# Groups whose proper subgroups are Baer-by-Chernikov or Baer-by-(finite rank)

Abdelhafid Badis; Nadir Trabelsi

Open Mathematics (2011)

- Volume: 9, Issue: 6, page 1344-1348
- ISSN: 2391-5455

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topAbdelhafid Badis, and Nadir Trabelsi. "Groups whose proper subgroups are Baer-by-Chernikov or Baer-by-(finite rank)." Open Mathematics 9.6 (2011): 1344-1348. <http://eudml.org/doc/269633>.

@article{AbdelhafidBadis2011,

abstract = {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.},

author = {Abdelhafid Badis, Nadir Trabelsi},

journal = {Open Mathematics},

keywords = {Locally graded; Locally (soluble-by-finite); Baer; Chernikov; Finite rank; locally graded groups; Baer groups; Baer-by-Chernikov groups; finitely generated subgroups; subgroups of finite index; subnormal subgroups},

language = {eng},

number = {6},

pages = {1344-1348},

title = {Groups whose proper subgroups are Baer-by-Chernikov or Baer-by-(finite rank)},

url = {http://eudml.org/doc/269633},

volume = {9},

year = {2011},

}

TY - JOUR

AU - Abdelhafid Badis

AU - Nadir Trabelsi

TI - Groups whose proper subgroups are Baer-by-Chernikov or Baer-by-(finite rank)

JO - Open Mathematics

PY - 2011

VL - 9

IS - 6

SP - 1344

EP - 1348

AB - 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.

LA - eng

KW - Locally graded; Locally (soluble-by-finite); Baer; Chernikov; Finite rank; locally graded groups; Baer groups; Baer-by-Chernikov groups; finitely generated subgroups; subgroups of finite index; subnormal subgroups

UR - http://eudml.org/doc/269633

ER -

## References

top- [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 Zbl0961.20031
- [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 Zbl0965.20019
- [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 Zbl1083.20034
- [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 Zbl1120.20030
- [6] Franciosi S., de Giovanni F., Sysak Y.P., Groups with many polycyclic-by-nilpotent subgroups, Ricerche Mat., 1999, 48(2), 361–378 Zbl0980.20023
- [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 Zbl0595.20027
- [8] Napolitani F., Pegoraro E., On groups with nilpotent by Černikov proper subgroups, Arch. Math. (Basel), 1997, 69(2), 89–94 Zbl0897.20021
- [9] Newman M.F., Wiegold J., Groups with many nilpotent subgroups, Arch. Math. (Basel), 1964, 15, 241–250 Zbl0134.26102
- [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 Zbl1172.20029
- [12] Wehrfritz B.A.F., Infinite Linear Groups, Ergeb. Math. Grenzgeb., 76, Springer, Berlin-Heidelberg-New York, 1973