Group generated by two elements of orders two and six acting on [formula omitted] and [formula omitted

Abstract

There is a well-known relation between the action z → ( az + b)/( cz + d) of the modular group on the real line R and continued fractions. In this paper we replace the modular group by M = 〈 x, y : x 2 = y 6 = 1〉 and R by rational projective line and real quadratic field. We define coset diagrams for the orbits of M acting on these fields and show that in the orbit pM, where p = ( a + √ n)/ c, the non-square positive integer n does not change its value and the numbers of the form p, where p and its algebraic conjugate p = (a − √n)/c have different signs, are finite in number and that part of the coset diagram containing such numbers forms a single closed path and it is the only closed path in the orbit pM.