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 $Gamma$‎. ‎Furt...

When working with numbers such as real numbers x ∈ R or complex numbers z ∈ C, there is an unambiguous notion of a magnitude |x| or |z| of a number, with which to measure which numbers are large and which are small. One can also use this notion of magnitude to define a distance |x − y| or |z − w| between two real numbers x, y ∈ R, or between two complex numbers z, w ∈ C, thus giving a quantitat...

0. Notations Throughout the paper we use the following notations: We write e(z) = e; C, R, Q, Z, N and N0, denote the set of complex numbers, real numbers, rational numbers, integers, positive integers, and positive integers including zero, respectively. Q(i) denotes the field of Gaussian numbers, and Z[i] the ring of Gaussian integers. We write tr(z) and N(z) for the trace and the norm of z ov...