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

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.