Skip to main content
SpringerLink
Log in

On CLT and non-CLT groups

  • Published:
Annali di Matematica Pura ed Applicata (1923 -) Aims and scope Submit manuscript

Abstract

In this note, we prove that for every integer \(d\ge 2\) which is not a prime power, there exists a finite solvable group G such that \(d\mid |G|\), \(\pi (G)=\pi (d)\) and G has no subgroup of order d. We also introduce the CLT-degree of a finite group and answer two questions about it.

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

  1. Baskaran, S.: CLT and non-CLT groups, I. Indian J. Math. 14, 81–82 (1972)

    MathSciNet  Google Scholar 

  2. Baskaran, S.: CLT and non-CLT groups of order \(p^2q^2\). Fund. Math. 92, 1–7 (1976)

    Article  MathSciNet  Google Scholar 

  3. Bray, H.G.: A note on CLT groups. Pac. J. Math. 27, 229–231 (1968)

    Article  MathSciNet  Google Scholar 

  4. Garonzi, M.: On divisors occuring as subgroup sizes, MathOverflow. https://mathoverflow.net/questions/115307/on-divisors-occurring-as-subgroup-sizes

  5. Grove, L.C.: Groups and Characters. Wiley, New York (1997)

    Book  Google Scholar 

  6. Huppert, B.: Endliche Gruppen, I. Springer, Berlin (1967)

    Book  Google Scholar 

  7. Isaacs, I.M.: Finite Group Theory. American Mathematical Society, Providence (2008)

    Google Scholar 

  8. Karpilovsky, G.: Group Representations, Volume 1, Part A: Background Material. Elsevier, Amsterdam (1992)

    Google Scholar 

  9. Lazorec, M.S., Tărnăuceanu, M.: A density result on the sum of element orders of a finite group. Arch. Math. 114, 601–607 (2020)

    Article  MathSciNet  Google Scholar 

  10. McCarthy, D.J.: A survey of partial converses to Lagrange’s theorem on finite groups. Trans. NY Acad. Sci. 33, 586–594 (1971)

    Article  MathSciNet  Google Scholar 

  11. McLain, D.H.: The existence of subgroups of given order in finite groups. Proc. Camb. Philos. Soc. 53, 278–285 (1957)

    Article  MathSciNet  Google Scholar 

  12. Nitecki, Z.: Cantorvals and subsum sets of null sequences. Am. Math. Mon. 122, 862–870 (2015)

    Article  MathSciNet  Google Scholar 

  13. Tărnăuceanu, M.: Non-CLT groups of order \(pq^3\). Math. Slovaca 64, 311–314 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is grateful to the reviewer for remarks which improve the previous version of the paper.

Author information

Authors and Affiliations

  1. Faculty of Mathematics, “Al.I. Cuza” University, Iaşi, Romania

    Marius Tărnăuceanu

Authors
  1. Marius Tărnăuceanu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marius Tărnăuceanu.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tărnăuceanu, M. On CLT and non-CLT groups. Annali di Matematica (2024). https://doi.org/10.1007/s10231-024-01450-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10231-024-01450-2

Keywords

Mathematics Subject Classification

Search

Navigation