Four Random Permutations Conjugated by an Adversary Generate $S_n$ with High Probability

Abstract

We prove a conjecture dating back to a 1978 paper of D.R.\ Musser~\cite{musserirred}, namely that four random permutations in the symmetric group $\mathcal{S}_n$ generate a transitive subgroup with probability $p_n > \epsilon$ for some $\epsilon > 0$ independent of $n$, even when an adversary is allowed to conjugate each of the four by a possibly different element of $\S_n$ (in other words, the cycle types already guarantee generation of $\mathcal{S}_n$). This is closely related to the following random set model. A random set $M \subseteq \mathbb{Z}^+$ is generated by including each $n \geq 1$ independently with probability $1/n$. The sumset $\text{sumset}(M)$ is formed. Then at most four independent copies of $\text{sumset}(M)$ are needed before their mutual intersection is no longer infinite.