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The Probability of Generating the Symmetric Group

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Abstract

We consider the probability p(S n ) that a pair of random permutations generates either the alternating group A n or the symmetric group S n . Dixon (1969) proved that p(S n ) approaches 1 as n→∞ and conjectured that p(S n ) = 1 − 1/n+o(1/n). This conjecture was verified by Babai (1989), using the Classification of Finite Simple Groups. We give an elementary proof of this result; specifically we show that p(S n ) = 1 − 1/n+O(n−2+ε). Our proof is based on character theory and character estimates, including recent work by Schlage-Puchta (2012).

Mathematics Subject Classification (2000)

20B30 20C15 

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References

  1. [1]
    L. Babai: The probability of generating the symmetric group, J. Combin. Theory Ser. A 52 (1989), 148–153.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    J. Bovey: The probability that some power of a permutation has small degree, Bull. Lond. Math. Soc. 12 (1980), 47–51.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    J. Bovey and A. Williamson: The probability of generating the symmetric group, Bull. Lond. Math. Soc. 10 (1978), 91–96.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    C. W. Curtis and I. Reiner: Methods of Representation Theory, Volume I, Wiley, New York (1990).MATHGoogle Scholar
  5. [5]
    J. D. Dixon: Asymptotics of generating the symmetric and alternating groups, Electron. J. Combin. 12 (2005), Research Paper #R56.Google Scholar
  6. [6]
    J. D. Dixon: The probability of generating the symmetric group, Math. Z. 110 (1969), 199–205.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    J. D. Dixon and B. Mortimer: Permutation Groups, Springer, New York (1996).CrossRefMATHGoogle Scholar
  8. [8]
    S. Eberhard, K. Ford and D. Koukoulopoulos: Permutations contained in transitive subgroups, Discrete Analysis 12 (2016).Google Scholar
  9. [9]
    S. Eberhard: The trivial lower bound for the girth of S n, arXiv:1706.09972 (2017).Google Scholar
  10. [10]
    J. S. Frame, G. de B. Robinson and R. M. Thrall: The hook graphs of the symmetric group, Canad. J. Math. 6 (1954), 316–324.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    G. H. Hardy and E. M. Wright: An Introduction to the Theory of Numbers, Clarendon, Oxford (1954).MATHGoogle Scholar
  12. [12]
    A. Kerber: Algebraic Combinatorics Via Finite Group Actions, BI-Wissenschaftsverlag, Mannheim-Wien-Zürich (1991).MATHGoogle Scholar
  13. [13]
    E. Manstavičius and R. Petuchovas: Permutations without long or short cycles, Electron. Notes Discrete Math. 49 (2015), 153–158.CrossRefMATHGoogle Scholar
  14. [14]
    T. W. Müller and J.-C. Schlage-Puchta: Character theory of symmetric groups, subgroup growth of Fuchsian groups, and random walks, Adv. Math. 213 (2007), 919–982.MATHGoogle Scholar
  15. [15]
    T. Nakayama: On some modular properties of irreducible representations of a symmetric group, I, Jap. J. Math. 17 (1940), 165–184.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    E. Netto: The Theory of Substitutions and its Applications to Algebra, The Inland Press, Ann Arbor (1892).MATHGoogle Scholar
  17. [17]
    R. Petuchovas: Asymptotic analysis of the cyclic structure of permutations, arXiv:1611.02934 (2016), 1–77.Google Scholar
  18. [18]
    B. E. Sagan: The Symmetric Group, Springer, New York (2001).CrossRefMATHGoogle Scholar
  19. [19]
    J.-C. Schlage-Puchta: Applications of character estimates to statistical problems for the symmetric group, Combinatorica 32 (2012), 309–323.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    N. J. A. Sloane: The On-Line Encyclopedia of Integer Sequences, http://oeis.org, Sequence A113869.Google Scholar
  21. [21]
    H. Wielandt: Finite Permutation Groups, Academic Press, New York (1964).MATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LondonUK
  2. 2.Institut für MathematikUniversität RostockRostockGermany

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