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On minimal non-(torsion-by-nilpotent) and non-((locally finite)-by-nilpotent) groups. (English. Abridged French version) Zbl 1113.20032

Summary: Let \(\Omega\) be a class of groups. A group is said to be minimal non-\(\Omega\) if it is not an \(\Omega\)-group, while all its proper subgroups belong to \(\Omega\). 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 \(G/\text{Frat}(G)\) is an infinite simple group, where \(\text{Frat}(G)\) stands for the Frattini subgroup of \(G\).

MSC:

20F19 Generalizations of solvable and nilpotent groups
20E15 Chains and lattices of subgroups, subnormal subgroups
20E07 Subgroup theorems; subgroup growth
20E32 Simple groups
20E28 Maximal subgroups
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