On the subgroups with non-trivial Mobius number

The paper is concerned with two questions of Avinoam Mann regarding the Möbius function μ(H,G), defined on the lattice of open subgroups H of a finitely generated profinite group G. These questions are related to a certain series, involving the Möbius function, which, when convergent, represents, for k large enough, the probability P(G,k) that G can be generated topologically by k random elements. The first question asks for the groups G for which |μ(H,G)| is bounded by a polynomial function in the index of H. The second question asks for the groups G in which the number b (n,G) of subgroups H of index n, for which μ(G,H)≠0, grows at most polynomially in n. Mann conjectured that if G is a positively finitely generated group (i.e. P(G,k)>0 for some k), then |μ(H,G)| is bounded by a polynomial function in the index of H, and b(n,G) grows at most polynomially in n. This conjecture is reduced to the study of finite monolithic groups with non-Abelian socle.