On conjugacy classes of the homomorphic images of a certain Bianchi group - Shoaib, Umar (Kuwait Journal of Science, 2017)
In this paper, we classify the conjugacy classes of the action of PSL₂(O₂) on the projective line over finite fields, PL(F_p), where p is the M-S prime, by using the method of parametrization and investigate the behavoior of coset diagrams of these actions. We prove that the action of PSL₂(O₂) on PL(F_p) is transitive for all conjugacy classes except for the conjugacy class in which 2 is a perfact square in F_p. We also prove that the homomorphic images of PSL₂(O₂) represented by these coset diagrams are isomorphic to the rank one Chevalley groups, L₂(p), for all p≥11. We also study the behavior of the coset diagram of the homomorphic images of PSL₂(O₂) for the conjugacy class in which 2 is a perfact square in F_p 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(O₂) on PL(F_p). This algebraic fact leads us to develop a formula to count the number of orbits occuring in each coset diagram of this particular class.