On minimal non-(torsion-by-nilpotent) and non-((locally finite)-by-nilpotent) groups

Abstract

Let Ω be a class of groups. A group is said to be minimal non- Ω if it is not an Ω-group, while all its proper subgroups belong to Ω. In this Note we prove that a minimal non-(torsion-by-nilpotent) (respectively, non-((locally finite)-by-nilpotent)) group G is a finitely generated perfect group which has no proper subgroup of finite index and such that G / Frat ( G ) is an infinite simple group, where Frat ( G ) stands for the Frattini subgroup of G. To cite this article: N. Trabelsi, C. R. Acad. Sci. Paris, Ser. I 344 (2007).