Invariable generation of the symmetric group

We say that permutations $\pi_1,\dots, \pi_r \in \mathcal{S}_n$ invariably generate $\mathcal{S}_n$ if, no matter how one chooses conjugates $\pi'_1,\dots,\pi'_r$ of these permutations, $\pi'_1,\dots,\pi'_r$ generate $\mathcal{S}_n$. We show that if $\pi_1,\pi_2,\pi_3$ are chosen randomly from $\mathcal{S}_n$ then, with probability tending to 1 as $n \rightarrow \infty$, they do not invariably generate $\mathcal{S}_n$. By contrast it was shown recently by Pemantle, Peres and Rivin that four random elements do invariably generate $\mathcal{S}_n$ with positive probability. We include a proof of this statement which, while sharing many features with their argument, is short and completely combinatorial.Comment: 15 pages. Corrections and clarifications suggested by the referees. Added a reference to a paper of Raouj and Stef which solves related problems about divisor

Invariable generation and the chebotarev invariant of a finite group

A subset S of a finite group G invariably generates G if G = <hsg(s) j s 2 Si > for each choice of g(s) 2 G; s 2 S. We give a tight upper bound on the minimal size of an invariable generating set for an arbitrary finite group G. In response to a question in [KZ] we also bound the size of a randomly chosen set of elements of G that is likely to generate G invariably. Along the way we prove that every finite simple group is invariably generated by two elements.Comment: Improved versio