Invariable generation of the symmetric group

Sean Eberhard, Kevin Ford, Ben Green

Research output: Contribution to journalArticlepeer-review


We say that permutations π1, . . ., πr ∈ Sn invariably generate Sn if, no matter how one chooses conjugates π'1, . . ., π'r of these permutations, the π'1, . . . m π'r permutations generate Sn. We show that if π12, 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.

Original languageEnglish (US)
Pages (from-to)1573-1590
Number of pages18
JournalDuke Mathematical Journal
Issue number8
StatePublished - Jun 1 2017

ASJC Scopus subject areas

  • General Mathematics


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