Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic Fields. (28th August 2020) - Chen, Tianlan; Bari, Muhammad Nadeem; Malik, Muhammad Aslam; Afzal Siddiqui, Hafiz Muhammad; Liu, Jia-Bao
Abstract : Reduced numbers play an important role in the study of modular group action on the PSL 2, ℤ -subset of Q m / 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 m / Q = ∪ k ∈ N Q ∗ 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 PSL 2, ℤ -orbits.