Defining relations of a group Γ=G3,4(2,Z) and its action on real quadratic field

In this paperý, ýwe have shown that the coset diagrams for theý ýaction of a linear-fractional group Γgenerated by the linear-fractionalý ýtransformations r:z→z−1z and s:z→−12(z) oný ýthe rational projective line is connected and transitiveý. ýBy using coset diagramsý, ýwe have shown that r3=s4=1 are defining relations for Γý. ýFurthermoreý, ýwe have studied some important results for the action of group Γ on realý ýquadratic field Q(n−−√)ý. ýAlsoý, ýwe have classified all the ambiguous numbers in the orbit.