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    Permutations contained in transitive subgroups

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    Permutations contained in transitive subgroups, Discrete Analysis 2016:12, 36 pp. This paper is part of a series. The previous paper in the series [2] concerned the following question. A property of elements x1,…,xtx_1,\dots,x_t of a group GG is said to hold _invariably_ if it holds even if x1,…,xtx_1,\dots,x_t are replaced by arbitrary conjugates. How many random permutations in SnS_n are needed for the probability that they invariably generate SnS_n to be bounded away from zero? It was known that four elements suffice [3], and the main result of [2] was that three elements did not suffice. This was achieved with the help of an estimate for the probability that a random permutation contains a fixed set (that is, a set that maps to itself) of given size kk, obtained in the first paper of the series [1]. Of course, a fixed set of size kk exists if and only if the cycle decomposition of the permutation contains cycles of lengths that add up to kk. The relevance of the sizes of fixed sets is that if permutations Ο€1,…,Ο€t\pi_1,\dots,\pi_t all have fixed sets of size kk, then we can find conjugates that have the same fixed set of size kk, and these conjugates clearly do not generate SnS_n. The estimate obtained for the probability that a random permutation has a fixed set of size kk is kβˆ’Ξ΄(1+log⁑k)βˆ’3/2k^{-\delta}(1+\log k)^{-3/2}, up to a constant, where Ξ΄=1βˆ’(1+log⁑log⁑2)/log⁑2\delta=1-(1+\log\log 2)/\log 2. This estimate holds for all kk (with the same constant) up to n/2n/2. Another consequence of the estimate in [1] is that when nn is even, the probability that a random permutation in SnS_n is contained in a transitive subgroup of SnS_n other than SnS_n or AnA_n is at least cnβˆ’Ξ΄(log⁑n)βˆ’3/2cn^{-\delta}(\log n)^{-3/2}. That is because with at least that probability there is a fixed set of size n/2n/2, so we can take the group of permutations that preserve the partition of {1,2,…,n}\{1,2,\dots,n\} into that set and its complement. The conjectures are made in that paper that there should be a matching upper bound, and a stronger upper bound when nn is odd. The current paper proves both these conjectures. The behaviour turns out to depend critically on the smallest prime factor pp of nn, with a change in behaviour as soon as pβ‰₯5p\geq 5. In order to prove these results, the authors obtain estimates for the probability that a random permutation Ο€βˆˆSn\pi\in S_n has disjoint fixed sets of sizes k1,…,kmk_1,\dots,k_m for given k1,…,kmk_1,\dots,k_m that add up to nn. Results of this kind have many applications, and this paper makes a significant contribution to our understanding of the subgroup structure of the symmetric group. [1] S. Eberhard, K. Ford and B. J. Green, _Permutations fixing a kk-set_, [arxiv:1507.04465](http://arxiv.org/abs/1507.04465) [2] S. Eberhard, K. Ford and B. J. Green, _Invariable generation of the symmetric group_, [arxiv:1508.01870](http://arxiv.org/abs/1508.01870) [3] R. Pemantle, Y. Peres and I. Rivin, _Four random permutations conjugated by an adversary generate SnS_n with high probability_, [arxiv:1412.3781](http://arxiv.org/abs/1412.3781
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