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1 June 2017 Invariable generation of the symmetric group
Sean Eberhard, Kevin Ford, Ben Green
Duke Math. J. 166(8): 1573-1590 (1 June 2017). DOI: 10.1215/00127094-0000007X

Abstract

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.

Citation

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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

Information

Received: 10 August 2015; Revised: 13 August 2016; Published: 1 June 2017
First available in Project Euclid: 10 February 2017

zbMATH: 06754739
MathSciNet: MR3659942
Digital Object Identifier: 10.1215/00127094-0000007X

Subjects:
Primary: 60C05
Secondary: 05E15 , 20B30

Keywords: invariable generation , random generators , Symmetric group

Rights: Copyright © 2017 Duke University Press

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Vol.166 • No. 8 • 1 June 2017
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