Hostname: page-component-f554764f5-nqxm9 Total loading time: 0 Render date: 2025-04-18T17:27:13.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounding the maximal size of independent generating sets of finite groups

Part of: Representation theory of groups Permutation groups

Published online by Cambridge University Press:  24 January 2020

Andrea Lucchini ,
Mariapia Moscatiello  and
Pablo Spiga
Andrea Lucchini
Affiliation:
Dipartimento di Matematica ‘Tullio Levi-Civita’, University of Padova, Via Trieste 53, 35121Padova, Italy (lucchini@math.unipd.it; mariapia.moscatiello@math.unipd.it)
Mariapia Moscatiello
Affiliation:
Dipartimento di Matematica ‘Tullio Levi-Civita’, University of Padova, Via Trieste 53, 35121Padova, Italy (lucchini@math.unipd.it; mariapia.moscatiello@math.unipd.it)
Pablo Spiga
Affiliation:
Dipartimento di Matematica Pura e Applicata, University of Milano-Bicocca, Via Cozzi 55, 20126Milano, Italy (pablo.spiga@unimib.it)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Denote by m(G) the largest size of a minimal generating set of a finite group G. We estimate m(G) in terms of $\sum _{p\in \pi (G)}d_p(G),$ where we are denoting by dp(G) the minimal number of generators of a Sylow p-subgroup of G and by π(G) the set of prime numbers dividing the order of G.

Keywords

Generating setsNumber of generators

MSC classification

Primary: 20D99: None of the above, but in this section
Secondary: 20B05: General theory for finite groups 20D20: Sylow subgroups, Sylow properties, $pi$-groups, $pi$-structure
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 1 , February 2021 , pp. 133 - 150
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

References

1Apisa, P. and Klopsch, B.. A generalization of the Burnside basis theorem. J. Algebra 400 (2014), 8–16.CrossRefGoogle Scholar
2Cameron, P. and Cara, P.. Independent generating sets and geometries for symmetric groups. J. Algebra 258 (2002), 641–650.CrossRefGoogle Scholar
3Guralnick, R.. On the number of generators of a finite group. Arch. Math. 53 (1989), 521–523.CrossRefGoogle Scholar
4Herzog, M.. On finite simple groups of order divisible by three primes only. J. Algebra 10 (1968), 383–388.CrossRefGoogle Scholar
5Huppert, B.. Endliche Gruppen. I. (German) Die Grundlehren der Mathematischen Wissenschaften, Band 134 (Springer-Verlag, Berlin-New York, 1967).CrossRefGoogle Scholar
6Keen, P. J., Independent sets in some classical groups of dimension three, Ph.D. Thesis. University of Birmingham, 2012.Google Scholar
7Kleidman, P. and Liebeck, M.. The subgroup structure of the finite classical groups. London Mathematical Society Lecture Note Series, vol. 129 (Cambridge: Cambridge University Press, 1990).CrossRefGoogle Scholar
8Kimmerle, W., Lyons, R., Sandling, R. and Teague, D. N.. Composition factors from the group ring and Artin's theorem on orders of simple groups. Proc. London Math. Soc. 60 (1990), 89–122.CrossRefGoogle Scholar
9Lucchini, A.. A bound on the number of generators of a finite group. Arch. Math. 53 (1989), 313–317.CrossRefGoogle Scholar
10Lucchini, A.. The largest size of a minimal generating set of a finite group. Arch. Math. 101 (2013), 1–8.CrossRefGoogle Scholar
11Lucchini, A.. Minimal generating sets of maximal size in finite monolithic groups. Arch. Math. 101 (2013), 401–410.CrossRefGoogle Scholar
12Lucchini, A.. A bound on the expected number of random elements to generate a finite group all of whose Sylow subgroups are d-generated. Arch. Math. 107 (2016), 1–8.CrossRefGoogle Scholar
13Ribenboin, P.. The book of prime number records, 2nd edn. (New York: Springer-Verlag, 1989).CrossRefGoogle Scholar
14Rosser, B.. Explicit bounds for some functions of prime numbers. Amer. J. Math. 63 (1941), 211–232.CrossRefGoogle Scholar
15Rosser, J. K. and Schoenfeld, L.. Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), 64–94.CrossRefGoogle Scholar
16Saxl, J. and Whiston, J.. On the maximal size of independent generating sets of PSL2(q). J. Algebra 258 (2002), 651–657.Google Scholar
17Whiston, J.. Maximal independent generating sets of the symmetric group. J. Algebra 232 (2000), 255–268.CrossRefGoogle Scholar
18Zsigmondy, K.. Zur Theorie der Potenzreste. Monatsch. Math. Phys. 3 (1892), 265–284.CrossRefGoogle Scholar