√ k 2 m) UNDER THE ACTION OF THE MODULAR GROUP PSL(2;Z)

We look at some ways in which coset diagrams have been used to nd the orbits, number of subgroups and structure of the nitely generated groups. In this paper we use coset diagrams and modular arithmetic to determine the exact num- ber of G-orbits of Q ' ∗ ( √ pk), Q ' ∗ ( √ 2pk), Q ' ∗ ( √ 22pk), and in general Q ' ∗ ( √ 2lpk), for each l ≥ 3 and k = 2h + 1 ≥ 3, for each odd prime p.