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 Δ (3, 3, k) =〈 u, v:u 3= v 3=( 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 Δ (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 ( u ¯ v ¯ ) k . In this way, they get Δ (3, 3, k) as permutation groups on PL( F q ).
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