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

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  • Tianlan Chen
  • Muhammad Nadeem Bari
  • Muhammad Aslam Malik
  • Hafiz Muhammad Afzal Siddiqui
  • Jia-Bao Liu
  • Shaofang Hong

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.

Suggested Citation

  • Tianlan Chen & Muhammad Nadeem Bari & Muhammad Aslam Malik & Hafiz Muhammad Afzal Siddiqui & Jia-Bao Liu & Shaofang Hong, 2020. "Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic Fields," Journal of Mathematics, Hindawi, vol. 2020, pages 1-10, August.
    • Handle: RePEc:hin:jjmath:9568254
    DOI: 10.1155/2020/9568254
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