On the minimal ramification problem for semiabelian groups

Abstract

It is now known that for any prime $p$ and any finite semiabelian $p$-group $G$, there exists a (tame) realization of $G$ as a Galois group over the rationals $\mathbb{Q}$ with exactly $d=d\left(G\right)$ ramified primes, where $d\left(G\right)$ is the minimal number of generators of $G$, which solves the minimal ramification problem for finite semiabelian $p$-groups. We generalize this result to obtain a theorem on finite semiabelian groups and derive the solution to the minimal ramification problem for a certain family of semiabelian groups that includes all finite nilpotent semiabelian groups $G$. Finally, we give some indication of the depth of the minimal ramification problem for semiabelian groups not covered by our theorem.