Abstract
Let n be a non-square positive integer and Q*(root n) the set {(a + root n)/c : a, (a(2) - n)/c, c are relatively prime integers}. Coset diagrams for orbits of the Hecke group H(lambda(6)) acting on projective line over the set Q*(root n) are known. If alpha is any real quadratic irrational number and alpha(H(lambda 6)) is an orbit of the group H(lambda(6)) under a then alpha(H(lambda 6)) subset of Q*(root n,). In this paper, we employ the coset diagrams to prove some of the results for action of H(lambda(6)) on real quadratic fields, which are known to hold in case of well known modular group H(lambda(3)). In fact, we investigate the question: when does an orbit of H(lambda(6)) containing a circuit (closed path) of a given type exist? We also determine a condition for existence of both a real quadratic irrational number gamma (= (a + root n)/3c') and its algebraic conjugate (gamma) over bar (= (a - root n)/3c') in the orbit.