Abstract It is known that each conjugacy class of actions of PGL(2, Z) on PL(Fq) = Fq O can be represented by a coset diagram D(o, q), where o E Fq and q is a power of a prime. We have associated each conjugacy class of actions of the infinite triangle group âˆ† (2, 3, 11) = < x, y : x2 = y3 = (x y)11 = 1> on PL(Fq) with a coset diagram D(o, q), where o E Fq. In other words, we have obtained conditions on o and q which guarantee only those coset diagrams which depict homomorphic images of âˆ† (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 PL(Fq) contain vertices on the vertical line of symmetry. It will enable us to show that for infinitely many values of the primepower q, the group PGL(2, q) has minimal genus, while also for infinitely many q, the group PSL(2, q) is an H'  group. We have also answered the following question. Given a fragment of a coset diagram, for what values of q and o can this fragment be found in the coset diagram representing the homomorphic images of âˆ† (2, 3, 11)? The condition for the existence of a fragment in D(o, q) is a polynomial f(z) in Z[z] such that f(o) = O. We have also devised a technique to stitch together small diagrams of homomorphic images of âˆ† (2, 3, 11) to obtain homomorphic images of âˆ† (2, 3, 11) for larger class of numbers not confined to q. By using Cebotarev's Density Theorem, we have also answered the question: with what frequency certain fragments of coset diagrams occur in homomorphic images of âˆ† (2, 3, 11)? With a subgroup of finite index in âˆ† (2, 3, 11) = < x, y: x2 = y3 = (xy)11= 1 >, we have associated a quintuple of nonnegative integers (n, g, e, f, h), with n = q + 1, and q = 11, or q = p, for some prime p with p =Â± 1 (mod 11), or q = p5, for some prime p =Â± 2 or Â± 3 or Â± 4 or Â± 5 (mod 11) and n is the number of vertices of a coset diagram, g is genus, e is the number of fixed points of x, f is the number of fixed points of y, and h is the number of fixed points of x y, which satisfies the genus formula 5n = 132(g 1) + 33e + 44 f + 60h. We have shown that all coset diagrams with vertices from PL(Fq) where q = 11, or q = p, for some prime p with p =Â± 1 (mod 11), or q = p5, for some prime p =Â± 2 or Â± 3 or Â± 4 or Â± 5 (mod 11) with specification (n, g, e, f, h), satisfying the genus formula 5n = 132 (g  1) + 33 e + 44 f + 60 h, correspond to a subgroup of finite index n in an infinite triangle group âˆ† (2, 3, 11). This is an important step towards classifying these groups. The groups Gk,l,m have been studied extensively by H. S. M. Coxeter. We have taken k = 3, l = 11, and answered G. Higman's question for the minimum values of m by using a diagrammatic argument. That is, we have shown that all but finitely many positive integers n, both alternating and symmetric groups occur as homomorphic images of G 3, 11,924. Finally, by using fragments, we have proved that for a family of positive integers n, all alternating and symmetric groups occur as a homomorphic images of G3,11,m where m = 2r s or m = 2r if r = s for some primes r and s.
