[Sur les groupes minimaux non-(périodiques-par-nilpotents) et non-((localement finis)-par-nilpotents)]
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 est un groupe simple infini, où 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 is an infinite simple group, where stands for the Frattini subgroup of G.
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@article{CRMATH_2007__344_6_353_0,
author = {Trabelsi, Nadir},
title = {On minimal non-(torsion-by-nilpotent) and non-((locally finite)-by-nilpotent) groups},
journal = {Comptes Rendus. Math\'ematique},
pages = {353--356},
publisher = {Elsevier},
volume = {344},
number = {6},
year = {2007},
doi = {10.1016/j.crma.2007.02.009},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2007.02.009/}
}
TY - JOUR AU - Trabelsi, Nadir TI - On minimal non-(torsion-by-nilpotent) and non-((locally finite)-by-nilpotent) groups JO - Comptes Rendus. Mathématique PY - 2007 SP - 353 EP - 356 VL - 344 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2007.02.009/ DO - 10.1016/j.crma.2007.02.009 LA - en ID - CRMATH_2007__344_6_353_0 ER -
%0 Journal Article %A Trabelsi, Nadir %T On minimal non-(torsion-by-nilpotent) and non-((locally finite)-by-nilpotent) groups %J Comptes Rendus. Mathématique %D 2007 %P 353-356 %V 344 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2007.02.009/ %R 10.1016/j.crma.2007.02.009 %G en %F CRMATH_2007__344_6_353_0
Trabelsi, Nadir. On minimal non-(torsion-by-nilpotent) and non-((locally finite)-by-nilpotent) groups. Comptes Rendus. Mathématique, Tome 344 (2007) no. 6, pp. 353-356. doi : 10.1016/j.crma.2007.02.009. http://www.numdam.org/articles/10.1016/j.crma.2007.02.009/
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