Groups whose proper subgroups are locally finite-by-nilpotent

Amel Dilmi[1]

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

Annales mathématiques Blaise Pascal (2007)

  • Volume: 14, Issue: 1, page 29-35
  • ISSN: 1259-1734

Abstract

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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 .

How to cite

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Dilmi, Amel. "Groups whose proper subgroups are locally finite-by-nilpotent." Annales mathématiques Blaise Pascal 14.1 (2007): 29-35. <http://eudml.org/doc/10539>.

@article{Dilmi2007,
abstract = {If $\mathcal\{X\}$ is a class of groups, then a group $G$ is said to be minimal non $\mathcal\{X\}$-group if all its proper subgroups are in the class $\mathcal\{X\}$, but $G$ itself is not an $\mathcal\{X\}$-group. The main result of this note is that if $c&gt;0$ is an integer and if $G$ is a minimal non $\mathcal\{(LF)N\}$ (respectively, $\mathcal\{(LF)N\}_\{c\}$)-group, then $G$ is a finitely generated perfect group which has no non-trivial finite factor and such that $G/Frat(G)$ is an infinite simple group; where $\mathcal\{N\}$ (respectively, $\mathcal\{N\}_\{c\}$, $\mathcal\{LF\}$) denotes the class of nilpotent (respectively, nilpotent of class at most $c$, locally finite) groups and $Frat(G)$ stands for the Frattini subgroup of $G$.},
affiliation = {Department of Mathematics Faculty of Sciences Ferhat Abbas University Setif 19000 ALGERIA},
author = {Dilmi, Amel},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Locally finite-by-nilpotent proper subgroups; Frattini factor group; locally finite-by-nilpotent groups; finitely generated perfect groups; infinite simple groups},
language = {eng},
month = {1},
number = {1},
pages = {29-35},
publisher = {Annales mathématiques Blaise Pascal},
title = {Groups whose proper subgroups are locally finite-by-nilpotent},
url = {http://eudml.org/doc/10539},
volume = {14},
year = {2007},
}

TY - JOUR
AU - Dilmi, Amel
TI - Groups whose proper subgroups are locally finite-by-nilpotent
JO - Annales mathématiques Blaise Pascal
DA - 2007/1//
PB - Annales mathématiques Blaise Pascal
VL - 14
IS - 1
SP - 29
EP - 35
AB - If $\mathcal{X}$ is a class of groups, then a group $G$ is said to be minimal non $\mathcal{X}$-group if all its proper subgroups are in the class $\mathcal{X}$, but $G$ itself is not an $\mathcal{X}$-group. The main result of this note is that if $c&gt;0$ is an integer and if $G$ is a minimal non $\mathcal{(LF)N}$ (respectively, $\mathcal{(LF)N}_{c}$)-group, then $G$ is a finitely generated perfect group which has no non-trivial finite factor and such that $G/Frat(G)$ is an infinite simple group; where $\mathcal{N}$ (respectively, $\mathcal{N}_{c}$, $\mathcal{LF}$) denotes the class of nilpotent (respectively, nilpotent of class at most $c$, locally finite) groups and $Frat(G)$ stands for the Frattini subgroup of $G$.
LA - eng
KW - Locally finite-by-nilpotent proper subgroups; Frattini factor group; locally finite-by-nilpotent groups; finitely generated perfect groups; infinite simple groups
UR - http://eudml.org/doc/10539
ER -

References

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  1. A.O. Asar, Nilpotent-by-Chernikov, J. London Math.Soc 61 (2000), 412-422 Zbl0961.20031MR1756802
  2. V.V. Belyaev, Groups of the Miller-Moreno type, Sibirsk. Mat. Z. 19 (1978), 509-514 Zbl0394.20025MR577067
  3. B. Bruno, R. E. Phillips, On minimal conditions related to Miller-Moreno type groups, Rend. Sem. Mat. Univ. Padova 69 (1983), 153-168 Zbl0522.20022MR716991
  4. G. Endimioni, G. Traustason, On Torsion-by-nilpotent groups, J. Algebra 241 (2001), 669-676 Zbl0984.20024MR1843318
  5. M. Kuzucuoglu, R. E. Phillips, Locally finite minimal non FC-groups, Math. Proc. Cambridge Philos. Soc. 105 (1989), 417-420 Zbl0686.20034MR985676
  6. M. F. Newman, J. Wiegold, Groups with many nilpotent subgroups, Arch. Math. 15 (1964), 241-250 Zbl0134.26102MR170949
  7. A. Y. Olshanski, An infinite simple torsion-free noetherian group, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 1328-1393 Zbl0431.20027MR567039
  8. J. Otal, J. M. Pena, Groups in which every proper subgroup is Cernikov-by-nilpotent or nilpotent-by-Cernikov, Arch.Math. 51 (1988), 193-197 Zbl0632.20018MR960393
  9. D. J. S. Robinson, Finiteness conditions and generalized soluble groups, (1972), Springer-Verlag Zbl0243.20032
  10. D. J. S. Robinson, A Course in the Theory of Groups, (1982), Springer-Verlag Zbl0483.20001MR648604
  11. H Smith, Groups with few non-nilpotent subgroups, Glasgow Math. J. 39 (1997), 141-151 Zbl0883.20018MR1460630
  12. M. Xu, Groups whose proper subgroups are Baer groups, Acta. Math. Sinica 40 (1996), 10-17 Zbl0840.20030MR1388572
  13. M. Xu, Groups whose proper subgroups are finite-by-nilpotent, Arch. Math. 66 (1996), 353-359 Zbl0857.20015MR1383898

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