Duke Math. J. 166 (8), 1573-1590, (1 June 2017) DOI: 10.1215/00127094-0000007X
Sean Eberhard, Kevin Ford, Ben Green
KEYWORDS: Symmetric group, invariable generation, random generators, 60C05, 20B30, 05E15
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.