TY - UNPB

T1 - The kernel generating condition and absolute Galois groups

AU - Efrat, Ido

PY - 2021/6/1

Y1 - 2021/6/1

N2 - For a list $\cal{L}$ 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 $\cal{L}$. We give a cohomological characterization of the
epimorphisms $\pi\colon S\to G$ of profinite groups (satisfying some
additional requirements) such that $\pi[T(S)]=T(G)$. For $p$ prime, this
is used to describe cohomologically the profinite groups $G$ whose $n$th
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 $\cal{L}$.

AB - For a list $\cal{L}$ 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 $\cal{L}$. We give a cohomological characterization of the
epimorphisms $\pi\colon S\to G$ of profinite groups (satisfying some
additional requirements) such that $\pi[T(S)]=T(G)$. For $p$ prime, this
is used to describe cohomologically the profinite groups $G$ whose $n$th
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 $\cal{L}$.

KW - Mathematics - Number Theory

KW - 12G05

KW - 12E30

KW - 16K50

M3 - ???researchoutput.researchoutputtypes.workingpaper.preprint???

T3 - arXiv:2106.11553 [math.NT]

BT - The kernel generating condition and absolute Galois groups

ER -