{"paper":{"title":"Invariable generation of the symmetric group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.GR","authors_text":"Ben Green, Kevin Ford, Sean Eberhard","submitted_at":"2015-08-08T08:36:02Z","abstract_excerpt":"We say that permutations $\\pi_1,\\dots, \\pi_r \\in \\mathcal{S}_n$ invariably generate $\\mathcal{S}_n$ if, no matter how one chooses conjugates $\\pi'_1,\\dots,\\pi'_r$ of these permutations, $\\pi'_1,\\dots,\\pi'_r$ generate $\\mathcal{S}_n$. We show that if $\\pi_1,\\pi_2,\\pi_3$ 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 positive probability. We include a proof of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.01870","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}