Homomorphic images of [formula omitted

Abstract

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