Profinite groups in which the probabilistic zeta function coincides with the subgroup zeta function

Abstract

We examine finitely generated profinite groups in which two formal Dirichlet series, the subgroup zeta function and the probabilistic zeta function, coincide; we call these groups ζ-reversible. In the class of prosolvable groups of finite rank we show some sufficient conditions for this property to hold and we produce a structural characterisation of torsion-free prosolvable groups of rank two which are ζ-reversible. For pro-p groups several results support the conjecture that ζ-reversibility is equivalent to the property that every open subgroup has the same minimal number of generators of the group itself; in particular this holds for powerful pro-p groups.