Acta mathematica scientia,Series B

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COSET DIAGRAMS FOR A HOMOMORPHIC IMAGE OF △(3,3,k)

M. Ashiq; Q. Mushtaq   

  1. Department of Basic Sciences and Humanities, College of E and ME
    National University of Sciences and Technology, Rawalpindi, Pakistan
  • Received:2005-12-21 Revised:2006-09-06 Online:2008-04-20 Published:2008-04-20
  • Contact: M. Ashiq

Abstract:

Let q be a prime power. By PL(Fq) the authors mean a projective line over the finite field Fq with the additional point ∞. In this article, the authors parametrize the conjugacy classes of nondegenerate homomorphisms which represent actions of △(3,3,k)=〈u,v:u3=v3=(uv)k=1〉on PL(Fq), where q≡±1(mod k). Also, for various values of k, they find the conditions for the existence of coset diagrams depicting the permutation actions of △(3,3,k) on PL(Fq). The conditions are polynomials with integer coefficients and the diagrams are such that every vertex in them is fixed by $(\overline{u}\,\overline{v})^{k}.$ In this way, they get △(3,3,k) as permutation groups on PL(Fq).

Key words: Coset diagrams, conjugacy classes, nondegenerate homomorphism, projective line and triangle groups

CLC Number: 

  • 20F05
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