  Let p be an odd prime number and let S be a finite set of prime numbers
congruent to 1 modulo p. We prove that the group G_S(Q)(p) has cohomological
dimension 2 if the linking diagram attached to S and p satisfies a certain
technical condition, and we show that G_S(Q)(p) is a duality group in these
cases. Furthermore, we investigate the decomposition behaviour of primes in the
extension Q_S(p)/Q and we relate the cohomology of G_S(Q)(p) to the etale
cohomology of the scheme Spec(Z)-S. Finally, we calculate the dualizing module.
