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On the minimal ramification problem for ℓ-groups

Part of: Representation theory of groups Algebraic number theory: global fields

Published online by Cambridge University Press:  18 March 2010

Hershy Kisilevsky  and
Jack Sonn
Hershy Kisilevsky
Affiliation:
Department of Mathematics and Statistics, Concordia University, Montreal, Quebec H3G 1M8, Canada (email: kisilev@mathstat.concordia.ca)
Jack Sonn
Affiliation:
Department of Mathematics, Technion, 32000 Haifa, Israel (email: sonn@math.technion.ac.il)
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Abstract

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Let ℓ be a prime number. It is not known whether every finite ℓ-group of rank n≥1 can be realized as a Galois group over ${\Bbb Q}$ with no more than n ramified primes. We prove that this can be done for the (minimal) family of finite ℓ-groups which contains all the cyclic groups of ℓ-power order and is closed under direct products, (regular) wreath products and rank-preserving homomorphic images. This family contains the Sylow ℓ-subgroups of the symmetric groups and of the classical groups over finite fields of characteristic not ℓ. On the other hand, it does not contain all finite ℓ-groups.

Keywords

Galois groupp-groupramified primeswreath productsemiabelian group

MSC classification

Secondary: 11R32: Galois theory 20D15: Nilpotent groups, $p$-groups

Information

Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 3 , May 2010 , pp. 599 - 606
Copyright
Copyright © Foundation Compositio Mathematica 2010

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