The Probability of Generating the Symmetric Group

2016/11/08 by Eberhard, Sean, Virchow, Stefan-Christoph
#20B30 (Primary) #20C15 (Secondary) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1611.02501

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+\mathcal 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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