Four random permutations conjugated by an adversary generate S_n with high probability

Abstract

ABSTRACT:We prove a conjecture dating back to a 1978 paper of D.R. Musser [11], namely that four random permutations in the symmetric group S n generate a transitive subgroup with probability p n > ε for some ε > 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 a transitive subgroup; by a well known argument, this implies generation of A n or S n except for probability 1 + o(1) as n → ∞. The analysis is closely related to the following random set model. A random set M ⊆ Z + is generated by including each n ≥ 1 independently with probability 1/n. The sumset sumset(M) is formed. Then at most four independent copies of sumset(M) are needed before their mutual intersection is no longer infinite.

Invariable generation of the symmetric group

“…By Lemma 1.3, as n → ∞, the number of cycles in π of length at most 2k approaches a Poisson distribution with parameter h 2k 1 + log 2k. Thus, with high probability (as n → ∞) the total number of points in cycles of π of length at most 2k is at most 2k log n, so with high probability each of these cycles is disjoint from (12). That is, the points 1 and 2 are both contained in cycles of π of length at least 2k + 1 with high probability.…”

“…Hence, with high probability, if 1 and 2 are in the same cycle they are a distance at least k +1 from each other. Thus, with high probability, c j (πσ) = c j (π) for each j k. Similarly πσ (12) is uniformly distributed over odd permutations, and with high probability c j (πσ(12)) = c j (π).…”

“…, π C ∈ S n invariably generate S n with probability bounded away from zero. Their method does not directly yield a reasonable value of C, but recently Pemantle, Peres, and Rivin [12] proved that we may take C = 4. Theorem 1.1 (Pemantle-Peres-Rivin [12]).…”

“…Their method does not directly yield a reasonable value of C, but recently Pemantle, Peres, and Rivin [12] proved that we may take C = 4. Theorem 1.1 (Pemantle-Peres-Rivin [12]). If π 1 , π 2 , π 3 , π 4 ∈ S n are chosen uniformly at random then the probability that π 1 , π 2 , π 3 , π 4 invariably generate S n is bounded away from zero.…”

Some connections between permutation cycles and Touchard polynomials and between permutations that fix a set and covers of multisets

“…A lower bound on P (E m ≥ 1) of the form A log m m was obtained in [5]. These results were dramatically improved in [9] where it was shown that P (E m ≥ 1) = m −δ+o (1) as m → ∞, where δ = 1 − 1+log log 2 log 2 ≈ 0.08607. And very recently, in [7], this latter bound has been refined to…”