Groups whose proper subgroups are locally finite-by-nilpotent

Dilmi, Amel

Annales mathématiques Blaise Pascal, Tome 14 (2007), p. 29-35 / Harvested from Numdam

Résumé

Si $\mathcal{X}$ est une classe de groupes, alors un groupe $G$ est dit minimal non $\mathcal{X}$-groupe si tous ses sous-groupes propres sont dans la classe $\mathcal{X}$, alors que $G$ lui-même n’est pas un $\mathcal{X}$-groupe. Le principal résultat de cette note affirme que si $c>0$ est un entier et si $G$ est un groupe minimal non $\mathcal{(LF)N}$ (respectivement, $\mathcal{(LF)N}_{c}$)-groupe, alors $G$ est un groupe parfait, de type fini, n’ayant pas de facteur fini non trivial et tel que $G/Frat(G)$ est un groupe simple infini ; où $\mathcal{N}$ (respectivement, $\mathcal{N}_{c}$, $\mathcal{LF}$) désigne la classe des groupes nilpotents (respectivement, nilpotents de classe égale au plus à $c$, localement finis) et $Frat(G)$ est le sous-groupe de Frattini de $G$.

Abstract

If $\mathcal{X}$ is a class of groups, then a group $G$ is said to be minimal non $\mathcal{X}$-group if all its proper subgroups are in the class $\mathcal{X}$, but $G$ itself is not an $\mathcal{X}$-group. The main result of this note is that if $c>0$ is an integer and if $G$ is a minimal non $\mathcal{(LF)N}$ (respectively, $\mathcal{(LF)N}_{c}$)-group, then $G$ is a finitely generated perfect group which has no non-trivial finite factor and such that $G/Frat(G)$ is an infinite simple group; where $\mathcal{N}$ (respectively, $\mathcal{N}_{c}$, $\mathcal{LF}$) denotes the class of nilpotent (respectively, nilpotent of class at most $c$, locally finite) groups and $Frat(G)$ stands for the Frattini subgroup of $G$.

Détail

Publié le : 2007-01-01

MR 2298722 | Zbl 1131.20023 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/ambp.225
Classification:  20F99

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