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On the minimal ramification problem for ℓ-groups
- Part of
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- Journal:
- Compositio Mathematica / Volume 146 / Issue 3 / May 2010
- Published online by Cambridge University Press:
- 18 March 2010, pp. 599-606
- Print publication:
- May 2010
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- Article
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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
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.
with no more than 