Skip to main content
SpringerLink
Log in

Large minimal invariable generating sets in the finite symmetric groups

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

Abstract

For a finite group G, let mI(G) denote the largest possible cardinality of a minimal invariable generating set of G. We prove an upper and a lower bound for mI(Sn), which show in particular that mI(Sn) is asymptotic to n/2 as n → ∞.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. T. M. Apostol, Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer, New York—Heidelberg, 1976.

    Book  MATH  Google Scholar 

  2. A. Blokhuis, A. Brouwer and B. De Weger, Binomial collisions and near collisions, Integers 17 (2017), Article no. a64.

  3. P. J. Cameron and P. Cara, Independent generating sets and geometries for symmetric groups, Journal of Algebra 258 (2002), 641–650.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. D. Dixon, Random sets which invariably generate the symmetric group, Discrete Mathematics 105 (1992), 25–39.

    Article  MathSciNet  MATH  Google Scholar 

  5. B. M. M. de Weger, Equal binomial coefficients: Some elementary considerations, Journal of Number Theory 63 (1997), 373–386.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Fulman and R. M. Guralnick, Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements, Transactions of the American Mathematical Society 364 (2012), 3023–3070.

    Article  MathSciNet  MATH  Google Scholar 

  7. The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.10.2, 2019.

  8. D. Garzoni and N. Gill, On the number of conjugacy classes of a primitive permutation group, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, to appear, https://doi.org/10.1017/prm.2021.74.

  9. D. Garzoni and A. Lucchini, Minimal invariable generating sets, Journal of Pure and Applied Algebra 224 (2020), 218–238.

    Article  MathSciNet  MATH  Google Scholar 

  10. D. M. Kane, Improved bounds on the number of ways of expressing t as a binomial coefficient, Integers 7 (2007), Article no. a53.

  11. W. M. Kantor, A. Lubotzky and A. Shalev, Invariable generation and the Chebotarev invariant of a finite group, Journal of Algebra 348 (2011), 302–314.

    Article  MathSciNet  MATH  Google Scholar 

  12. M. W. Liebeck, C. E. Praeger and J. Saxl, On the O’Nan-Scott theorem for finite primitive permutation groups, Australian Mathematical Society. Journal. Series A. Pure Mathematics and Statistics 44 (1988), 389–396.

    Article  MathSciNet  MATH  Google Scholar 

  13. M. W. Liebeck and A. Shalev, Maximal subgroups of symmetric groups, Journal of Combinatorial Theory. Series A 75 (1996), 341–352.

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Lucchini, The largest size of a minimal generating set of a finite group, Archiv der Mathematik 101 (2013), 1–8.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Lucchini, Minimal generating sets of maximal size in finite monolithic groups, Archiv der Mathematik 101 (2013), 401–410.

    Article  MathSciNet  MATH  Google Scholar 

  16. G. Robin, Estimation de la fonction de Tchebychef θ sur le k-ièmenombrepremier et grandes valeurs de la fonction ω(n) nombre de diviseurs premiers de n, Acta Arithmetica 42 (1983), 367–389.

    Article  MathSciNet  MATH  Google Scholar 

  17. D. Singmaster, How often does an integer occur as a binomial coefficient?, American Mathematical Monthly 78 (1971), 385–386.

    Article  MathSciNet  Google Scholar 

  18. J. Whiston, Maximal independent generating sets of the symmetric group, Journal of Algebra 232 (2000), 255–268.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica “Tullio Levi-Civita”, Università degli Studi di Padova, Via Trieste 63, 35131, Padova, Italy

    Daniele Garzoni

  2. School of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA, UK

    Nick Gill

Authors
  1. Daniele Garzoni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nick Gill
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniele Garzoni.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Garzoni, D., Gill, N. Large minimal invariable generating sets in the finite symmetric groups. Isr. J. Math. 255, 581–598 (2023). https://doi.org/10.1007/s11856-023-2467-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-023-2467-y

Search

Navigation