{"paper":{"title":"The probability that a pair of elements of a finite group are conjugate","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"John R. Britnell, Mark Wildon, Simon R. Blackburn","submitted_at":"2011-08-08T19:24:51Z","abstract_excerpt":"Let $G$ be a finite group, and let $\\kappa(G)$ be the probability that elements $g$, $h\\in G$ are conjugate, when $g$ and $h$ are chosen independently and uniformly at random. The paper classifies those groups $G$ such that $\\kappa(G) \\geq 1/4$, and shows that $G$ is abelian whenever $\\kappa(G)|G| < 7/4$. It is also shown that $\\kappa(G)|G|$ depends only on the isoclinism class of $G$.\n  Specialising to the symmetric group $S_n$, the paper shows that $\\kappa(S_n) \\leq C/n^2$ for an explicitly determined constant $C$. This bound leads to an elementary proof of a result of Flajolet \\emph{et al},"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.1784","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"}