Israel Journal of Mathematics

, Volume 211, Issue 1, pp 481–491 | Cite as

Invariable generation of prosoluble groups

Article
First Online:
Received:
Revised:

Abstract

A group G is invariably generated by a subset S of G if G = 〈s g(s) | s ∈ S〉 for each choice of g(s) ∈ G, s ∈ S. Answering two questions posed by Kantor, Lubotzky and Shalev in [8], we prove that the free prosoluble group of rank d ≥ 2 cannot be invariably generated by a finite set of elements, while the free solvable profinite group of rank d and derived length l is invariably generated by precisely l(d − 1) + 1 elements.

Keywords

Normal Subgroup Semidirect Product Wreath Product Minimal Normal Subgroup Soluble Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    E. Detomi and A. Lucchini, Invariable generation with elements of coprime prime-power order, Journal of Algebra 423 (2015), 683–701.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    J. D. Dixon, Random sets which invariably generate the symmetric group, Discrete Mathematics 105 (1992), 25–39.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    K. Doerk and T. Hawkes, Finite Soluble Groups, de Gruyter Expositions in Mathematics, Vol. 4, Walter de Gruyter & Co, Berlin, 1992.CrossRefMATHGoogle Scholar
  4. [4]
    W. Gaschütz, Die Eulersche Funktion endlicher auflösbarer Gruppen, Illinois Journal of Mathematics 3 (1959), 469–476.MathSciNetMATHGoogle Scholar
  5. [5]
    K. Gruenberg, Relation Modules of Finite Groups, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 25, American Mathematical Society, Providence, RI, 1976.MATHGoogle Scholar
  6. [6]
    T.W. Hungerford, Algebra, Reprint of the 1974 original, Graduate Texts inMathematics, Vol. 73, Springer-Verlag, New York-Berlin, 1980.CrossRefMATHGoogle Scholar
  7. [7]
    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.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    W. M. Kantor, A. Lubotzky and A. Shalev, Invariable generation of infinite groups, Journal of Algebra 421 (2015), 296–310.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    A. Lucchini, Generating wreath products, Archiv der Mathematik 62 (1994), 481–490.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    B. C. Oltikar and L. Ribes, On prosupersolvable groups, Pacific Journal of Mathematics 77 (1978), 183–188.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    L. Ribes and P. Zalesskii, Profinite Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Vol. 40, Springer-Verlag, Berlin, 2000.Google Scholar
  12. [12]
    U. Stammbach, Cohomological characterisations of finite solvable and nilpotent groups, Journal of Pure and Applied Algebra 11 (1977/78), 293–301.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    J. Wiegold, Transitive groups with fixed-point-free permutations, Archiv der Mathematik 27 (1976), 473–475.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    J. S. Wilson, Profinite Groups, London Mathematical Society Monographs. New Series, Vol. 19, Clarendon Press, Oxford University Press, New York, 1998.MATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2016

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di PadovaPadovaItaly

Personalised recommendations