Skip to main content
SpringerLink
Log in

The expected number of random elements to generate a finite abelian group

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

Suppose G is a finite abelian group with minimal number of generators r. It is shown that the expected number of elements from G (chosen independently and with the uniform distribution) so that the elements chosen generate G is less than r + where= 2118456563...The constant is explicitly described in terms of the Riemann zeta-function and is best possible.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

REFERENCES

  1. V. Acciaro, The probability of generating some common families of finite groups, Util. Math. 49 (1996), 243–254.

    Google Scholar 

  2. J. P. Buhler, H. W. Lenstra, Jr., and C. Pomerance, Factoring integers with the number field sieve, in: The development of the number field sieve, A. K. Lenstra and H. W. Lenstra, Jr., eds., Lecture Notes in Math. 1554, pp. 50–94, Springer-Verlag, Berlin, 1993.

    Google Scholar 

  3. J. D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969), 199–205.

    Google Scholar 

  4. P. ErdŐs and G. Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged) 7 (1935), 95–102.

    Google Scholar 

  5. A. IviĆ, The Riemann zeta-function, Wiley, New York, 1985.

    Google Scholar 

  6. W. M. Kantor and A. Lubotzky, The probability of generating a finite classical group, Geom. Dedicata 36 (1990), 67–87.

    Google Scholar 

  7. S. Konyagin and C. Pomerance, On primes recognizable in deterministic polynomial time, in: The mathematics of Paul Erdős, vol. 1, R. L. Graham and J. Nešetřil, eds., pp. 176–198, Springer-Verlag, Berlin, 1997.

    Google Scholar 

  8. H. W. Lenstra, Jr. and C. Pomerance, A rigorous time bound for factoring integers, J. Amer. Math. Soc. 5 (1992), 483–516.

    Google Scholar 

  9. M. W. Liebeck and A. Shalev, The probability of generating a _nite simple group, Geom. Dedicata 56 (1995), 103–113.

    Google Scholar 

  10. I. Pak, On probability of generating a finite group, preprint, 1999.

  11. G. Poupard and J. Stern, Short proofs of knowledge of factoring, in: Proc. PKC2000, Lecture Notes in Comput. Sci. 1751, pp. 147–166, Springer-Verlag, Berlin, 2000.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fundamental Mathematics Research, Bell Labs - Lucent Technologies, Murray Hill, NJ, 07974, USA

    Carl Pomerance

Authors
  1. Carl Pomerance
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pomerance, C. The expected number of random elements to generate a finite abelian group. Periodica Mathematica Hungarica 43, 191–198 (2002). https://doi.org/10.1023/A:1015250102792

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1015250102792

Keywords

Search

Navigation