Invariable generation of the symmetric group

TL;DR

This paper investigates the probability that random permutations in the symmetric group invariably generate the entire group, showing that three are insufficient with high probability while four suffice with positive probability.

Contribution

It proves that three random permutations almost never invariably generate the symmetric group, and provides a concise combinatorial proof that four do, extending understanding of generation properties.

Findings

Three random permutations do not invariably generate $ ext{S}_n$ with high probability.

Four random permutations do invariably generate $ ext{S}_n$ with positive probability.

The proof for four permutations is short, combinatorial, and builds on recent results.

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 $\pi'_1,\dots,\pi'_r$ 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.