Galois groups of p-extensions with two ramification points

Let Kp(p, q) be the maximal p-extension of the field ℚ of rational numbers with ramification points p and q. Let Gp(p, q) be the Galois group of the extension Kp(p, q)/ℚ. It is known that G p(p, q) can be presented by two generators which satisfy a single relation. The form of this relation is known only modulo the second member of the descending central series of Gp(p, q). In this paper, we find an arithmetical-type condition on which the form of the relation modulo the third member of the descending central series of Gp(p,q) depends. We also consider two examples with p = 3, q = 19 and p = 3, q = 37. © 2000 Kluwer Academic/Plenum Publishers.