On the minimal ramification problem for semiabelian groups

TL;DR

This paper extends the minimal ramification problem solution for semiabelian groups, including all finite nilpotent semiabelian groups, and discusses the problem's depth for other semiabelian groups.

Contribution

It generalizes existing results to broader classes of semiabelian groups and provides solutions for finite nilpotent semiabelian groups.

Findings

Solution for finite semiabelian groups including nilpotent cases

Extension of minimal ramification problem to broader semiabelian groups

Insights into the complexity for groups not covered by the main theorem

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 Q with exactly d = d(G) ramified primes, where d(G) 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.