{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:EYCBDOFTIBIRF2E4IOGRDWBC2N","short_pith_number":"pith:EYCBDOFT","schema_version":"1.0","canonical_sha256":"260411b8b3405112e89c438d11d822d36a8c6f14b076fb81a014bc26402e1de6","source":{"kind":"arxiv","id":"2506.01638","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2506.01638","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2025-06-02T13:15:15Z","cross_cats_sorted":[],"title_canon_sha256":"c2e86a6809aae0c584624c9fced8ffa52b0643938f21d92b711608bf6f4627d5","abstract_canon_sha256":"a5e7dad6f03f199382fe3d1f4501a34ce7579744bbb35a91476eec710725303c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:14:17.142389Z","signature_b64":"X0aJf4W//uYZGsV0QYd3np3+JLknTirUGxnq79bpdZbYIe0GyR2CRjyzJZPl4LA2u9D9QbDXL6wb/+c+XMzcBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"260411b8b3405112e89c438d11d822d36a8c6f14b076fb81a014bc26402e1de6","last_reissued_at":"2026-07-05T11:14:17.141967Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:14:17.141967Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2506.01638","created_at":"2026-07-05T11:14:17.142019+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.01638v1","created_at":"2026-07-05T11:14:17.142019+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.01638","created_at":"2026-07-05T11:14:17.142019+00:00"},{"alias_kind":"pith_short_12","alias_value":"EYCBDOFTIBIR","created_at":"2026-07-05T11:14:17.142019+00:00"},{"alias_kind":"pith_short_16","alias_value":"EYCBDOFTIBIRF2E4","created_at":"2026-07-05T11:14:17.142019+00:00"},{"alias_kind":"pith_short_8","alias_value":"EYCBDOFT","created_at":"2026-07-05T11:14:17.142019+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/EYCBDOFTIBIRF2E4IOGRDWBC2N","json":"https://pith.science/pith/EYCBDOFTIBIRF2E4IOGRDWBC2N.json","graph_json":"https://pith.science/api/pith-number/EYCBDOFTIBIRF2E4IOGRDWBC2N/graph.json","events_json":"https://pith.science/api/pith-number/EYCBDOFTIBIRF2E4IOGRDWBC2N/events.json","paper":"https://pith.science/paper/EYCBDOFT"},"agent_actions":{"view_html":"https://pith.science/pith/EYCBDOFTIBIRF2E4IOGRDWBC2N","download_json":"https://pith.science/pith/EYCBDOFTIBIRF2E4IOGRDWBC2N.json","view_paper":"https://pith.science/paper/EYCBDOFT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.01638&json=true","fetch_graph":"https://pith.science/api/pith-number/EYCBDOFTIBIRF2E4IOGRDWBC2N/graph.json","fetch_events":"https://pith.science/api/pith-number/EYCBDOFTIBIRF2E4IOGRDWBC2N/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EYCBDOFTIBIRF2E4IOGRDWBC2N/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EYCBDOFTIBIRF2E4IOGRDWBC2N/action/storage_attestation","attest_author":"https://pith.science/pith/EYCBDOFTIBIRF2E4IOGRDWBC2N/action/author_attestation","sign_citation":"https://pith.science/pith/EYCBDOFTIBIRF2E4IOGRDWBC2N/action/citation_signature","submit_replication":"https://pith.science/pith/EYCBDOFTIBIRF2E4IOGRDWBC2N/action/replication_record"}},"created_at":"2026-07-05T11:14:17.142019+00:00","updated_at":"2026-07-05T11:14:17.142019+00:00"}