# 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

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topGherbi, 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 -

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