The Probability of Generating the Symmetric Group
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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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