The Probability of Generating the Symmetric Group

Sean Eberhard*, Stefan-Christoph Virchow

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


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).

Original languageEnglish
Pages (from-to)273-288
Number of pages16
Issue number2
Publication statusPublished - 01 Apr 2019
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2018, János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics


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