Badis, Abdelhafid; Trabelsi, Nadir Groups whose proper subgroups are Baer-by-Chernikov or Baer-by-(finite rank). (English) Zbl 1247.20044 Cent. Eur. J. Math. 9, No. 6, 1344-1348 (2011). It has been proved by F. Napolitani and E. Pegoraro [Arch. Math. 69, No. 2, 89-94 (1997; Zbl 0897.20021)] and A. O. Asar [J. Lond. Math. Soc., II. Ser. 61, No. 2, 412-422 (2000; Zbl 0961.20031)] that if \(G\) is a locally graded group whose proper subgroups are nilpotent-by-Chernikov, then \(G\) itself is nilpotent-by-Chernikov. Recall that a group \(G\) is said to be ‘locally graded’ if every finitely generated non-trivial subgroup of \(G\) contains a proper subgroup of finite index; thus locally graded groups form a wide class of generalized soluble groups, containing in particular all locally (soluble-by-finite) groups. The main result of the paper under review shows that if \(G\) is any locally graded group whose proper subgroups are Baer-by-Chernikov, then \(G\) itself is a Baer-by-Chernikov group. Here a group is called a ‘Baer group’ if all its cyclic subgroups are subnormal. Reviewer: Francesco de Giovanni (Napoli) Cited in 1 ReviewCited in 2 Documents MSC: 20F19 Generalizations of solvable and nilpotent groups 20E07 Subgroup theorems; subgroup growth 20E25 Local properties of groups Keywords:locally graded groups; Baer groups; Baer-by-Chernikov groups; finitely generated subgroups; subgroups of finite index; subnormal subgroups Citations:Zbl 0897.20021; Zbl 0961.20031 PDFBibTeX XMLCite \textit{A. Badis} and \textit{N. Trabelsi}, Cent. Eur. J. Math. 9, No. 6, 1344--1348 (2011; Zbl 1247.20044) Full Text: DOI References: [1] Asar A.O., Locally nilpotent p-groups whose proper subgroups are hypercentral or nilpotent-by-Chernikov, J. London Math. Soc., 2000, 61(2), 412-422 http://dx.doi.org/10.1112/S0024610799008479; · Zbl 0961.20031 [2] Chernikov N.S., Theorem on groups of finite special rank, Ukrainian Math. J., 1990, 42(7), 855-861 http://dx.doi.org/10.1007/BF01062091; · Zbl 0751.20030 [3] Dixon M.R., Evans M.J., Smith H., Groups with all proper subgroups nilpotent-by-finite rank, Arch. Math. (Basel), 2000, 75(2), 81-91; · Zbl 0965.20019 [4] Dixon M.R., Evans M.J., Smith H., Groups with all proper subgroups soluble-by-finite rank, J. Algebra, 2005, 289(1), 135-147 http://dx.doi.org/10.1016/j.jalgebra.2005.01.047; · Zbl 1083.20034 [5] Dixon M.R., Evans M.J., Smith H., Embedding groups in locally (soluble-by-finite) simple groups, J. Group Theory, 2006, 9(3), 383-395 http://dx.doi.org/10.1515/JGT.2006.026; · Zbl 1120.20030 [6] Franciosi S., de Giovanni F., Sysak Y.P., Groups with many polycyclic-by-nilpotent subgroups, Ricerche Mat., 1999, 48(2), 361-378; · Zbl 0980.20023 [7] Kleidman P.B., Wilson R.A., A characterization of some locally finite simple groups of Lie type, Arch. Math. (Basel), 1987, 48(1), 10-14; · Zbl 0595.20027 [8] Napolitani F., Pegoraro E., On groups with nilpotent by Černikov proper subgroups, Arch. Math. (Basel), 1997, 69(2), 89-94; · Zbl 0897.20021 [9] Newman M.F., Wiegold J., Groups with many nilpotent subgroups, Arch. Math. (Basel), 1964, 15, 241-250; · Zbl 0134.26102 [10] Robinson D.J.S., Finiteness Conditions and Generalized Soluble Groups. I, Ergeb. Math. Grenzgeb., 62, Springer, Berlin-Heidelberg-New York, 1972; [11] Smith H., More countably recognizable classes of groups, J. Pure Appl. Algebra, 2009, 213(7), 1320-1324 http://dx.doi.org/10.1016/j.jpaa.2008.11.030; · Zbl 1172.20029 [12] Wehrfritz B.A.F., Infinite Linear Groups, Ergeb. Math. Grenzgeb., 76, Springer, Berlin-Heidelberg-New York, 1973; · Zbl 0261.20038 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.