The Probability of Generating the Symmetric Group

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 $n\to \infty$ 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 $p(S_n)=1-1/n+\mathcal{O}(n^{-2+\epsilon })$. Our proof is based on character theory and character estimates, including recent work by Schlage-Puchta (2012).

Revised and improved version with co-author Sean Eberhard