Coset Diagrams in PSL(2, O2)-Space

Abstract

This thesis contains four chapters, the first chapter contains basic concepts and examples related to one of the Euclidean Bianchi groups PSL(2, 0 2) which is the central part of our attention in present work. The second chapter is associated with the action of PSL(2, 0 2) on proj ective lines over the finite field, PL(Fp) , where all p 's are the M-S primes. We draw coset diagrams for our investigations about PSL(2,02)' We prove that the permutation subgroups of the action of PSL(2,02) on PL(Fp) are subgroups of Ap+1 ' These coset diagrams do not admit symmetry in the 2- dimensional space due to the amalgamated structure of these subgroups. We define the symmetry spoilers for these coset diagrams and create formu lae to count the number of symmetry spoilers occurring in each coset diagrams of the action of PSL(2, 0 2) on PL(Fp). In this connection, we draw coset diagrams of the action of PSL(2,02) on PL(Fp) after removing all symmetry spoilers and prove that the group depicted by each component is isomorphic to A4 . In other words we gave a graphical explanation of the amalgamated structure of these subgroups. We also prove that these diagrams admit symmetry about the vertical line of axis in this case. We also obtain a formula to count the number of orbits occurring as copies of At!- in each coset diagram. In the last section of this chapter, we study algebraic properties of the action of PSL(2,02) on PL(Fp). We prove that the action is transitive and primitive. By using the primitive characteristic of this action, we prove that the permutation subgroups of P SL(2, 0 2) are simple for all p ~ 11, where p is M-S prime. The third chapter is devoted to study the conjugacy classes of the actions of PSL(2,02) on P L(Fp) tlu'ough the process of parametrization. In this cOlmection, we prove that there exists a conjugacy class of the non-degenerate homomorphisms from PSL(2,02) into PSL(2, p) corresponding to each perfect square in Fp. We subdivide the study in four cases in this connection. This subdivision is based on the classification of the sequence of M-S primes. We discuss in detail behavior of the coset diagrams of the actions of PSL(2,02) on PL(Fp) in each case. Each action yields non-Abelian and simple Lie type permutation subgroups of rank 1 which are in fac t isomorphic to the rank one Chevalley groups, L2 (p) , for all p ~ 11. We also note that each coset diagram is cotmected, for all the conjugacy classes except for the conjugacy class in which 2 is a perfect square inFp . This graphical information leads us to note the point that the action is transitive. These algebraic facts are proved in the last section of the chapter. In the last chapter, we consider a special case to study behavior of the coset diagrams corresponding to the conjugacy classes in which 2 is a perfect square in Fp . In this case we obtain symmetric group of degree 4 as a homomorphic image of PSL(2, 02) in this case. We prove that these coset diagrams admit symmetry about the vertical line of axis. We prove that each coset diagram represents an intransitive action of PSL(2, 0 2) on PL(Fp) in this case. In the last section we count the number of orbits of a coset diagram of an intransitive action