G-subsets of an invariant subset Q*(√k2m) of Q(√m)\Q under the Modular Group Action

PSL2(Z) is the well known modular group with the presentation < x, y: x(2) = y(3) = 1 > where x : C' -> C' and y : C' -> C' are the Mobius transformations defined by: x(z) = -1/z, y(z) = z-1/z. Let n = k(2)m, where m is a square free positive integer and k is any non zero integer. Then Q*(root n) = {a+root n/c : a, c not equal 0, b = a(2)-n/c is an element of Z and (a, b, c) = 1} is a G-subset of Q(root m)\Q. In this paper we are interested in finding the cardinality of the set E(p)r, r >= 1, consisting of all classes [a, b,c](mod p(r)) of the elements of Q*(root n). Also we determine, for each non-square n, the all G-subsets of Q*(root n) under the modular group action by using classes [a, b, c](mod n).