Hyper–(Abelian–by–finite) groups with many subgroups of finite depth
Annales mathématiques Blaise Pascal, Volume 14 (2007) no. 1, p. 17-28

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 ${\gamma }_{n}\left(〈x\text{,}\phantom{\rule{4pt}{0ex}}{x}^{y}〉\right)$ $={\gamma }_{n+1}\left(〈x\text{,}\phantom{\rule{4pt}{0ex}}{x}^{y}〉\right)$ for some positive integer $n=n\left(x,y\right)$ (respectively, $〈x,{x}^{y}〉$ is an extension of a group satisfying the minimal condition on normal subgroups by an Engel group).

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 ${\gamma }_{n}\left(〈x,{x}^{y}〉\right)={\gamma }_{n+1}\left(〈x,{x}^{y}〉\right)$ pour un certain entier positif $n=n\left(x,y\right)$ (respectivement, $〈x,{x}^{y}〉$ est une extension d’un groupe vérifiant la condition minimale sur les sous-groupes normaux par un groupe d’Engel).

DOI : https://doi.org/10.5802/ambp.224
Classification:  20F22,  20F99
Keywords: Infinite subsets, finite depth, Engel groups, minimal condition on normal subgroups, finite-by-nilpotent groups, finitely generated hyper-(Abelian-by-finite) groups
@article{AMBP_2007__14_1_17_0,
author = {Gherbi, Fares and Rouabhi, Tarek},
title = {Hyper--(Abelian--by--finite) groups with many subgroups of finite depth},
journal = {Annales math\'ematiques Blaise Pascal},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {14},
number = {1},
year = {2007},
pages = {17-28},
doi = {10.5802/ambp.224},
zbl = {1131.20024},
language = {en},
url = {http://www.numdam.org/item/AMBP_2007__14_1_17_0}
}

Gherbi, Fares; Rouabhi, Tarek. Hyper–(Abelian–by–finite) groups with many subgroups of finite depth. Annales mathématiques Blaise Pascal, Volume 14 (2007) no. 1, pp. 17-28. doi : 10.5802/ambp.224. http://www.numdam.org/item/AMBP_2007__14_1_17_0/

 Abdollahi, A. Finitely generated soluble groups with an Engel condition on infinite subsets, Rend. Sem. Mat. Univ. Padova, Tome 103 (2000), pp. 47-49 | Numdam | MR 1789531 | Zbl 0966.20019

 Abdollahi, A. Some Engel conditions on infinite subsets of certain groups, Bull. Austral. Math. Soc., Tome 62 (2000), pp. 141-148 | Article | MR 1775895 | Zbl 0964.20019

 Abdollahi, A.; Taeri, B. A condition on finitely generated soluble groups, Comm. Algebra, Tome 27 (1999), pp. 5633-5638 | Article | MR 1713058 | Zbl 0942.20014

 Abdollahi, A.; Trabelsi, N. Quelques extensions d’un problème de Paul Erdos sur les groupes, Bull. Belg. Math. Soc., Tome 9 (2002), pp. 205-215 | MR 2017077 | Zbl 1041.20022

 Boukaroura, A. Characterisation of finitely generated finite-by-nilpotent groups, Rend. Sem. Mat. Univ. Padova, Tome 111 (2004), pp. 119-126 | Numdam | MR 2076735 | Zbl 05058721

 Delizia, C.; Rhemtulla, A. H.; Smith, H. Locally graded groups with a nilpotence condition on infinite subsets, J. Austral. Math. Soc. (series A), Tome 69 (2000), pp. 415-420 | Article | MR 1793472 | Zbl 0982.20019

 Endimioni, G. Groups covered by finitely many nilpotent subgroups, Bull. Austral. Math. Soc., Tome 50 (1994), pp. 459-464 | Article | MR 1303902 | Zbl 0824.20034

 Endimioni, G. Groups in which certain equations have many solutions, Rend. Sem. Mat. Univ. Padova, Tome 106 (2001), pp. 77-82 | Numdam | MR 1876214 | Zbl 1072.20035

 Golod, E. S. Some problems of Burnside type, Amer. Math. Soc. Transl. Ser. 2, Tome 84 (1969), pp. 83-88 | MR 238880 | Zbl 0206.32402

 Hall, P. Finite-by-nilpotent groups, Proc. Cambridge Philos. Soc., Tome 52 (1956), pp. 611-616 | Article | MR 80095 | Zbl 0072.25801

 Lennox, J. C. Finitely generated soluble groups in which all subgroups have finite lower central depth, Bull. London Math. Soc., Tome 7 (1975), pp. 273-278 | Article | MR 382448 | Zbl 0314.20029

 Lennox, J. C. Lower central depth in finitely generated soluble-by-finite groups, Glasgow Math. J., Tome 19 (1978), p. 153-154 | Article | MR 486159 | Zbl 0394.20027

 Lennox, J. C.; Wiegold, J. Extensions of a problem of Paul Erdos on groups, J. Austral. Math. Soc. Ser. A, Tome 31 (1981), pp. 459-463 | Article | MR 638274 | Zbl 0492.20019

 Longobardi, P. On locally graded groups with an Engel condition on infinite subsets, Arch. Math., Tome 76 (2001), pp. 88-90 | Article | MR 1811284 | Zbl 0981.20027

 Longobardi, P.; Maj, M. Finitely generated soluble groups with an Engel condition on infinite subsets, Rend. Sem. Mat. Univ. Padova, Tome 89 (1993), pp. 97-102 | Numdam | MR 1229046 | Zbl 0797.20031

 Neumann, B. H. A problem of Paul Erdos on groups, J. Austral. Math. Soc. ser. A, Tome 21 (1976), pp. 467-472 | Article | MR 419283 | Zbl 0333.05110

 Robinson, D. J. S. Finiteness conditions and generalized soluble groups, Springer-Verlag, Berlin, Heidelberg, New York (1972) | Zbl 0243.20032

 Robinson, D. J. S. A course in the theory of groups, Springer-Verlag, Berlin, Heidelberg, New York (1982) | MR 648604 | Zbl 0483.20001

 Segal, D. A residual property of finitely generated abelian by nilpotent groups, J. Algebra, Tome 32 (1974), pp. 389-399 | Article | MR 419612 | Zbl 0293.20029

 Segal, D. Polycyclic groups, Cambridge University Press, Cambridge, London, New York, New Rochelle, Melbourne, Sydney (1984) | MR 713786 | Zbl 0516.20001

 Taeri, B. A question of P. Erdos and nilpotent-by-finite groups, Bull. Austral. Math. Soc., Tome 64 (2001), pp. 245-254 | Article | MR 1860061 | Zbl 0995.20020

 Trabelsi, N. Finitely generated soluble groups with a condition on infinite subsets, Algebra Colloq., Tome 9 (2002), pp. 427-432 | MR 1933851 | Zbl 1035.20030

 Trabelsi, N. Soluble groups with many 2-generator torsion-by-nilpotent subgroups, Publ. Math. Debrecen, Tome 67/1-2 (2005), pp. 93-102 | MR 2163117 | Zbl 02201546