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

Tianlan Chen,Muhammad Aslam Malik,Jia-Bao Liu,Muhammad Nadeem Bari,Hafiz Muhammad Afzal Siddiqui

doi:10.1155/2020/9568254

Abstract

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.

Highlights

Lci􏼈a(nne>attde1+Fgnoea√ rrn. ndkict√)N2s−/m c o:1,w(a
A quadratic irrational number c is said to be reduced if c > 1 and −1 < c < 0
Coset diagrams for the G-orbit acting on the real quadratic field give some interesting information

Summary

Research Article

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( m )/Q. In we define new particular, we notions classify oPfSeLq(u2i,vZal)e-nort,bcityscloicfaQlly(√eq m ui)v/aQle nt,∪akn∈dNQsi∗m(i√la kr Gm -c)irccounittasiinninPgSLG(-2c,irZcu)-iotsrboiftsleonf grtehal6quaanddradteictermine that the number of equivalence classes of G-circuits of length 6 is ten. We 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 helps us in classifying reduced numbers lying in the PSL(2, Z)-orbits We 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 helps us in classifying reduced numbers lying in the PSL(2, Z)-orbits

Introduction

Lemma tained in

Ti is denoted by equivalent classes

Number of classes of the cyclically equivalent circuits