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

摘要

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(modk). 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 (uv)k. In this way, they get △(3, 3, k) as permutation groups on PL(Fq).