Group generated by two elements of orders 2 and 4 acting on real quadratic fields

Coset diagrams for the orbit of the groupG=〈x,y∶x ²=y ⁴ =1〉 acting on real quadratic fields give some interesting information. By using these coset diagrams, we show that in the orbitpG, where , the non-square positive integern does not change its value and the real quadratic irrational numbers of the formp, wherep and its algebraic conjugate 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 ofp.