On Certain Closed Normal Subgroups of Free Profinite Groups of Countably Infinite Rank#

Abstract

Let F be a free profinite group of countably infinite rank and 𝒞(Δ) the class of all finite groups whose composition factors are in Δ for a non-empty class Δ of finite simple groups. Let R Δ(F) be the intersection of all open normal subgroups N of F such that F/N is in 𝒞(Δ). Then we prove that, if 𝒩 is the class of finite groups which have no non-trivial 𝒞(Δ)-quotient, then R Δ(F) is a pro-𝒩 group of countable rank and every finite 𝒩-embedding problem for R Δ(F) is solvable.