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 + root 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) over bar=(a-root 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.