Our main result is that a locally graded group whose proper subgroups are Baer-by-Chernikov is itself Baer-by-Chernikov. We prove also that a locally (soluble-by-finite) group whose proper subgroups are Baer-by-(finite rank) is itself Baer-by-(finite rank) if either it is locally of finite rank but not locally finite or it has no infinite simple images.
@article{bwmeta1.element.doi-10_2478_s11533-011-0077-0,
author = {Abdelhafid Badis and Nadir Trabelsi},
title = {Groups whose proper subgroups are Baer-by-Chernikov or Baer-by-(finite rank)},
journal = {Open Mathematics},
volume = {9},
year = {2011},
pages = {1344-1348},
zbl = {1247.20044},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0077-0}
}
Abdelhafid Badis; Nadir Trabelsi. Groups whose proper subgroups are Baer-by-Chernikov or Baer-by-(finite rank). Open Mathematics, Tome 9 (2011) pp. 1344-1348. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0077-0/
[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
[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