# Hyper–(Abelian–by–finite) groups with many subgroups of finite depth

Fares Gherbi[1]; Tarek Rouabhi[1]

• [1] Department of Mathematics Faculty of Sciences Ferhat Abbas University Setif, 19000 ALGERIA
• Volume: 14, Issue: 1, page 17-28
• ISSN: 1259-1734

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## Abstract

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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).

## How to cite

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Gherbi, Fares, and Rouabhi, Tarek. "Hyper–(Abelian–by–finite) groups with many subgroups of finite depth." Annales mathématiques Blaise Pascal 14.1 (2007): 17-28. <http://eudml.org/doc/10537>.

@article{Gherbi2007,
abstract = {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\langle x\text\{, \}x^\{y\}\right\rangle )$$=\gamma _\{n+1\}(\left\langle x\text\{, \}x^\{y\}\right\rangle ) for some positive integer n=n(x,y) (respectively, \left\langle x,x^\{y\}\right\rangle is an extension of a group satisfying the minimal condition on normal subgroups by an Engel group).}, affiliation = {Department of Mathematics Faculty of Sciences Ferhat Abbas University Setif, 19000 ALGERIA; Department of Mathematics Faculty of Sciences Ferhat Abbas University Setif, 19000 ALGERIA}, author = {Gherbi, Fares, Rouabhi, Tarek}, journal = {Annales mathématiques Blaise Pascal}, keywords = {Infinite subsets; finite depth; Engel groups; minimal condition on normal subgroups; finite-by-nilpotent groups; finitely generated hyper-(Abelian-by-finite) groups; hyper-Abelian-by-finite groups; Chernikov groups; combinatorial conditions on infinite subsets; groups of finite depth; finitely generated groups; lower central series}, language = {eng}, month = {1}, number = {1}, pages = {17-28}, publisher = {Annales mathématiques Blaise Pascal}, title = {Hyper–(Abelian–by–finite) groups with many subgroups of finite depth}, url = {http://eudml.org/doc/10537}, volume = {14}, year = {2007}, } TY - JOUR AU - Gherbi, Fares AU - Rouabhi, Tarek TI - Hyper–(Abelian–by–finite) groups with many subgroups of finite depth JO - Annales mathématiques Blaise Pascal DA - 2007/1// PB - Annales mathématiques Blaise Pascal VL - 14 IS - 1 SP - 17 EP - 28 AB - The main result of this note is that a finitely generated hyper-(Abelian-by-finite) group G is finite-by-nilpotent if and only if every infinite subset contains two distinct elements x, y such that \gamma _{n}(\left\langle x\text{, }x^{y}\right\rangle )$$=\gamma _{n+1}(\left\langle x\text{, }x^{y}\right\rangle )$ for some positive integer $n=n(x,y)$ (respectively, $\left\langle x,x^{y}\right\rangle$ is an extension of a group satisfying the minimal condition on normal subgroups by an Engel group).
LA - eng
KW - Infinite subsets; finite depth; Engel groups; minimal condition on normal subgroups; finite-by-nilpotent groups; finitely generated hyper-(Abelian-by-finite) groups; hyper-Abelian-by-finite groups; Chernikov groups; combinatorial conditions on infinite subsets; groups of finite depth; finitely generated groups; lower central series
UR - http://eudml.org/doc/10537
ER -

## References

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1. A. Abdollahi, Finitely generated soluble groups with an Engel condition on infinite subsets, Rend. Sem. Mat. Univ. Padova 103 (2000), 47-49 Zbl0966.20019MR1789531
2. A. Abdollahi, Some Engel conditions on infinite subsets of certain groups, Bull. Austral. Math. Soc. 62 (2000), 141-148 Zbl0964.20019MR1775895
3. A. Abdollahi, B. Taeri, A condition on finitely generated soluble groups, Comm. Algebra 27 (1999), 5633-5638 Zbl0942.20014MR1713058
4. A. Abdollahi, N. Trabelsi, Quelques extensions d’un problème de Paul Erdos sur les groupes, Bull. Belg. Math. Soc. 9 (2002), 205-215 Zbl1041.20022MR2017077
5. A. Boukaroura, Characterisation of finitely generated finite-by-nilpotent groups, Rend. Sem. Mat. Univ. Padova 111 (2004), 119-126 Zbl1119.20039MR2076735
6. C. Delizia, A. H. Rhemtulla, H. Smith, Locally graded groups with a nilpotence condition on infinite subsets, J. Austral. Math. Soc. (series A) 69 (2000), 415-420 Zbl0982.20019MR1793472
7. G. Endimioni, Groups covered by finitely many nilpotent subgroups, Bull. Austral. Math. Soc. 50 (1994), 459-464 Zbl0824.20034MR1303902
8. G. Endimioni, Groups in which certain equations have many solutions, Rend. Sem. Mat. Univ. Padova 106 (2001), 77-82 Zbl1072.20035MR1876214
9. E. S. Golod, Some problems of Burnside type, Amer. Math. Soc. Transl. Ser. 2 84 (1969), 83-88 Zbl0206.32402MR238880
10. P. Hall, Finite-by-nilpotent groups, Proc. Cambridge Philos. Soc. 52 (1956), 611-616 Zbl0072.25801MR80095
11. J. C. Lennox, Finitely generated soluble groups in which all subgroups have finite lower central depth, Bull. London Math. Soc. 7 (1975), 273-278 Zbl0314.20029MR382448
12. J. C. Lennox, Lower central depth in finitely generated soluble-by-finite groups, Glasgow Math. J. 19 (1978), 153-154 Zbl0394.20027MR486159
13. J. C. Lennox, J. Wiegold, Extensions of a problem of Paul Erdos on groups, J. Austral. Math. Soc. Ser. A 31 (1981), 459-463 Zbl0492.20019MR638274
14. P. Longobardi, On locally graded groups with an Engel condition on infinite subsets, Arch. Math. 76 (2001), 88-90 Zbl0981.20027MR1811284
15. P. Longobardi, M. Maj, Finitely generated soluble groups with an Engel condition on infinite subsets, Rend. Sem. Mat. Univ. Padova 89 (1993), 97-102 Zbl0797.20031MR1229046
16. B. H. Neumann, A problem of Paul Erdos on groups, J. Austral. Math. Soc. ser. A 21 (1976), 467-472 Zbl0333.05110MR419283
17. D. J. S. Robinson, Finiteness conditions and generalized soluble groups, (1972), Springer-Verlag, Berlin, Heidelberg, New York Zbl0243.20032
18. D. J. S. Robinson, A course in the theory of groups, (1982), Springer-Verlag, Berlin, Heidelberg, New York Zbl0483.20001MR648604
19. D. Segal, A residual property of finitely generated abelian by nilpotent groups, J. Algebra 32 (1974), 389-399 Zbl0293.20029MR419612
20. D. Segal, Polycyclic groups, (1984), Cambridge University Press, Cambridge, London, New York, New Rochelle, Melbourne, Sydney Zbl0516.20001MR713786
21. B. Taeri, A question of P. Erdos and nilpotent-by-finite groups, Bull. Austral. Math. Soc. 64 (2001), 245-254 Zbl0995.20020MR1860061
22. N. Trabelsi, Finitely generated soluble groups with a condition on infinite subsets, Algebra Colloq. 9 (2002), 427-432 Zbl1035.20030MR1933851
23. N. Trabelsi, Soluble groups with many 2-generator torsion-by-nilpotent subgroups, Publ. Math. Debrecen 67/1-2 (2005), 93-102 Zbl1094.20018MR2163117

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