Skip to main content
Log in

Groups where the centers of the irreducible characters form a chain II

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

In this paper, we determine new characterizations of nested and nested GVZ-groups, including character-free characterizations, but we additionally show that nested groups and nested GVZ-groups can be defined in terms of the existence of certain normal series.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Berkovich, Y.: Groups of Prime Power Order. De Gruyter Expositions in Mathematics, vol. 1. Walter de Gruyter GmbH & Co. KG, Berlin (2008)

    Book  Google Scholar 

  2. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comput. 24, 235–265 (1997)

    Article  MathSciNet  Google Scholar 

  3. Burkett, S.T., Lewis, M.L.: Characterizations of nested GVZ-groups by central series. arXiv:1907.04795

  4. Burkett, S.T., Lewis, M.L.: GVZ-groups. arXiv:1909.05841. Submitted for publication

  5. Burkett, S.T., Lewis, M.L.: A characterization of nested groups in terms of conjugacy classes. Comptes rendus Mathematique. arXiv:1909.05849. Submitted for publication

  6. Camina, A.R.: Some conditions which almost characterize Frobenius groups. Isr. J. Math. 31, 153–160 (1978)

    Article  MathSciNet  Google Scholar 

  7. Erné, M., Koslowski, J., Melton, A., Strecker, G.E.: A primer on Galois connections in Papers on general topology and applications. Ann. N.Y. Acad. Sci. 1993, 103–125 (1991)

    MATH  Google Scholar 

  8. Fernández-Alcober, G.A., Moretó, A.: Groups with two extreme character degrees and their normal subgroups. Trans. Am. Math. Soc. 353, 2171–2192 (2001)

    Article  MathSciNet  Google Scholar 

  9. Isaacs, I.M.: Character Theory of Finite Groups. Dover Publications Inc, New York (1994)

    MATH  Google Scholar 

  10. Isaacs, I.M.: Subgroups generated by small classes in finite groups. Proc. Am. Math. Soc. 136, 2299–2301 (2008)

    Article  MathSciNet  Google Scholar 

  11. Lewis, M.L.: Character Tables of Groups Where All Nonlinear Irreducible Characters Vanish Off the Center, Ischia Group Theory 2008, pp. 174–182. World Sci. Publ, Hackensack, NJ (2009)

    MATH  Google Scholar 

  12. Lewis, M.L.: The vanishing-off subgroup. J. Algebra 321, 1313–1325 (2009)

    Article  MathSciNet  Google Scholar 

  13. Lewis, M.L.: Groups where the centers of the irreducible characters form a chain. Monatshefte für Mathematik. arXiv:1902.10689. Submitted for publication

  14. Longobardi, P., Maj, M., Mann, A.: Minimal classes and maximal class in \(p\)-groups. Isr. J. Math. 110, 93–102 (1999)

    Article  MathSciNet  Google Scholar 

  15. Mann, A.: Elements of minimal breadth in finite p-groups and Lie algebras. J. Aust. Math. Soc. 81, 209–214 (2006)

    Article  MathSciNet  Google Scholar 

  16. Mann, A.: Conjugacy class sizes in finite groups. J. Aust. Math. Soc. 85, 251–255 (2008)

    Article  MathSciNet  Google Scholar 

  17. Mattarei, S.: Retrieving information about a group from its character table. Ph.D. dissertation University of Warwick (1992)

  18. Mlaiki, N.M.: Camina triples. Can. Math. Bull. 57, 125–131 (2014)

    Article  MathSciNet  Google Scholar 

  19. Nenciu, A.: Isomorphic character tables of nested GVZ-groups. J. Algebra Appl. 11, 12 (2012)

    Article  MathSciNet  Google Scholar 

  20. Nenciu, A.: Nested GVZ-groups. J. Group Theory 19, 693–704 (2016)

    Article  MathSciNet  Google Scholar 

  21. Verardi, L.: Gruppi semiextraseciali di esponente \(p\). Ann. Mat. Pura Appl. (4) 148, 131–171 (1987)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mark L. Lewis.

Additional information

Communicated by John S. Wilson.

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

Burkett, S.T., Lewis, M.L. Groups where the centers of the irreducible characters form a chain II. Monatsh Math 192, 783–812 (2020). https://doi.org/10.1007/s00605-019-01362-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-019-01362-x

Keywords

Mathematics Subject Classification

Navigation