Comptes Rendus
no. 6
Group Theory
On minimal non-(torsion-by-nilpotent) and non-((locally finite)-by-nilpotent) groups
[Sur les groupes minimaux non-(périodiques-par-nilpotents) et non-((localement finis)-par-nilpotents)]

Présenté par : Christophe Soulé

Nadir Trabelsi1
1 Department of Mathematics, Faculty of Sciences, University Ferhat Abbas, Setif 19000, Algeria
Comptes Rendus. Mathématique, Volume 344 (2007) no. 6, pp. 353-356.

Soit Ω une classe de groupes. Un groupe est dit minimal non-Ω s'il n'est pas un Ω-groupe alors que tous ses sous-groupes propres le sont. Dans cette Note, nous prouvons que si G est un groupe minimal non-(périodique-par-nilpotent) (respectivement, non-((localement fini)-par-nilpotent)), alors G est un groupe parfait de type fini qui n'admet pas de sous-groupe propre d'indice fini et tel que G/Frat(G) est un groupe simple infini, où Frat(G) désigne le sous-groupe de Frattini de G.

Let Ω be a class of groups. A group is said to be minimal non-Ω if it is not an Ω-group, while all its proper subgroups belong to Ω. In this Note we prove that a minimal non-(torsion-by-nilpotent) (respectively, non-((locally finite)-by-nilpotent)) group G is a finitely generated perfect group which has no proper subgroup of finite index and such that G/Frat(G) is an infinite simple group, where Frat(G) stands for the Frattini subgroup of G.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2007.02.009
Nadir Trabelsi 1

1 Department of Mathematics, Faculty of Sciences, University Ferhat Abbas, Setif 19000, Algeria 
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Nadir Trabelsi. On minimal non-(torsion-by-nilpotent) and non-((locally finite)-by-nilpotent) groups. Comptes Rendus. Mathématique, Volume 344 (2007) no. 6, pp. 353-356. doi : 10.1016/j.crma.2007.02.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.02.009/

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