Groups whose proper subgroups are locally finite-by-nilpotent

Dilmi, Amel

Annales mathématiques Blaise Pascal, Tome 14 (2007) no. 1, pp. 29-35.

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 $\left(\mathrm{ℒℱ}\right)𝒩$ (respectivement, $\left(\mathrm{ℒℱ}\right){𝒩}_c$)-groupe, alors $G$ est un groupe parfait, de type fini, n’ayant pas de facteur fini non trivial et tel que $G/Frat\left(G\right)$ est un groupe simple infini ; où $𝒩$ (respectivement, ${𝒩}_c$, $\mathrm{ℒℱ}$) désigne la classe des groupes nilpotents (respectivement, nilpotents de classe égale au plus à $c$, localement finis) et $Frat\left(G\right)$ est le sous-groupe de Frattini de $G$.

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 $\left(\mathrm{ℒℱ}\right)𝒩$ (respectively, $\left(\mathrm{ℒℱ}\right){𝒩}_c$)-group, then $G$ is a finitely generated perfect group which has no non-trivial finite factor and such that $G/Frat\left(G\right)$ is an infinite simple group; where $𝒩$ (respectively, ${𝒩}_c$, $\mathrm{ℒℱ}$) denotes the class of nilpotent (respectively, nilpotent of class at most $c$, locally finite) groups and $Frat\left(G\right)$ stands for the Frattini subgroup of $G$.

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