Rocky Mountain Journal of Mathematics

The expected number of elements to generate a finite group with $d$-generated Sylow subgroups

Andrea Lucchini and Mariapia Moscatiello

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Given a finite group $G,$ let $e(G)$ be the expected number of elements of $G$ which have to be drawn at random, with replacement, before a set of generators is found. If all of the Sylow subgroups of $G$ can be generated by $d$ elements, then $e(G)\leq d+\kappa $, where $\kappa $ is an absolute constant that is explicitly described in terms of the Riemann zeta function and is the best possible in this context. Approximately, $\kappa $ equals 2.752394. If $G$ is a permutation group of degree $n,$ then either $G={Sym} (3)$ and $e(G)=2.9$ or $e(G)\leq \lfloor n/2\rfloor +\kappa ^*$ with $\kappa ^* \sim 1.606695.$ These results improve weaker bounds recently obtained by Lucchini.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 6 (2018), 1963-1982.

Dates
First available in Project Euclid: 24 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1543028448

Digital Object Identifier
doi:10.1216/RMJ-2018-48-6-1963

Mathematical Reviews number (MathSciNet)
MR3879312

Zentralblatt MATH identifier
06987235

Subjects
Primary: 20P05: Probabilistic methods in group theory [See also 60Bxx]

Keywords
Groups generation waiting time Sylow subgroups permutation groups

Citation

Lucchini, Andrea; Moscatiello, Mariapia. The expected number of elements to generate a finite group with $d$-generated Sylow subgroups. Rocky Mountain J. Math. 48 (2018), no. 6, 1963--1982. doi:10.1216/RMJ-2018-48-6-1963. https://projecteuclid.org/euclid.rmjm/1543028448


Export citation

References

  • H. Cohen, High precision computation of Hardy-Littlewood constants, preprint, https://www.math.u-bordeaux.fr/~hecohen/.
  • D. Easdown and C. Praeger, On minimal faithful permutation representations of finite groups, Bull. Australian Math. Soc. 38 (1988), 207–220.
  • The GAP Group, GAP–Groups, algorithms, and programming, version 4.7.7, http://www.gap-system.org (2015).
  • W. Gaschütz, Die Eulersche Funktion endlicher auflösbarer Gruppen, Illinois J. Math. 3 (1959), 469–476.
  • R. Guralnick, On the number of generators of a finite group, Arch. Math. 53 (1989), 521–523.
  • D. Holt, Representing quotients of permutation groups, Quart. J. Math. Oxford 48 (1997), 347–350.
  • L.G. Kovács and C.E. Praeger, Finite permutation groups with large abelian quotients, Pacific J. Math. 136 (1989), 283–292.
  • A. Lubotzky, The expected number of random elements to generate a finite group, J. Algebra 257 (2002), 452–Â-495.
  • A. Lubotzky and D. Segal, Subgroup growth, Progr. Math. 212 (2003).
  • A. Lucchini, A bound on the number of generators of a finite group, Arch. Math. 53 (1989), 313–317.
  • ––––, The expected number of random elements to generate a finite group, Monatsh. Math. 181 (2016), 123–142.
  • ––––, 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.
  • N.E. Menezes, Random generation and chief length of finite groups, Ph.D. dissertation, http://hdl.handle.net/10023/3578.
  • The PARI Group, PARI/GP version 2.9.0, University of Bordeaux, 2016, http://pari.math.u-bordeaux.fr/.
  • C. Pomerance, The expected number of random elements to generate a finite abelian group, Period. Math. Hungar. 43 (2001), 191–198.
  • D. Robinson, A course in the theory of groups, Grad. Texts Math. 80 (1993).