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

Document Type: Research Paper

Authors

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.

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### References

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### History

• Receive Date: 07 April 2016
• Revise Date: 19 October 2016
• Accept Date: 21 October 2016