The Probability of Generating the Symmetric Group

TL;DR

This paper provides an elementary proof that the probability of two random permutations generating the symmetric group approaches 1 at a rate of 1-1/n, refining previous results with character theory techniques.

Contribution

It offers a new elementary proof of the asymptotic probability for generating the symmetric group, improving the error term estimate.

Findings

Proves p(S_n)=1-1/n+O(n^{-2+ε})

Uses character theory and recent estimates

Simplifies previous proofs relying on classification

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).