A finiteness condition on the coefficients of the probabilistic zeta function

We discuss whether finiteness properties of a profinite group G can be deduced from the coefficients of the probabilistic‎ ‎zeta function PG(s)‎. ‎In particular we prove that if PG(s) is rational and all but finitely many non abelian composition factors of G are isomorphic to PSL(2,p) for some prime p‎, ‎then G contains only finitely many maximal subgroups‎.