On conjugacy classes of the homomorphic images of a certain Bianchi group

In this paper, we classify the conjugacy classes of the action of PSL2(O-2) on the projective line over finite fields, PL(F-p) where p is the M - S prime, by using the method of parameterization and investigate the behavior of coset diagrams of these actions. We prove that the action of PSL2(O-2) on PL(Fp) is transitive for all conjugacy classes except for the conjugacy class in which 2 is a perfect square in Fp. We also prove that the homomorphic images of PSL2 (O-2) represented by these coset diagrams are isomorphic to the rank one Chevalley groups, L-2 (p) for all p >= 11. We also study the behavior of the coset diagram of the homomorphic images of PSL2 (O-2) for the conjugacy class in which 2 is a perfact square in Fp and prove that these coset diagrams admit symmetry about the vertical line of axis in two dimensional space. We also prove that these coset diagrams depict intransitive action of PSL2(O-2) on PL(F-p) in this case. This algebraic fact leads us to develop a formula to count the number of orbits occurring in each coset diagram of this particular class.