Vol. 4, No. 8, 2010

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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 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/