, Volume 39, Issue 2, pp 273–288 | Cite as

The Probability of Generating the Symmetric Group

  • Sean EberhardEmail author
  • Stefan-Christoph Virchow


We consider the probability p(Sn) that a pair of random permutations generates either the alternating group An or the symmetric group Sn. Dixon (1969) proved that p(Sn) approaches 1 as n→∞ and conjectured that p(Sn) = 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(Sn) = 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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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2018

Authors and Affiliations

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

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