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

Authors

  • Umar Shoaib

Abstract

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.

References

Ashiq, M. (2006). Action of a two generator group on a real quadratic

field. Southeast Asian Bulletin of Mathematics, 30(3):399-404.

Cameron, P.J. (2000). Notes on classical groups, Queen Mary and

Westfield College, London.

Carter, R.W. (1972). Simple groups of Lie type, John Wiley and Sons,

London.

Everitt, B. (1997). Alternating quotients of the (3,q,r) triangle groups.

Communication in Algebra, 26(6):1817-1832.

Fine, B. (1989). Algebraic theory of the Bianchi groups, Marcel Dekker

Inc, New-York.

Higman, G. & Mushtaq, Q. (1983). Coset diagrams and relations for

PSL(2,Z). The Arab Gulf Journal of Scientific Research, 1(1):159-164.

Moghaddamfar, A.R. (2008). Recognizing finite groups through order

and degree pattern. Algebra Colloquium, 15(3):449-456.

Mushtaq, Q. (1992). Parameterization of all homomorphisms from

PGL(2,Z) into PGL(2,q). Communications in Algebra, 20(4):1023-1040.

Mushtaq, Q. & Shuaib, U. (2013). Coset diagrams of the action of

a certain Bianchi group on PL(Fp). Quasi groups and related systems,

(2):245-254.

Sengun, M.H. (2011). On the integral cohmology of the Bianchi

groups. Kuwait Journal of Science, 43(2):487-505.

Shen, R. & Zhou, Y. (2016). Finite simple groups with some abelian

Sylow subgroups. Experimental Mathematics, 20(4):1-15.

Torstensson, A. (2010). Coset diagrams in the study of finitely

presented groups with an application to quotients of the modular group.

Journal of Commutative Algebra, 2(4):501-514.

Wilson, J.S. (1998). Conjugacy separability of Bianchi groups and HNN

extensions. Mathematical Proceeding of the Cambridge Philosophical

Society, 123( 2): 227-242.

Downloads

Published

03-05-2017