1 research outputs found

    Four Random Permutations Conjugated by an Adversary Generate SnS_n with High Probability

    Full text link
    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 Sn\mathcal{S}_n generate a transitive subgroup with probability pn>ϵp_n > \epsilon for some ϵ>0\epsilon > 0 independent of nn, even when an adversary is allowed to conjugate each of the four by a possibly different element of §n\S_n (in other words, the cycle types already guarantee generation of Sn\mathcal{S}_n). This is closely related to the following random set model. A random set M⊆Z+M \subseteq \mathbb{Z}^+ is generated by including each n≥1n \geq 1 independently with probability 1/n1/n. The sumset sumset(M)\text{sumset}(M) is formed. Then at most four independent copies of sumset(M)\text{sumset}(M) are needed before their mutual intersection is no longer infinite.Comment: 19pages, 1 figur
    corecore