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


  • Umar Shoaib


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.


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