A^2 = B^2 = -Id, then there is an integer matrix C such that CAC^-1 = B. We went over a similar procedure in class for rational matrices using Module Theory - make Q^2n into a Q[t]-module and jump through some hoops to decompose it as a product of Q[t] / (d1) x ... x Q[t] / (d2n) to come up with a nice basis for the transformation given by A (or B). Of course, the result follows for Q^2n using the Jordan Normal Form of a vector space endomorphism, but the point was we could get at it in a different and more general way using modules.

Meanwhile, preparation for the Algebra qualifier continues in earnest. It's amazing how instructive this preparation is - in looking at past qualifiers, I've run into a number of ideas and problems that I haven't necessarily had to deal with - such as solvable groups and diagram chasing - and so I've been able to expand my horizons as I also become more comfortable with useful tools I've seen before, such as the Sylow theorems and the minimal and characteristic polynomial of a matrix.

I've been thinking a bit today about hypergraphs and how various graph theoretic notions might generalize for them - but I'll post more later when it's percolated more.