Comptes Rendus
Algebra/Group theory
Metric ultraproducts of finite simple groups
Comptes Rendus. Mathématique, Volume 352 (2014) no. 6, pp. 463-466.

Some new results on metric ultraproducts of finite simple groups are presented. Suppose that G is such a group, defined in terms of a non-principal ultrafilter ω on N and a sequence (Gi)iN of finite simple groups, and that G is neither finite nor a Chevalley group over an infinite field. Then G is isomorphic to an ultraproduct of alternating groups or to an ultraproduct of finite simple classical groups. The isomorphism type of G determines which of these two cases arises, and, in the latter case, the ω-limit of the characteristics of the groups Gi. Moreover, G is a complete path-connected group with respect to the natural metric on G.

Nous présentons de nouveaux résultats relatifs aux ultraproduits des groupes finis simples. Soit G un tel groupe, associé à un ultrafiltre ω sur N et une suite (Gi)iN de groupes finis simples, et supposons que G n'est ni fini ni un groupe de Chevalley sur un corps infini. Un tel groupe G est alors isomorphe, soit à un ultraproduit de groupes alternés, soit à un ultraproduit de groupes finis simples classiques. La classe d'isomorphisme de G nous permet de distinguer ces deux cas et, dans le second cas, de déterminer la ω-limite des charactéristiques des groupes Gi. Le groupe G est, de plus, complet et connexe par arcs pour la métrique naturelle sur G.

Published online:
DOI: 10.1016/j.crma.2014.03.015

Andreas Thom 1; John S. Wilson 2

1 Mathematisches Institut, U Leipzig, PF 100920, 04009 Leipzig, Germany
2 Mathematical Institute, Radcliffe Observatory Quarter, Oxford OX2 6GG, England, United Kingdom
     author = {Andreas Thom and John S. Wilson},
     title = {Metric ultraproducts of finite simple groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {463--466},
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     number = {6},
     year = {2014},
     doi = {10.1016/j.crma.2014.03.015},
     language = {en},
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%A John S. Wilson
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Andreas Thom; John S. Wilson. Metric ultraproducts of finite simple groups. Comptes Rendus. Mathématique, Volume 352 (2014) no. 6, pp. 463-466. doi : 10.1016/j.crma.2014.03.015.

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