• 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: 14, Pages: 1458-1469

Original Article

Properties of circuits in coset diagrams by modular group

Received Date:26 March 2020, Accepted Date:24 April 2020, Published Date:28 May 2020


Background/Objectives: Graham Higman gave the idea of coset diagrams for the action of modular group PSL(2;Z) on real quadratic irrationals. These special types of graphical figures are composed of closed paths known as Circuits. These circuits can be classified into certain types of even length with respect to the number of inside\outside triangles. This study is to discuss different properties of reduced numbers in coset diagrams of the type (p;q). Methods: In this study, we have investigated different properties of type (p; q) using reduced quadratic irrationals and continued fractions. We have categorized reduced numbers in accordance with their position in the real line. Distance between two ambiguous numbers and reduced numbers is introduced in this article which will help the reader to understand the structural significance of reduced numbers in a circuit. We have explored different conditions under which certain reduced numbers have the same circuit. Moreover, continued fractions have been used to assist the foundation laid by modular group action and different general results have been derived in this context. Findings: It was possible to define new notions of equivalent, cyclically equivalent and similar circuits using partitions of n and discuss various properties of reduced numbers included in coset diagrams of circuits with length up to four.


  1. Cayley A. The theory of groups, graphical representations. American Journal of Mathematics. 1878;p. 174–176.
  2. Higman G, Mushtaq Q. Coset diagrams and relations for PSL(2,Z) Arab Gulf Journal of. Scientific Research. 1983;1(1):159–164.
  3. Mushtaq Q. Modular group acting on real quadratic fields. Bulletin of the Australian Mathematical Society. 1988;37(2):303–309. doi: 10.1017/s000497270002685x
  4. Mushtaq Q. On word structure of the modular group over finite and real quadratic fields. Discrete Mathematics. 1998;178(1-3):155–164. doi: 10.1016/s0012-365x(97)81824-9
  5. Malik MA, HSM, Majeed A. Modular Group Action on Certain Quadratic Fields. 1995;28:47–68.
  6. Muhammad A, Malik S, Ali. Reduced quadratic irrational numbers and types of G-circuits with length four by modular group. Indian Journal of Science and technology. 2018;11:1–7.
  7. Ali S, Malik MA. Classification of PSL(2, Z) circuits having length six. Indian Journal of Science and technology. 2018;11(42):1–18.
  8. Mushtaq Q, Razaq A, Yousaf A. On contaction of vertices of the circuits in coset diagrams for PSL(2,Z) Proceedings Mathematical Sciences. 2018;129(1).
  9. Andrew A, John EC. Jones and Barlett Publishers, Inc. London. 1995.


© 2020 Javed, 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.