Trabelsi, Nadir On minimal non-(torsion-by-nilpotent) and non-((locally finite)-by-nilpotent) groups. (English. Abridged French version) Zbl 1113.20032 C. R., Math., Acad. Sci. Paris 344, No. 6, 353-356 (2007). 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 \(G\). Cited in 4 Documents 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 Keywords:minimal non-(torsion-by-nilpotent) groups; non-((locally finite)-by-nilpotent) groups; finitely generated perfect groups; subgroups of finite index; infinite simple groups; Frattini subgroup PDFBibTeX XMLCite \textit{N. Trabelsi}, C. R., Math., Acad. Sci. Paris 344, No. 6, 353--356 (2007; Zbl 1113.20032) Full Text: DOI References: [1] Asar, A. O., Nilpotent-by-Chernikov, J. London Math. Soc., 61, 412-422 (2000) · Zbl 0961.20031 [2] Bruno, B.; Phillips, R. E., On minimal conditions related to Miller-Moreno type groups, Rend. Sem. Mat. Univ. Padova, 69, 153-168 (1983) · Zbl 0522.20022 [3] Franciosi, S.; De Giovanni, F.; Sysak, Y. P., Groups with many polycyclic-by-nilpotent subgroups, Ricerche Mat., 48, 361-378 (1999) · Zbl 0980.20023 [4] Newman, M. F.; Wiegold, J., Groups with many nilpotent subgroups, Arch. Math., 15, 241-250 (1964) · Zbl 0134.26102 [5] Ol’shanskii, A. Y., An infinite simple torsion-free Noetherian group, Izv. Akad. Nauk SSSR Ser. Mat., 43, 1328-1393 (1979) · Zbl 0431.20027 [6] Otal, J.; Pena, J. M., Minimal non-CC groups, Comm. Algebra, 16, 1231-1242 (1988) · Zbl 0644.20025 [7] Robinson, D. J.S., Finiteness Conditions and Generalized Soluble Groups (1972), Springer-Verlag · Zbl 0243.20032 [8] Smith, H., Groups with few non-nilpotent subgroups, Glasgow Math. J., 39, 141-151 (1997) · Zbl 0883.20018 [9] N. Trabelsi, Locally graded groups with few non-(torsion-by-nilpotent) subgroups, Ischia Group Theory 2006, World Sci. Publ., in press; N. Trabelsi, Locally graded groups with few non-(torsion-by-nilpotent) subgroups, Ischia Group Theory 2006, World Sci. Publ., in press [10] Xu, M., Groups whose proper subgroups are finite-by-nilpotent, Arch. Math., 66, 353-359 (1996) · Zbl 0857.20015 [11] Zaicev, D. I., Stably nilpotent groups, Mat. Zametki, 2, 337-346 (1967) · Zbl 0167.28802 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.