Section 9.4 & 7.1 Due May 24

1. The definition of addition where [a,b] + [c,d] = [ad+bc, bd] was very random. Theorem 9.31 made no sense. So with groups can you just define * as whatever you want it to be? Because composition doesn't seem like just a standard operation like multiplication of addition or something. I don't get what the actual group is. Is it the set of the bijective compositions? Or the set of the permutations?
2. It was interesting how on the proof of lemma 9.29 they proved it backwards and went from f(a)+f(b) to f(a+b) instead of the other way around. I haven't seen it like that before. Groups seem similar to rings, but with less operations.