Abstract
We say that permutations invariably generate if, no matter how one chooses conjugates of these permutations, the permutations generate . We show that if , and are chosen randomly from , then, with probability tending to as , they do not invariably generate . By contrast, it was shown recently by Pemantle, Peres, and Rivin that four random elements do invariably generate with probability bounded away from zero. We include a proof of this statement which, while sharing many features with their argument, is short and completely combinatorial.
Citation
Sean Eberhard. Kevin Ford. Ben Green. "Invariable generation of the symmetric group." Duke Math. J. 166 (8) 1573 - 1590, 1 June 2017. https://doi.org/10.1215/00127094-0000007X
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