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

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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 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

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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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