On the minimal ramification problem for semiabelian groups

Kisilevsky, Hershy; Neftin, Danny; Sonn, Jack

doi:10.2140/ant.2010.4.1077

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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

DOI: 10.2140/ant.2010.4.1077

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.

Keywords

Galois group, nilpotent group, ramified primes, wreath product, semiabelian group

Mathematical Subject Classification 2000

Primary: 11R32

Secondary: 20D15

Milestones

Received: 20 December 2009

Revised: 24 June 2010

Accepted: 1 August 2010

Published: 24 February 2011

Authors

Hershy Kisilevsky
Department of Mathematics and Statistics Concordia University 1455 de Maisonneuve Blvd West Montreal, QC H3G 1M8 Canada

Danny Neftin
Department of Mathematics Technion – Israel Institute of Technology 32000 Haifa Israel
http://www.technion.ac.il/~neftind/
Jack Sonn
Department of Mathematics Technion – Israel Institute of Technology 32000 Haifa Israel
http://www.math.technion.ac.il/~sonn/

Vol. 4, No. 8, 2010