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 $p$.