G-subsets of an invariant subset ℚ*(√k 2m) of ℚ(√m) ℚ under the modular group action - Aslam Malik M.; Asim Zafar M. (Utilitas Mathematica, 2013)
PSL2(Z) is the well known modular group with the presentation(x, y: x2 = y3 = 1) where x: C' → C' and y: C' → C' are the Möbius transformations defined by: x(z) =-1/z, y(z) = x-1/xLet n = k2m, where m is a square free positive integer and k is any non zero integer. Then ℚ*(√n) = a+√n/c: a, c ≠0, b = a2-n/c ∈ Z and(a, b, c) = 1 is a G-subset of ℚ(√m). In this paper we are interested in finding the cardinality of the set Epr, r ≥ 1, consisting of all classes [a, b, c](mod pr) of the elements of Q*(√n). Also we determine, for each non-square n, the all G-subsets of Q *(√n) under the modular group action by using classes [a, b, c](mod n).