Duke Mathematical Journal

Invariable generation of the symmetric group

Sean Eberhard, Kevin Ford, and Ben Green

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We say that permutations π1,,πrSn invariably generate Sn if, no matter how one chooses conjugates π'1,,π'r of these permutations, the π'1,,π'r permutations generate Sn. We show that if π1,π2, and π3 are chosen randomly from Sn, then, with probability tending to 1 as n, they do not invariably generate Sn. By contrast, it was shown recently by Pemantle, Peres, and Rivin that four random elements do invariably generate Sn 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

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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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 20B30: Symmetric groups 05E15: Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80]

symmetric group invariable generation random generators


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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