Invariable generation of the symmetric group

Abstract

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 $\pi {\text{'}}_1,\dots ,\pi {\text{'}}_r$ of these permutations, the $\pi {\text{'}}_1,\dots ,\pi {\text{'}}_r$ permutations generate ${\mathcal{S}}_n$. We show that if ${\pi }_1,{\pi }_2$, and ${\pi }_3$ are chosen randomly from ${\mathcal{S}}_n$, then, with probability tending to $1$ as $n\to \infty$, 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 probability bounded away from zero. We include a proof of this statement which, while sharing many features with their argument, is short and completely combinatorial.