Skip to main content
SpringerLink
Log in

Groups with the weak minimal condition on non-normal non-abelian subgroups

  • Original Paper
  • Published:
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry Aims and scope Submit manuscript

Abstract

A group is called metahamiltonian if all its non-abelian subgroups are normal. It is proved here that a (generalized) soluble group satisfying the weak minimal condition on non-normal non-abelian subgroups is either minimax or metahamiltonian.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Baer, R.: Polyminimaxgruppen. Math. Ann. 175, 1–43 (1968)

    Article  MathSciNet  Google Scholar 

  • De Falco, M., de Giovanni, F., Musella, C., Sysak, Y.P.: On metahamiltonian groups of infinite rank. J. Algebra 407, 135–148 (2014)

    Article  MathSciNet  Google Scholar 

  • De Mari, F.: Groups satisfying weak chain conditions on certain non-normal subgroups. Boll. Un. Mat. Ital. 10B, 853–866 (2007)

    MathSciNet  MATH  Google Scholar 

  • De Mari, F.: Groups satisfying weak chain conditions on non-modular subgroups. Commun. Algebra 46, 1709–1715 (2018)

    Article  MathSciNet  Google Scholar 

  • De Mari, F., de Giovanni, F.: Groups with few normalizer subgroups. Ir. Math. Soc. Bull. 56, 103–113 (2005)

    MathSciNet  MATH  Google Scholar 

  • Dixon, M.R., Evans, M.J.: Locally soluble-by-finite groups with the weak minimal condition on non-nilpotent subgroups. J. Algebra 249, 226–246 (2002)

    Article  MathSciNet  Google Scholar 

  • Kazarin, L.S., Kurdachenko, L.A., Subbotin, I.Ya.: On groups saturated with abelian subgroups. Int. J. Algebra Comput. 8, 443–466 (1998)

    Article  MathSciNet  Google Scholar 

  • Kurdachenko, L.A.: Some classes of groups with the weak minimality and maximality conditions for normal subgroups. Ukr. Math. J. 42, 1050–1056 (1990)

    MathSciNet  Google Scholar 

  • Kurdachenko, L.A., Gorestkiĭ, V.È.: Groups with weak minimality and maximality conditions for subgroups that are not normal. Ukr. Math. J. 41, 1474–1477 (1989)

    Article  MathSciNet  Google Scholar 

  • Lennox, J.C., Robinson, D.J.S.: The Theory of Infinite Soluble Groups. Clarendon Press, Oxford (2004)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Phillips, R.E., Wilson, J.S.: On certain minimal conditions for infinite groups. J. Algebra 51, 41–68 (1978)

    Article  MathSciNet  Google Scholar 

  • Romails, G.M., Sesekin, N.F.: Metahamiltonian groups. Ural. Gos. Mat. Zap. 5, 101–106 (1966)

    MathSciNet  Google Scholar 

  • Romails, G.M., Sesekin, N.F.: Metahamiltonian groups II. Ural. Gos. Mat. Zap. 6, 52–58 (1968)

    MathSciNet  Google Scholar 

  • Romails, G.M., Sesekin, N.F.: Metahamiltonian groups III. Ural. Gos. Mat. Zap. 7, 195–199 (1969/1970)

  • Zaǐcev, D.I.: Groups satisfying the weak minimal condition. Ukr. Math. J. 20, 408–416 (1968)

    Google Scholar 

  • Zaǐcev, D.I.: The groups which satisfy a weak minimality condition. Math. Sb. 78, 323–331 (1969)

    MathSciNet  Google Scholar 

  • Zaičev, D.I.: On the theory of minimax groups. Ukr. Math. J. 23, 536–542 (1971)

    MathSciNet  Google Scholar 

  • Zaičev, D.I.: Groups satisfying the weak minimal condition for non-abelian subroups. Ukr. Math. J. 23, 543–546 (1971)

    Google Scholar 

Download references

Acknowledgements

We wish to thank the referee for his useful comments on a previous version of this paper.

Author information

Authors and Affiliations

  1. Università degli Studi di Napoli Federico II, Naples, Italy

    Fausto De Mari

Authors
  1. Fausto De Mari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fausto De Mari.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

De Mari, F. Groups with the weak minimal condition on non-normal non-abelian subgroups. Beitr Algebra Geom 61, 1–7 (2020). https://doi.org/10.1007/s13366-019-00450-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13366-019-00450-1

Keywords

Mathematics Subject Classification

Search

Navigation