The probability of generating the symmetric group

In [2], Dixon considered the following question: " Suppose two permutations are chosen at random from the symmetric group Sn of degree n. What is the probability that they will generate Sn? " Actually, Netto conjectured last century that almost all pairs of elements from Sn will generate Sn or An. Dixon showed that this is true in the following sense: The proportion of ordered pairs (x, y) (x, y e Sn) which generate either An or Sn is greater than 1 — 2/(log log n) 2 for all sufficiently large n. In fact this bound is only a very rough result, as Dixon himself suggests. In this paper we shall prove