• P-ISSN 0974-6846 E-ISSN 0974-5645

Indian Journal of Science and Technology


Indian Journal of Science and Technology

Year: 2020, Volume: 13, Issue: 27, Pages: 2797-2810

Review Article

Symmetries of Icosahedral group and classification of G-circuits of length six

Received Date:26 May 2020, Accepted Date:11 June 2020, Published Date:31 July 2020


Objectives: To determine the exact number of equivalence classes of G-circuits of length q≥2:Methods/Statistical Analysis: To classify G-orbits of Q(m)/Q containing G-circuits of length 6. Findings: The equivalence classes of G-circuits of length 6 is ten in number and determine the exact number of G-orbits and structure of G-orbits corresponding to each of ten equivalence classes of Gcircuits. Furthermore, we describe some generalized G-circuits of length 2t corresponding to each of these ten equivalence classes and the structure of these G-circuits with conditions on t. Applications/Improvements: We employ Symmetries of Icosahedral group to explore cyclically equivalence classes of Gcircuits and similar G-circuits of length 6 corresponding to each of these ten equivalence classes. This study helps us in classifying reduced numbers lying in PSL(2, Z)-orbits. These results are verified by some suitable example.

Keywords: Rotational symmetries of icosahedral group; partition function;

equivalence classes of G-circuits; reduced quadratic irrational numbers


  1. Adler A, John EC. The Theory of Numbers. Jones and Bartlett Publishers, Inc: London. 1995.
  2. Razaq A, Mushtaq Q, Yousaf A. The number of circuits of length 4 in PSL(2,ℤ)-space. Communications in Algebra. 2018;46:5136–5145. Available from: https://dx.doi.org/10.1080/00927872.2018.1461880
  3. Mushtaq Q, Razaq A, Yousaf A. On contraction of vertices of the circuits in coset diagrams for\varvecPSL\varvec(2,ZZ\varvec)PSL ( 2 , Z ) Proceedings - Mathematical Sciences. 2019;129(1):1–26. Available from: https://dx.doi.org/10.1007/s12044-018-0450-z
  4. Malik MA, Zafar A. On subsets of Q(√m)/Q under the Action of Hecke Groups H(〖 〗_q ) Applied Mathematics. 2014;5:1284–1291.
  5. Aslam MA, Zafar A. On Orbits of Q(√m)/Q under the Action of Hecke Group H(√2) Middle East Journal of Scientific Research. 2013;15(12):1641–1650.
  6. Mushtaq Q. Modular group acting on real quadratic fields. Bulletin of the Australian Mathematical Society. 1988;37(2):303–309. Available from: https://dx.doi.org/10.1017/s000497270002685x
  7. Javed MA, Aslam MA. Properties of Circuits in Coset diagram by Modular Group. Indian Journal of Science and Technology. 2020;13(14):1458–1469. Available from: https://doi.org/10.17485/IJST/v13i14.7
  8. Aslam MA, Zafar A. G-subsets of an invariant subset of under the modular group action. Utilitas Mathematica. 2013;p. 377–387.
  9. Sajjad A, Aslam MA. Classification of PSL(2, Z) Circuits Having Length Six. Indian Journal of Science and Technology. 2018;p. 1–18. Available from: https://doi.org/10.17485/ijst/2018/v11i42/132011
  10. Aslam MA, Sajjad A. Reduced Quadratic Irrational Numbers and Types of G-circuits with Length Four by Modular Group. Indian Journal of Science and Technology. 2018;p. 1–7.
  11. Spiegel M, Stephens L. Schaums’s out lines Statistics. 1960.


© 2020 Bari, Malik.This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Published By Indian Society for Education and Environment (iSee).


Subscribe now for latest articles and news.