On the minimal ramification problem for ℓ-groups

Publisher： Cambridge University Press

E-ISSN： 1570-5846|146|3|599-606

ISSN： 0010-437x

Source： Compositio Mathematica, Vol.146, Iss.3, 2010-05, pp. : 599-606

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Abstract

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.

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