Abstract:

Let q be a prime power. By PL(F_q) the authors mean a projective line over the finite field F_q 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(F_q), 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(F_q). 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(F_q).

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