Icosahedral group and classification of \(\mathrm{PSL}(2, Z)\)-orbits of real quadratic fields (Q827229)

Summary: Reduced numbers play an important role in the study of modular group action on the \(\mathrm{PSL}(2,\mathbb{Z})\)-subset of \(Q(\sqrt{m})/Q\). For this purpose, we define new notions of equivalent, cyclically equivalent, and similar \(G\)-circuits in \(\mathrm{PSL}(2,\mathbb{Z})\)-orbits of real quadratic fields. In particular, we classify \(\text{PSL}(2,\mathbb{Z})\)-orbits of \(Q(\sqrt{m})/Q=\cup_{k \in N} Q^* (\sqrt{k^2 m})\) 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. This study also helps us in classifying reduced numbers lying in the \(\mathrm{PSL}(2,\mathbb{Z})\)-orbits.