The probability of generating the symmetric group when one of the generators is random

A classical result of JOHN DIXON (1969) asserts that a pair of random permutations of a set of n elements almost surely generates either the symmetric or the alternating group of degree n. We answer the question, "For what permutation groups G ≤ S_n do G and a random permutation σ ∈ S_n almost surely generate the symmetric or the alternating group?" Extending Dixon's result, we prove that this is the case if and only if G fixes o(n) elements of the permutation domain. The question arose in connection with the study of the diameter of Cayley graphs of the symmetric group. Our proof is based on a result by Luczak and Pyber on the structure of random permutations.