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Four Random Permutations Conjugated by an Adversary Generate with High Probability
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 generate a transitive subgroup with probability
for some independent of , even when an
adversary is allowed to conjugate each of the four by a possibly different
element of (in other words, the cycle types already guarantee generation
of ). This is closely related to the following random set model.
A random set is generated by including each independently with probability . The sumset is
formed. Then at most four independent copies of are needed
before their mutual intersection is no longer infinite.Comment: 19pages, 1 figur