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Archiv der Mathematik

, Volume 105, Issue 3, pp 201–204 | Cite as

A note on the probability of generating alternating or symmetric groups

  • Luke MorganEmail author
  • Colva M. Roney-Dougal
Article

Abstract

We improve on recent estimates for the probability of generating the alternating and symmetric groups A n and S n . In particular, we find the sharp lower bound if the probability is given by a quadratic in n −1. This leads to improved bounds on the largest number h(A n ) such that a direct product of h(A n ) copies of A n can be generated by two elements.

Keywords

Symmetric group Alternating group Generation Probability 

Mathematics Subject Classification

20B30 20P05 

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References

  1. 1.
    Dixon J.D.: The probability of generating the symmetric group. Math. Z 110, 199–205 (1969)MathSciNetCrossRefGoogle Scholar
  2. 2.
    J. D. Dixon, Asymptotics of generating the symmetric and alternating groups, Electron. J. Combin. 12 (2005), Research paper 56, 5 pp.Google Scholar
  3. 3.
    Hall P.: The Eulerian function of a group. J. Math. Oxford 7, 133–141 (1936)Google Scholar
  4. 4.
    Liebeck M.W., Shalev A.: Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotszky. J. Algebra 184, 31–57 (1996)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Maróti A., Tamburini M.C.: Bounds for the probability of generating the symmetric and alternating groups. Arch. Math. (Basel) 96(2), 115–121 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Menezes N.E., Quick M., Roney-Dougal C.M.: The probability of generating a finite simple group. Israel J. Math 198, 371–392 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    N. E. Menezes, Random generation and chief length of finite groups, PhD thesis, University of St Andrews (2013).Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of Western AustraliaCrawleyAustralia
  2. 2.School of Mathematics and StatisticsUniversity of St AndrewsSt Andrews, FifeUK

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