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

Annales mathématiques Blaise Pascal (2007)

  • Volume: 14, Issue: 1, page 17-28
  • ISSN: 1259-1734

Abstract

top
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 γ n ( x , x y ) = γ n + 1 ( x , x y ) for some positive integer n = n ( x , y ) (respectively, x , x y is an extension of a group satisfying the minimal condition on normal subgroups by an Engel group).

How to cite

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.