Dilmi, Amel Groups whose proper subgroups are locally finite-by-nilpotent. (English) Zbl 1131.20023 Ann. Math. Blaise Pascal 14, No. 1, 29-35 (2007). Let \(\mathcal X\) be a class of groups. If a group \(G\) does not belong to \(\mathcal X\), but every of its proper subgroups belongs to \(\mathcal X\), then \(G\) is said to be a minimal non-\(\mathcal X\)-group. One of the two main results of the article yields that if \(G\) is a minimal non-(locally finite-by-nilpotent)-group, then \(G\) is a finitely generated perfect group which has no non-trivial finite factor and its factor-group by the Frattini subgroup is an infinite simple group (Theorem 1). Theorem 2 is similar concerning minimal non-(locally finite-by-nilpotent of class at most \(c\))-groups. Reviewer: Igor Subbotin (Los Angeles) Cited in 4 Documents MSC: 20F19 Generalizations of solvable and nilpotent groups 20E25 Local properties of groups 20E32 Simple groups Keywords:locally finite-by-nilpotent groups; Frattini factor group; finitely generated perfect groups; infinite simple groups PDFBibTeX XMLCite \textit{A. Dilmi}, Ann. Math. Blaise Pascal 14, No. 1, 29--35 (2007; Zbl 1131.20023) Full Text: DOI Numdam EuDML References: [1] Asar, A. O., Nilpotent-by-Chernikov, J. London Math.Soc, 61, 2, 412-422 (2000) · Zbl 0961.20031 · doi:10.1112/S0024610799008479 [2] Belyaev, V. V., Groups of the Miller-Moreno type, Sibirsk. Mat. Z., 19, 3, 509-514 (1978) · Zbl 0394.20025 [3] 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 [4] Endimioni, G.; Traustason, G., On Torsion-by-nilpotent groups, J. Algebra, 241, 2, 669-676 (2001) · Zbl 0984.20024 · doi:10.1006/jabr.2001.8772 [5] Kuzucuoglu, M.; Phillips, R. E., Locally finite minimal non FC-groups, Math. Proc. Cambridge Philos. Soc., 105, 417-420 (1989) · Zbl 0686.20034 · doi:10.1017/S030500410007777X [6] Newman, M. F.; Wiegold, J., Groups with many nilpotent subgroups, Arch. Math., 15, 241-250 (1964) · Zbl 0134.26102 · doi:10.1007/BF01589192 [7] Olshanski, A. Y., An infinite simple torsion-free noetherian group, Izv. Akad. Nauk SSSR Ser. Mat., 43, 1328-1393 (1979) · Zbl 0431.20027 [8] Otal, J.; Pena, J. M., Groups in which every proper subgroup is Cernikov-by-nilpotent or nilpotent-by-Cernikov, Arch.Math., 51, 193-197 (1988) · Zbl 0632.20018 · doi:10.1007/BF01207469 [9] Robinson, D. J. S., Finiteness conditions and generalized soluble groups (1972) · Zbl 0243.20032 [10] Robinson, D. J. S., A Course in the Theory of Groups (1982) · Zbl 0483.20001 [11] Smith, H., Groups with few non-nilpotent subgroups, Glasgow Math. J., 39, 141-151 (1997) · Zbl 0883.20018 · doi:10.1017/S0017089500032031 [12] Xu, M., Groups whose proper subgroups are Baer groups, Acta. Math. Sinica, 40, 10-17 (1996) · Zbl 0840.20030 [13] Xu, M., Groups whose proper subgroups are finite-by-nilpotent, Arch. Math., 66, 353-359 (1996) · Zbl 0857.20015 · doi:10.1007/BF01781552 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.