Modular group acting on real quadratic fields

Q. Mushtaq

doi:10.1017/S000497270002685X

Abstract

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Coset diagrams for the orbit of the modular group G = 〈x, y: x2 = y3 = 1〉 acting on real quadratic fields give some interesting information. By using these coset diagrams, we show that for a fixed value of n, a non-square positive integer, there are only a finite number of real quadratic irrational numbers of the form , where θ and its algebraic conjugate have different signs, and that part of the coset diagram containing such numbers forms a single circuit (closed path) and it is the only circuit in the orbit of θ.

References

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[3]Mushtaq, Q., Coset Diagrams for the Modular Group, D.Phil.Thesis (Oxford University, 1983).Google Scholar

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Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Mushtaq, Q. and Aslam, M. 1993. Group generated by two elements of orders 2 and 4 acting on real quadratic fields. Acta Mathematica Sinica, Vol. 9, Issue. 1, p. 48.

Mushtaq, Q. and Shaheen, F. 1994. Factor groups of G6,6,6 through coset diagrams for an action on PL (Fq). Discrete Mathematics, Vol. 126, Issue. 1-3, p. 225.

Mushtaq, Qaiser 1998. On word structure of the modular group over finite and real quadratic fields. Discrete Mathematics, Vol. 178, Issue. 1-3, p. 155.

Mushtaq, Q. and Aslam, M. 1998. Group generated by two elements of orders two and six acting on R and Q(√n). Discrete Mathematics, Vol. 179, Issue. 1-3, p. 145.

Malik, M. Aslam Husnine, S. M. and Majeed, Abdul 2005. Properties of real quadratic irrational numbers under the action of the group H = áx, y: x2 = y4 = 1ñ. Studia Scientiarum Mathematicarum Hungarica, Vol. 42, Issue. 4, p. 371.

Husnine, S. M. Malik, M. Aslam and Majeed, Abdul 2005. On ambiguous numbers of an invariant subset \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $Q^{*} \left(\sqrt {k^2m}\,\right)$ \end{document} of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $Q\left(\sqrt m\,\right)$ \end{document} under the action of the modular group \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} ${\rm PSL}\, (2,Z)$ \end{document}. Studia Scientiarum Mathematicarum Hungarica, Vol. 42, Issue. 4, p. 401.

Mushtaq, Q. and Hayat, U. 2007. Pell Numbers, Pell–Lucas Numbers and Modular Group. Algebra Colloquium, Vol. 14, Issue. 01, p. 97.

Afzal, Farkhanda Afzal, Qamar and Malik, M. Aslam 2013. Quadratic Fields under the Action of Subgroups of M. International Journal of Modeling and Optimization, p. 283.

Malik, M. Aslam and Zafar, M. Asim 2014. On Subsets of Q(&radic;<span style=" text-decoration:overline;">m</span>) Q under the Action of Hecke Groups H(&lambda;<sub>q</sub>). Applied Mathematics, Vol. 05, Issue. 08, p. 1284.

Mushtaq, Qaiser and Anis, Saima 2016. Coset Diagram for the Action of Picard Group on $\mathbb{Q}(i,\sqrt{3})$. Algebra Colloquium, Vol. 23, Issue. 01, p. 33.

Razaq, Abdul Mushtaq, Qaiser and Yousaf, Awais 2018. The number of circuits of length 4 inPSL(2,ℤ)-space. Communications in Algebra, Vol. 46, Issue. 12, p. 5136.

Rafiq, Ayesha and Mushtaq, Qaiser 2019. Coset diagrams of the modular group and continued fractions. Comptes Rendus Mathematique, Vol. 357, Issue. 8, p. 655.

Mushtaq, Qaiser Razaq, Abdul and Yousaf, Awais 2019. On contraction of vertices of the circuits in coset diagrams for $${\varvec{PSL}}\varvec{(2}, \pmb {\mathbb {Z}} \varvec{)}$$ PSL ( 2 , Z ) . Proceedings - Mathematical Sciences, Vol. 129, Issue. 1,

Rafiq, Ayesha and Khan, Majid 2019. Construction of new S-boxes based on triangle groups and its applications in copyright protection. Multimedia Tools and Applications, Vol. 78, Issue. 11, p. 15527.

Shahzad, Imran Mushtaq, Qaiser and Razaq, Abdul 2019. Construction of New S-Box Using Action of Quotient of the Modular Group for Multimedia Security. Security and Communication Networks, Vol. 2019, Issue. , p. 1.

Chen, Tianlan Bari, Muhammad Nadeem Malik, Muhammad Aslam Afzal Siddiqui, Hafiz Muhammad and Liu, Jia-Bao 2020. Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic Fields. Journal of Mathematics, Vol. 2020, Issue. , p. 1.

Yousaf, Awais Alolaiyan, Hanan Razaq, Abdul and Younis, Muhammad 2020. Evolution of ambiguous numbers under the actions of a Bianchi group. Journal of Taibah University for Science, Vol. 14, Issue. 1, p. 615.

Alolaiyan, Hanan Razaq, Abdul Yousaf, Awais Zahra, Rida and Yildirim, Kenan 2021. A Comprehensive Overview on the Formation of Homomorphic Copies in Coset Graphs for the Modular Group. Journal of Mathematics, Vol. 2021, Issue. , p. 1.

Deajim, Abdulaziz and Hong, Shaofang 2021. The Hecke Group H λ 4 Acting on Imaginary Quadratic Number Fields. Journal of Mathematics, Vol. 2021, Issue. , p. 1.

Aamir, Muhammad Awais Yousaf, Muhammad Razaq, Abdul and Guardo, Elena 2021. Number of Distinct Homomorphic Images in Coset Diagrams. Journal of Mathematics, Vol. 2021, Issue. , p. 1.

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Modular group acting on real quadratic fields

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