[Sous-groupes d'indice fini des groupes profinis]
Le résultat principal est que tout sous-groupe d'indice fini dans un groupe profini de type fini est ouvert. Par conséquent, la topologie d'un tel groupe est uniquement déterminée par la structure de groupe sous-jacente. Ce résultat se déduit d'un « théorème d'uniformité » pour les groupes finis : soit w un mot tel que la variété de groupes associée est localement finie, et soit d un entier. Si G est un groupe fini ayant d générateurs, alors chaque élément du sous-groupe verbal w(G) est produit de fw(d) valeurs de w dans G. On obtient des résultats analogues pour le sous-groupe dérivé.
We prove that every subgroup of finite index in a (topologically) finitely generated profinite group is open. This implies that the topology in such a group is uniquely determined by the group structure. The result follows from a ‘uniformity theorem’ about finite groups: given a group word w that defines a locally finite variety and a natural number d, there exists f=fw(d) such that in every finite d-generator group G, each element of the verbal subgroup w(G) is a product of f w-values. Similar methods show that in a finite d-generator group, each element of the derived group is a product of g(d) commutators; this implies that the (abstract) derived group in any finitely generated profinite group is closed.
Accepté le :
Publié le :
Nikolay Nikolov 1 ; Dan Segal 2
@article{CRMATH_2003__337_5_303_0,
author = {Nikolay Nikolov and Dan Segal},
title = {Finite index subgroups in profinite groups},
journal = {Comptes Rendus. Math\'ematique},
pages = {303--308},
publisher = {Elsevier},
volume = {337},
number = {5},
year = {2003},
doi = {10.1016/S1631-073X(03)00349-2},
language = {en},
}
Nikolay Nikolov; Dan Segal. Finite index subgroups in profinite groups. Comptes Rendus. Mathématique, Volume 337 (2003) no. 5, pp. 303-308. doi : 10.1016/S1631-073X(03)00349-2. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00349-2/
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