Groups with some restrictions on non-Baer subgroups

Abstract

It is proved that if G is an \(\mathfrak {X}\)-group of infinite rank whose proper subgroups of infinite rank are Baer groups, then so are all proper subgroups of G, where \(\mathfrak {X}\) is the class defined by N.S. Černikov as the closure of the class of periodic locally graded groups by the closure operations \(\varvec{\acute{P}}\), \(\varvec{\grave{P}}\) and \( \varvec{L}\). We prove also that if a locally graded group, which is neither Baer nor Černikov, satisfies the minimal condition on non-Baer subgroups, then it is a Baer-by-Černikov group which is a direct product of a p-subgroup containing a minimal non-Baer subgroup of infinite rank, by a Černikov nilpotent \(p^{\prime }\)-subgroup, for some prime p. Our last result states that a group is locally graded and has only finitely many conjugacy classes of non-Baer subgroups if, and only if, it is Baer-by-finite and has only finitely many non-Baer subgroups.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Abdollahi, A., Trabelsi, N., Zitouni, A.: Groups with an Engel restriction on proper subgroups of infinite rank. J. Algebra Appl. 19, 8 (2020)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Arikan, A., Trabelsi, N.: On minimal non-Baer-groups. Commun. Algebra 39, 2489–2497 (2011)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Badis, A., Trabelsi, N.: Soluble minimal non-(finite-by-Baer) groups. Ricerche Mat. 59, 129–135 (2010)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Černikov, N.S.: Theorem on groups of finite special rank. J. Ukrain. Math. 42, 855–861 (1990)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Černikov, S.N.: Infinite groups with certain prescribed properties of systems of infinite subgroups of them. Dokl. Akad. Nauk. SSSR 159, 759–760 (1964)

    MathSciNet  Google Scholar 

  6. 6.

    De Falco, M., de Giovanni, F., Musella, C., Trabelsi, N.: Groups with restrictions on subgroups of infinite rank. Rev. Mat. Iberoam. 30, 537–550 (2014)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Dixon, M.R., Evans, M.J., Smith, H.: Locally (soluble-by-finite) groups with various restrictions on subgroups of infinite rank. Glasg. Math. J. 47, 309–317 (2005)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Dixon, M.R., Evans, M.J., Smith, H.: Groups with some minimal condition on non-nilpotent subgroups. J. Group Theory 68, 207–215 (2001)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Dixon, M.R., Evans, M.J., Smith, H.: Locally (soluble-by-finite) groups with all proper non-nilpotent subgroups of finite rank. J. Pure Appl. Algebra 135, 33–43 (1999)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Dixon, M.R., Evans, M.J., Smith, H.: Locally (soluble-by-finite) groups with all proper insoluble subgroups of finite rank. Arch. Math. 68, 100–109 (1997)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Endimioni, G., Sica, C.: Centralizer of Engel elements in a group. Algebra Colloq. 17, 487–494 (2010)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Franciosi, S., de Giovanni, F., Sysak, Y.P.: Groups with many FC-subgroups. J. Algebra 218, 165–182 (1999)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Otal, J., Peña, J.M.: locally graded groups with certain minimal conditions for subgroups. II. Publ. Mat. 32, 151–157 (1988)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Phillips, R.E., Wilson, J.S.: On certain minimal conditions for infinite groups. J. Algebra 51, 41–48 (1978)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Robinson, D.J.S.: Finiteness Conditions and Generalized Soluble Groups. Springer, Berlin (1972)

    Google Scholar 

  16. 16.

    Smith, H.: Groups with few non-nilpotent subgroups. Glasgow Math. J. 39, 141–151 (1997)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Xu, M.: Groups whose proper subgroups are Baer groups. Acta Math. Sinica 12, 10–17 (1996)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

Both authors are grateful to the referee of a previous version of this work for careful reading and many suggestions improving the presentation of this paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Nadir Trabelsi.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was completed by the support of the General Directorate of Scientific Research and Technological Development (DGRSDT, Algeria).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Badis, A., Trabelsi, N. Groups with some restrictions on non-Baer subgroups. Ricerche mat (2021). https://doi.org/10.1007/s11587-021-00557-5

Download citation

Keywords

  • Baer
  • Locally (soluble-by-finite)
  • Minimal condition
  • Rank

Mathematics Subject Classification

  • 20F19
  • 20F99