#### Vol. 4, No. 8, 2010

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On the minimal ramification problem for semiabelian groups

### Hershy Kisilevsky, Danny Neftin and Jack Sonn

Vol. 4 (2010), No. 8, 1077–1090
##### 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 $d=d\left(G\right)$ 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.

##### Keywords
Galois group, nilpotent group, ramified primes, wreath product, semiabelian group
Primary: 11R32
Secondary: 20D15