{"paper":{"title":"Finite groups with the minimal generating set exchange property","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Andrea Lucchini, Patricia Medina Capilla","submitted_at":"2025-06-02T13:15:15Z","abstract_excerpt":"Let $d(G)$ be the smallest cardinality of a generating set of a finite group $G.$ We give a complete classification of the finite groups with the property that, whenever $ \\langle x_1, \\dots, x_{d(G)} \\rangle = \\langle y_1, \\dots, y_{d(G)} \\rangle = G$, for any $1 \\leq i \\leq d(G)$ there exists $1 \\leq j \\leq d(G)$ such that $\\langle x_1, \\dots, x_{i-1}, y_j, x_{i+1}, \\dots, x_{d(G)} \\rangle = G.$ We also prove that for every finite group $G$ and every maximal subgroup $M$ of $G$, there exists a generating set for $G$ of minimal size in which at least $d(G)-2$ elements belong to $M$. We conjec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.01638","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.01638/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}