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

Abstract EN

Reduced numbers play an important role in the study of modular group action on the PSL2,ℤ-subset of Qm/Q.

For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in PSL2,ℤ-orbits of real quadratic fields.

In particular, we classify PSL2,ℤ-orbits of Qm/Q=∪k∈NQ∗k2m 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 PSL2,ℤ-orbits.