On the probabilistic zeta-function of pro(finite-soluble) groups

In this paper we extend Gaschutz' theorem to profinite groups (cf. Thm. A) and use this result to prove two theorems on the minimal number of generators of a pro(finite-soluble) group (cf. Thm. B, Thm. C). In section 3 we give a parameterization of the set of isomorphism classes of irreducible discrete F-p [G] -modules, where G is either a finitely generated free pro(finite-abelian) group or a tame ramification group I-q, q = p(f) (cf. Prop. 3.3, Prop. 3.6). This parameterization together with Theorem A and B allows us to calculate the probabilistic zeta-function for finitely generated free pro(finite-metabelian) groups and the absolute Galois groups of l-adic number fields explicitly (cf. Thm. D, Thm. E).