Skip to main content
SpringerLink
Log in

A bound on the diameter of the Brauer graph of a block of a solvable group

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Let G be a solvable group and p a prime. If B is a p-block of G, then there is a graph associated to B, called the Brauer graph of B, that captures information about the decomposition numbers of B. By the definition of a block, this graph is connected. In a previous paper [1], it was shown that if the defect group of B is abelian, or if G has odd order, then there is an absolute bound on the diameter of this connected graph. In this paper we show that in fact there is an absolute bound of 35 for the diameter of the Brauer graph of any p-block of any solvable group. Moreover, if p ≥ 11, we use a new large orbit theorem (which may be of independent interest) to show that the diameter is at most 15.

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. J. P. Cossey and M. L. Lewis, Bounding the diameter of the Brauer graph of a block of a solvable group, Proceedings of the Edinburgh Mathematical Society 59 (2016), 591–603.

    Article  MathSciNet  Google Scholar 

  2. I. M. Isaacs, Characters of solvable and symplectic groups, American Journal of Mathematics 95 (1973), 594–635.

    Article  MathSciNet  Google Scholar 

  3. I. M. Isaacs, Characters of π-separable groups, Journal of Algebra 86 (1984), 98–128.

    Article  MathSciNet  Google Scholar 

  4. A. Laradji, Relative π-blocks of π-separable groups. II, Journal of Algebra 237 (2001), 521–532.

    Article  MathSciNet  Google Scholar 

  5. O. Manz and T. R. Wolf, Representations of Solvable Groups, London Mathematical Society Lecture Note Series, Vol. 185, Cambridge University Press, Cambridge, 1993.

    Book  Google Scholar 

  6. G. Navarro, Characters and Blocks of Finite Groups, London Mathematical Society Lecture Note Series, Vol. 250, Cambridge University Press, Cambridge, 1998.

    Book  Google Scholar 

  7. Y. Yang, Orbits of the actions of finite solvable groups, Journal of Algebra 321 (2009), 2012–2021.

    Article  MathSciNet  Google Scholar 

  8. Y. Yang, Regular orbits of finite primitive solvable groups, Journal of Algebra 323 (2010), 2735–2755.

    Article  MathSciNet  Google Scholar 

  9. Y. Yang, Regular orbits of finite primitive solvable groups, II, Journal of Algebra 341 (2011), 23–34.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgement

The project was supported by a grant from the Simons Foundation (No. 499532). The authors are grateful to the referee for the valuable suggestions which improved the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Akron, Akron, OH, 44325, USA

    James Cossey

  2. Department of Mathematics, Texas State University, San Marcos, TX, 78666, USA

    Yong Yang

Authors
  1. James Cossey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cossey, J., Yang, Y. A bound on the diameter of the Brauer graph of a block of a solvable group. Isr. J. Math. 250, 115–138 (2022). https://doi.org/10.1007/s11856-022-2333-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-022-2333-3

Search

Navigation