The kernel generating condition and absolute Galois groups

Abstract

For a list L of finite groups and for a profinite group G, we consider the intersection T (G) of all open normal subgroups N of G with G/N in L. We give a cohomological characterization of the epimorphisms π : S → G of profinite groups (satisfying some additional requirements) such that π[T (S)] = T (G). For p prime, this is used to describe cohomologically the profinite groups G whose nth term G (n,p) (resp., G (n,p) ) in the p-Zassenhaus filtration (resp., lower p-central filtration) is an intersection of this form. When G = G F is the absolute Galois group of a field F containing a root of unity of order p, we recover as special cases results by Mináč, Spira and the author, describing G (3,p) and G (3,p) as T (G) for appropriate lists L.