A little math problem

This is a little math problem that I once thought trivial. I suppose it is, but it can be a puzzler. Just for fun the answer (or at least an answer) will appear next month.

Let T(k) be the transformation x -> k/(1-x). For certain values of k T(k) defines a cyclic group. For example, for k = 1 we have the sequence {x, 1/(1-x), x-1/x}. Oddly enough the only rational values of k for which T(k) defines a cyclic group are 1, 1/2, and 1/3. Please explain why.