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).
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Eberhard, S., Virchow, S. The Probability of Generating the Symmetric Group. Combinatorica 39, 273–288 (2019). https://doi.org/10.1007/s00493-017-3629-5
Mathematics Subject Classification (2000)