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

Abstract

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 alter- nating 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 ar esult byLuczak and Pyber on the structure of random permutations.