Group Theory
On minimal non-(torsion-by-nilpotent) and non-((locally finite)-by-nilpotent) groups
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.

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

1 Department of Mathematics, Faculty of Sciences, University Ferhat Abbas, Setif 19000, Algeria
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Trabelsi, Nadir. 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. http://www.numdam.org/articles/10.1016/j.crma.2007.02.009/

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

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

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

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

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

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

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

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

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

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