Hyper–(Abelian–by–finite) groups with many subgroups of finite depth

[Groupes hyper-(Abelien-par-fini) ayant beaucoup de sous-groupes de profondeur finie]

Gherbi, Fares ; Rouabhi, Tarek

Annales mathématiques Blaise Pascal, Tome 14 (2007), p. 17-28 / Harvested from Numdam

Résumé

Le principal résultat de cet article est qu’un groupe $G$ hyper-(Abélien-par-fini) de type fini est fini-par-nilpotent si, et seulement si, toute partie infinie de $G$ contient deux éléments distincts $x,y$ tels que $\gamma _{n}(\left\langle x,x^{y}\right\rangle )=\gamma _{n+1}(\left\langle x,x^{y}\right\rangle )$ pour un certain entier positif $n=n(x,y)$ (respectivement, $\left\langle x,x^{y}\right\rangle $ est une extension d’un groupe vérifiant la condition minimale sur les sous-groupes normaux par un groupe d’Engel).

Abstract

The main result of this note is that a finitely generated hyper-(Abelian-by-finite) group $G$ is finite-by-nilpotent if and only if every infinite subset contains two distinct elements $x$, $y$ such that $\gamma _{n}(\left\langle x\text{, }x^{y}\right\rangle )$ $=\gamma _{n+1}(\left\langle x\text{, }x^{y}\right\rangle )$ for some positive integer $n=n(x,y)$ (respectively, $\left\langle x,x^{y}\right\rangle $ is an extension of a group satisfying the minimal condition on normal subgroups by an Engel group).

Détail

Publié le : 2007-01-01

Zbl 1131.20024

DOI : https://doi.org/10.5802/ambp.224
Classification:  20F22,  20F99
Mots clés: Parties infinies, profondeur finie, Les groupes d’Engel, La condition minimale sur les sous-groupes normaux, les groupes fini-par-nilpotents, les groupes hyper-(Abelien-par-fini) de type fini

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