Abstract
A subset {g 1,..., g d } of a finite group G invariably generates \(\left\{ {g_1^{{x_1}}, \ldots ,g_d^{{x_d}}} \right\}\) generates G for every choice of x i ∈ G. The Chebotarev invariant C(G) of G is the expected value of the random variable n that is minimal subject to the requirement that n randomly chosen elements of G invariably generate G. The first author recently showed that \(C\left( G \right) \leqslant \beta \sqrt {\left| G \right|} \) for some absolute constant β. In this paper we show that, when G is soluble, then β is at most 5/3. We also show that this is best possible. Furthermore, we show that, in general, for each ε > 0 there exists a constant c ε such that \(C\left( G \right) \leqslant \left( {1 + \in } \right)\sqrt {\left| G \right|} + {c_ \in }\).
Similar content being viewed by others
References
A. Ballester-Bolinches and L. M. Ezquerro, Classes of Finite Groups, Mathematics and Its Applications, Vol. 584, Springer, Dordrecht, 2006.
P. J. Cameron, Permutation Groups, London Mathematical Society Student Texts, Vol. 45, Cambridge University Press, Cambridge, 1999.
F. Dalla Volta and A. Lucchini, Finite groups that need more generators than any proper quotient, Journal of the Australian Mathematical Society 64 (1998), 82–91.
E. Detomi and A. Lucchini, Crowns and factorization of the probabilistic zeta function of a finite group, Journal of Algebra 265 (2003), 651–668.
J. D. Dixon, Random sets which invariably generate the symmetric group, Discrete Mathematics 105 (1992), 25–39.
W. Gaschütz, Praefrattinigruppen, Archiv der Mathematik 13 (1962), 418–426.
P. Jiménez-Seral and J. Lafuente, On complemented nonabelian chief factors of a finite group, Israel Journal of Mathematics 106 (1998), 177–188.
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.
E. Kowalski and D. Zywina, The Chebotarev invariant of a finite group, Experimental Mathematics 21 (2012), 38–56.
A. Lucchini, The Chebotarev invariant of a finite group: A conjecture of Kowalski and Zywina, arXiv:1509.05859v2.
C. Pomerance, The expected number of random elements to generate a finite abelian group, Periodica Mathematica Hungarica 43 (2001), 191–198
U. Stammbach, Cohomological characterisations of finite solvable and nilpotent groups, Journal of Pure and Applied Algebra 11 (1977/78), 293–301.
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by Università di Padova (Progetto di Ricerca di Ateneo: “Invariable generation of groups”).
Rights and permissions
About this article
Cite this article
Lucchini, A., Tracey, G. An upper bound on the Chebotarev invariant of a finite group. Isr. J. Math. 219, 449–467 (2017). https://doi.org/10.1007/s11856-017-1507-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-017-1507-x