Duke Mathematical Journal
- Duke Math. J.
- Volume 166, Number 8 (2017), 1573-1590.
Invariable generation of the symmetric group
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.
Duke Math. J., Volume 166, Number 8 (2017), 1573-1590.
Received: 10 August 2015
Revised: 13 August 2016
First available in Project Euclid: 10 February 2017
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Eberhard, Sean; Ford, Kevin; Green, Ben. Invariable generation of the symmetric group. Duke Math. J. 166 (2017), no. 8, 1573--1590. doi:10.1215/00127094-0000007X. https://projecteuclid.org/euclid.dmj/1486695668