{"paper":{"title":"Limit points in the range of the commuting probability function on finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Peter Hegarty","submitted_at":"2012-07-03T17:32:53Z","abstract_excerpt":"If G is a finite group, then Pr(G) denotes the fraction of ordered pairs of elements of G which commute. We show that, if l \\in (2/9,1] is a limit point of the function Pr on finite groups, then l \\in \\Q and there exists an e = e_l > 0 such that Pr(G) \\not\\in (l - e_l, l) for any finite group G. These results lend support to some old conjectures of Keith Joseph."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.0760","kind":"arxiv","version":1},"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"}