A Galois dichotomy
Jochen Königsmann
Generalizing a theorem of Neukirch-Geyer-Pop we show that for any non-henselian
field $K$ either every finite group occurs as subquotient of the absolute
Galois group $G_K$ of $K$ or all decomposition subgroups of $G_K$ are pro-$p$
groups for a fixed prime $p$.
A consequence of this is that there are exactly two possibilities for a
non-trivial class $\mathcal{C}$ of finite groups closed under forming
subgroups, quotients and extensions for which the fa mily of pro-$\mathcal{C}$
absolute Galois groups is closed under free pro-$\mathcal{C}$ products: either
$\mathcal{C}$ is the class of all finite groups or $\mathcal{C}$ is the class
of all finite $p$-groups for a fixed prime $p$.