The kernel generating condition and absolute Galois groups

2021/06/22 by Efrat, Ido
#12E30 #12G05 #16K50 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2106.11553

Abstract

For a list \calL 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 \calL. We give a cohomological characterization of the epimorphisms π\colon 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 Minac, Spira and the author, describing G_(3,p) and G^(3,p) as T(G) for appropriate lists \calL.

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