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

摘要

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 ofΔ(3,3,k) =〈u,v:u^3=v^3=(uv)~k=1〉on PL(F_q),where q≡±1(modk).Also,for various values of k,they find the conditions for the existence of coset diagrams depicting the permutation actions ofΔ(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(■)~k.In this way,they getΔ(3,3,k)as permutation groups on PL(F_q).