Probabilistic properties of profinite groups

Let ℭ be a class of finite groups which is closed for subgroups, quotients and direct products. Given a profinite group G and an element x ∈ G, we denote by Pℭ(x, G) the probability that x and a randomly chosen element of G generate a pro-ℭ subgroup. We say that a profinite group G is ℭ-positive if Pℭ(x, G) > 0 for all x ∈ G. We establish several equivalent conditions for a profinite group to be ℭ-positive when ℭ is the class of finite soluble groups or of finite nilpotent groups. In particular, for the above classes, the profinite ℭ-positive groups are virtually prosoluble (resp., virtually nilpotent). We also draw some consequences on the prosoluble (resp. pronilpotent) graph of a profinite group.