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

Document Type : Research Paper


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.


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.


Main Subjects

M. Ashiq and Q. Mushtaq, Finite presentation of a linear-fractional group, Algebra Colloq. 12 (2005), no. 4, 585--589.
H.S.M. Coxeter, The abstract group Gm;n;p, Trans. Amer. Math. Soc. 45 (1939), no. 1, 73--150.
Q. Mushtaq, Modular group acting on real quadratic fields, Bull. Aust. Math. Soc. 37 (1988), no. 2, 303--309.
Q. Mushtaq, On word structure of the modular group over finite and real quadratic fields, Discrete Math. 178 (1998), no. 1-3, 155--164.
Q. Mushtaq and M. Aslam, Group generated by two elements of orders two and six acting on R and Q(p,n), Discrete Math. 179 (1998), no. 1-3, 145--154.
Q. Mushtaq and G.C.Rota, Alternating Groups as Quotients of two generator groups, Adv. Math. 96 (1992), no. 1, 113--121.
W.W. Stothers, Subgroup of the (2,3,7)-triangle group, Manuscripta Math. 20 (1977), no. 4, 323--334.