Real Quadratic Irrational Numbers and Modular Group Action

It is well known that G = (x, y : x(2) = y(3) = 1) represents the modular group, where x(z) = y(z) = are linear fractional transformations. Let n = k(2)m, where k is any non zero integer and m is square free positive integer. Then Q* (root n) := {a +root n/c : a,c,b = a(2)-n/c epsilon Z and (a, b, c) = 1} is a G-subset of Q(root m) [11]. We denote alpha = a+root n/c by alpha(a,b, c). We say that two elements az (a, b, c), (a', b', c') of Q* (n) are s-equivalent if and only if a a' (mod s), b b' (mod s) and c c' (mod s). In this paper we investigate that the cardinality of the set E-p, p a prime, consisting of all classes [a, b, c](mod p) of the elements of Q* (root n) is p(3) - 1. Also we obtain two proper G-subsets of Q* (root n) corresponding to each odd prime divisor of n.