Comptes Rendus
no. 6
Group Theory
On minimal non-(torsion-by-nilpotent) and non-((locally finite)-by-nilpotent) groups

Presented by: 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.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.02.009

Nadir Trabelsi 1

1 Department of Mathematics, Faculty of Sciences, University Ferhat Abbas, Setif 19000, Algeria 
@article{CRMATH_2007__344_6_353_0,
     author = {Nadir Trabelsi},
     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},
}
TY  - JOUR
AU  - Nadir Trabelsi
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
DO  - 10.1016/j.crma.2007.02.009
LA  - en
ID  - CRMATH_2007__344_6_353_0
ER  - 
%0 Journal Article
%A Nadir Trabelsi
%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
%R 10.1016/j.crma.2007.02.009
%G en
%F CRMATH_2007__344_6_353_0
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/

[1] A.O. Asar Nilpotent-by-Chernikov, J. London Math. Soc., Volume 61 (2000), pp. 412-422

[2] B. Bruno; R.E. Phillips On minimal conditions related to Miller–Moreno type groups, Rend. Sem. Mat. Univ. Padova, Volume 69 (1983), pp. 153-168

[3] S. Franciosi; F. De Giovanni; Y.P. Sysak Groups with many polycyclic-by-nilpotent subgroups, Ricerche Mat., Volume 48 (1999), pp. 361-378

[4] M.F. Newman; J. Wiegold Groups with many nilpotent subgroups, Arch. Math., Volume 15 (1964), pp. 241-250

[5] A.Y. Ol'shanskii An infinite simple torsion-free Noetherian group, Izv. Akad. Nauk SSSR Ser. Mat., Volume 43 (1979), pp. 1328-1393

[6] J. Otal; J.M. Pena Minimal non-CC groups, Comm. Algebra, Volume 16 (1988), pp. 1231-1242

[7] D.J.S. Robinson Finiteness Conditions and Generalized Soluble Groups, Springer-Verlag, 1972

[8] H. Smith Groups with few non-nilpotent subgroups, Glasgow Math. J., Volume 39 (1997), pp. 141-151

[9] N. Trabelsi, Locally graded groups with few non-(torsion-by-nilpotent) subgroups, Ischia Group Theory 2006, World Sci. Publ., in press

[10] M. Xu Groups whose proper subgroups are finite-by-nilpotent, Arch. Math., Volume 66 (1996), pp. 353-359

[11] D.I. Zaicev Stably nilpotent groups, Mat. Zametki, Volume 2 (1967), pp. 337-346

Cited by Sources:

Comments - Policy