• 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 ≤ S_n do G and a random permutation σ 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 Łuczak and Pyber on the structure of random permutations.

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