Groups whose proper subgroups are locally finite-by-nilpotent

Amel Dilmi

doi:10.5802/ambp.225

Amel Dilmi ¹

¹ Department of Mathematics Faculty of Sciences Ferhat Abbas University Setif 19000 ALGERIA

Annales mathématiques Blaise Pascal, Tome 14 (2007) no. 1, pp. 29-35.

Résumé
Abstract

Si $𝒳$ est une classe de groupes, alors un groupe $G$ est dit minimal non $𝒳$ -groupe si tous ses sous-groupes propres sont dans la classe $𝒳$ , alors que $G$ lui-même n’est pas un $𝒳$ -groupe. Le principal résultat de cette note affirme que si $c > 0$ est un entier et si $G$ est un groupe minimal non $(ℒℱ) 𝒩$ (respectivement, $(ℒℱ) 𝒩_{c}$ )-groupe, alors $G$ est un groupe parfait, de type fini, n’ayant pas de facteur fini non trivial et tel que $G / F r a t (G)$ est un groupe simple infini ; où $𝒩$ (respectivement, $𝒩_{c}$ , $ℒℱ$ ) désigne la classe des groupes nilpotents (respectivement, nilpotents de classe égale au plus à $c$ , localement finis) et $F r a t (G)$ est le sous-groupe de Frattini de $G$ .

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

MR Zbl

DOI : 10.5802/ambp.225

Classification : 20F99
Mots clés : Locally finite-by-nilpotent proper subgroups, Frattini factor group.

Affiliations des auteurs :

Amel Dilmi ¹

¹ Department of Mathematics Faculty of Sciences Ferhat Abbas University Setif 19000 ALGERIA

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Amel Dilmi. Groups whose proper subgroups are locally finite-by-nilpotent. Annales mathématiques Blaise Pascal, Tome 14 (2007) no. 1, pp. 29-35. doi : 10.5802/ambp.225. https://ambp.centre-mersenne.org/articles/10.5802/ambp.225/

Bibliographie
Cité par

[1] A.O. Asar Nilpotent-by-Chernikov, J. London Math.Soc, Volume 61 (2000) no. 2, pp. 412-422 | DOI | MR | Zbl

[2] V.V. Belyaev Groups of the Miller-Moreno type, Sibirsk. Mat. Z., Volume 19 (1978) no. 3, pp. 509-514 | MR | Zbl

[3] 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 | Numdam | MR | Zbl

[4] G. Endimioni; G. Traustason On Torsion-by-nilpotent groups, J. Algebra, Volume 241 (2001) no. 2, pp. 669-676 | DOI | MR | Zbl

[5] M. Kuzucuoglu; R. E. Phillips Locally finite minimal non FC-groups, Math. Proc. Cambridge Philos. Soc., Volume 105 (1989), pp. 417-420 | DOI | MR | Zbl

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

[7] A. Y. Olshanski An infinite simple torsion-free noetherian group, Izv. Akad. Nauk SSSR Ser. Mat., Volume 43 (1979), pp. 1328-1393 | MR | Zbl

[8] J. Otal; J. M. Pena Groups in which every proper subgroup is Cernikov-by-nilpotent or nilpotent-by-Cernikov, Arch.Math., Volume 51 (1988), pp. 193-197 | DOI | MR | Zbl

[9] D. J. S. Robinson Finiteness conditions and generalized soluble groups, Springer-Verlag, 1972

[10] D. J. S. Robinson A Course in the Theory of Groups, Springer-Verlag, 1982 | MR | Zbl

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

[12] M. Xu Groups whose proper subgroups are Baer groups, Acta. Math. Sinica, Volume 40 (1996), pp. 10-17 | MR | Zbl

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

Cité par Sources :