Defining relations of a group $Gamma= G^{3,4}(2,Z)$ and its action on real quadratic field

Authors

  • M. A. Zaighum Department of Basic Sciences Riphah International University Islamabad‎, ‎Pakistan.
  • M. Ashiq National University of Sciences and Technology‎, ‎MCS Campus‎, ‎Rawalpindi,\ Pakistan.
  • T. Imran Department of mathematics and statistics, Riphah International University, Islamabad‎, ‎Pakistan.
Abstract:

In this paper‎, ‎we have shown that the coset diagrams for the‎ ‎action of a linear-fractional group $Gamma$ generated by the linear-fractional‎ ‎transformations $r:zrightarrow frac{z-1}{z}$ and $s:zrightarrow frac{-1}{2(z+1)}$ on‎ ‎the rational projective line is connected and transitive‎. ‎By using coset diagrams‎, ‎we have shown that $r^{3}=s^{4}=1$ are defining relations for $Gamma$‎. ‎Furthermore‎, ‎we have studied some important results for the action of group $Gamma$ on real‎ ‎quadratic field $Q(sqrt{n})$‎. ‎Also‎, ‎we have classified all the ambiguous numbers in the orbit.

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Journal title

volume 43  issue 6

pages  1811- 1820

publication date 2017-11-30

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