Groups whose proper subgroups are locally finite-by-nilpotent
Annales mathématiques Blaise Pascal, Volume 14 (2007) no. 1, p. 29-35

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/Frat(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 Frat(G) stands for the Frattini subgroup of G.

Si 𝒳 est une classe de groupes, alors un groupe G est dit minimal non 𝒳-groupe si tous ses sous-groupes propres sont dans la classe 𝒳, alors que G lui-même n’est pas un 𝒳-groupe. Le principal résultat de cette note affirme que si c>0 est un entier et si G est un groupe minimal non (ℒℱ)𝒩 (respectivement, (ℒℱ)𝒩 c )-groupe, alors G est un groupe parfait, de type fini, n’ayant pas de facteur fini non trivial et tel que G/Frat(G) est un groupe simple infini ; où 𝒩 (respectivement, 𝒩 c , ℒℱ) désigne la classe des groupes nilpotents (respectivement, nilpotents de classe égale au plus à c, localement finis) et Frat(G) est le sous-groupe de Frattini de G.

DOI : https://doi.org/10.5802/ambp.225
Classification:  20F99
Keywords: Locally finite-by-nilpotent proper subgroups, Frattini factor group.
@article{AMBP_2007__14_1_29_0,
     author = {Dilmi, Amel},
     title = {Groups whose proper subgroups are locally finite-by-nilpotent},
     journal = {Annales math\'ematiques Blaise Pascal},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {14},
     number = {1},
     year = {2007},
     pages = {29-35},
     doi = {10.5802/ambp.225},
     mrnumber = {2298722},
     zbl = {1131.20023},
     language = {en},
     url = {http://www.numdam.org/item/AMBP_2007__14_1_29_0}
}
Dilmi, Amel. Groups whose proper subgroups are locally finite-by-nilpotent. Annales mathématiques Blaise Pascal, Volume 14 (2007) no. 1, pp. 29-35. doi : 10.5802/ambp.225. http://www.numdam.org/item/AMBP_2007__14_1_29_0/

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