Group Generated by Two Elements of Orders 2 and 4 Acting on Real Quadratic Fields

Coset diagrams for the orbit of the group G= (x,y : x² = y.= 1) acting on real quadratic fields give some interesting information. By using these coset diagrams, we show that in the orbit pG, where p = (a + .n)/c, the non-square positive integer n does not change its value and the real quadratic irrational numbers of the form p, where p and its algebraic conjugate. (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, which is the only closed path in the orbit of p.