Invariable Generation of the Symmetric Group

Abstract. We say that permutations π1, . . . , πr ∈ Sn invariably generate Sn if, no matter how one chooses conjugates π 1 , . . . , π r of these permutations, π 1 , . . . , π r generate Sn. We show that if π1, π2, π3 are chosen randomly from Sn then, with probability tending to 1 as n → ∞, they do not invariably generate Sn. By contrast it was shown recently by Pemantle, Peres and Rivin that four random elements do invariably generate Sn with probability bounded away from zero. We include a proof of this statement which, while sharing many