Ashiq, M., Imran, T., Zaighum, M. (2017). Defining relations of a group $\Gamma= G^{3,4}(2,Z)$ and its action on real quadratic field. Bulletin of the Iranian Mathematical Society, 43(6), 1811-1820.

M. Ashiq; T. Imran; M. A. Zaighum. "Defining relations of a group $\Gamma= G^{3,4}(2,Z)$ and its action on real quadratic field". Bulletin of the Iranian Mathematical Society, 43, 6, 2017, 1811-1820.

Ashiq, M., Imran, T., Zaighum, M. (2017). 'Defining relations of a group $\Gamma= G^{3,4}(2,Z)$ and its action on real quadratic field', Bulletin of the Iranian Mathematical Society, 43(6), pp. 1811-1820.

Ashiq, M., Imran, T., Zaighum, M. Defining relations of a group $\Gamma= G^{3,4}(2,Z)$ and its action on real quadratic field. Bulletin of the Iranian Mathematical Society, 2017; 43(6): 1811-1820.

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

^{1}National University of Sciences and Technology, MCS Campus, Rawalpindi, Pakistan.

^{2}Department of mathematics and statistics, Riphah International University, Islamabad, Pakistan.

^{3}Department of Basic Sciences 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 $r:z\rightarrow \frac{z-1}{z}$ and $s:z\rightarrow \frac{-1}{2(z+1)}$ 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.