{"paper":{"title":"Generating sets of finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Andrea Lucchini, Colva M. Roney-Dougal, Peter J. Cameron","submitted_at":"2016-09-20T10:06:19Z","abstract_excerpt":"We investigate the extent to which the exchange relation holds in finite groups $G$. We define a new equivalence relation $\\equiv_{\\mathrm{m}}$, where two elements are equivalent if each can be substituted for the other in any generating set for $G$. We then refine this to a new sequence $\\equiv_{\\mathrm{m}}^{(r)}$ of equivalence relations by saying that $x \\equiv_{\\mathrm{m}}^{(r)}y$ if each can be substituted for the other in any $r$-element generating set. The relations $\\equiv_{\\mathrm{m}}^{(r)}$ become finer as $r$ increases, and we define a new group invariant $\\psi(G)$ to be the value o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.06077","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"}