Icosahedral Group and Classification of PSL(2,Z)-Orbits of Real Quadratic Fields

Reduced numbers play an important role in the study of modular group action on the PSL(2, Z)-subset of Q(root mv)/Q. For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in PSL(2, Z)-orbits of real quadratic fields. In particular, we classify PSL(2, Z)-orbits of Q(root mv)/Q =UkNQ*(root k(2)m) containing G-circuits of length 6 and determine that the number of equivalence classes of G-circuits of length 6 is ten. We also employ the icosahedral group to explore cyclically equivalence classes of circuits and similar G-circuits of length 6 corresponding to each of these ten circuits. +is study also helps us in classifying reduced numbers lying in the PSL(2, Z)-orbits.