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 ≤ Sn do G and a random permutation σ ∈ Sn 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.