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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???