I am not really clear on what is meant by commutators. I know that the commutator of G is ABA^-1B^-1, but I am not sure how to check if a group is solvable by having the commutator eventually equal the trivial group. For example, I know that the Heisenberg group of 3x3 upper triangular matrices is two-step solvable, but am not sure how to SHOW that. I know that it means that the first commutator doesn't equal the identity matrix and that the second one does... but how do I show this??? Also, how do I show that the group GL(2,R) (2x2 invertible matrices) IS NOT solvable???