Hyper–(Abelian–by–finite) groups with many subgroups of finite depth
Annales mathématiques Blaise Pascal, Volume 14 (2007) no. 1, p. 17-28

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).

Le principal résultat de cet article est qu’un groupe G hyper-(Abélien-par-fini) de type fini est fini-par-nilpotent si, et seulement si, toute partie infinie de G contient deux éléments distincts x,y tels que γ n (x,x y )=γ n+1 (x,x y ) pour un certain entier positif n=n(x,y) (respectivement, x,x y est une extension d’un groupe vérifiant la condition minimale sur les sous-groupes normaux par un groupe d’Engel).

DOI : https://doi.org/10.5802/ambp.224
Classification:  20F22,  20F99
Keywords: Infinite subsets, finite depth, Engel groups, minimal condition on normal subgroups, finite-by-nilpotent groups, finitely generated hyper-(Abelian-by-finite) groups
@article{AMBP_2007__14_1_17_0,
     author = {Gherbi, Fares and Rouabhi, Tarek},
     title = {Hyper--(Abelian--by--finite) groups with many subgroups of finite depth},
     journal = {Annales math\'ematiques Blaise Pascal},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {14},
     number = {1},
     year = {2007},
     pages = {17-28},
     doi = {10.5802/ambp.224},
     zbl = {1131.20024},
     language = {en},
     url = {http://www.numdam.org/item/AMBP_2007__14_1_17_0}
}
Gherbi, Fares; Rouabhi, Tarek. Hyper–(Abelian–by–finite) groups with many subgroups of finite depth. Annales mathématiques Blaise Pascal, Volume 14 (2007) no. 1, pp. 17-28. doi : 10.5802/ambp.224. http://www.numdam.org/item/AMBP_2007__14_1_17_0/

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