## Duke Mathematical Journal

### Invariable generation of the symmetric group

#### Abstract

We say that permutations $\pi_{1},\ldots,\pi_{r}\in\mathcal{S}_{n}$ invariably generate $\mathcal{S}_{n}$ if, no matter how one chooses conjugates $\pi'_{1},\ldots,\pi'_{r}$ of these permutations, the $\pi'_{1},\ldots,\pi'_{r}$ permutations generate $\mathcal{S}_{n}$. We show that if $\pi_{1},\pi_{2}$, and $\pi_{3}$ are chosen randomly from $\mathcal{S}_{n}$, then, with probability tending to $1$ as $n\rightarrow\infty$, they do not invariably generate $\mathcal{S}_{n}$. By contrast, it was shown recently by Pemantle, Peres, and Rivin that four random elements do invariably generate $\mathcal{S}_{n}$ 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.

#### Article information

Source
Duke Math. J., Volume 166, Number 8 (2017), 1573-1590.

Dates
Revised: 13 August 2016
First available in Project Euclid: 10 February 2017

https://projecteuclid.org/euclid.dmj/1486695668

Digital Object Identifier
doi:10.1215/00127094-0000007X

Mathematical Reviews number (MathSciNet)
MR3659942

Zentralblatt MATH identifier
06754739

#### Citation

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

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