Homomorphic images of image Original Research Article

It is known that each conjugacy class of actions of PGL(2,Z)PGL(2,Z) on Fq∪{∞}Fq∪{∞} can be represented by a coset diagram D(θ,q),D(θ,q), where θ∈Fqθ∈Fq and q is a power of a prime p. In this paper, we are interested in parametrizing the conjugacy classes of actions of the infinite triangle group △(2,3,11)=〈x,y:x2=y3=(xy)11=1〉△(2,3,11)=〈x,y:x2=y3=(xy)11=1〉 on Fq∪{∞}.Fq∪{∞}. For each θ∈Fqθ∈Fq we then associate a coset diagram D(θ,q)D(θ,q) depicting the conjugacy class of actions of △(2,3,11)△(2,3,11) on Fq∪{∞}.Fq∪{∞}. We have obtained conditions on θθ and q which guarantee only those coset diagrams which depict homomorphic images of △(2,3,11)△(2,3,11) in PGL(2,q).PGL(2,q). We are interested in finding also when the coset diagrams for the actions of PGL(2,Z)PGL(2,Z) on Fq∪{∞}Fq∪{∞} contain vertices on the vertical line of symmetry. It will enable us to show that for infinitely many values of q,q, the group PGL(2,q)PGL(2,q) has minimal genus, while also for infinitely many q,q, the group PSL(2,q)PSL(2,q) is an H*H*-group.