{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:FFFNSU32SDGZLIZZR4FLDORVLZ","short_pith_number":"pith:FFFNSU32","schema_version":"1.0","canonical_sha256":"294ad9537a90cd95a3398f0ab1ba355e60c38bc10371c9793ca32654c6b76673","source":{"kind":"arxiv","id":"1609.06077","version":1},"attestation_state":"computed","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"},"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":"1609.06077","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-09-20T10:06:19Z","cross_cats_sorted":[],"title_canon_sha256":"6bf03965f42ed65ad9b4b710a2b7a7914f888aad0bd2182632f11376a4ee9930","abstract_canon_sha256":"8ac08569f10b0fa1c8f319716ab01f5bb0fc1f75dec1dde772d761a1b4ad709a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:43.636154Z","signature_b64":"ZqAqm/t3B8HTC2i4BQsQrgQu4FOeR1xpuLAWyMHX6IVDj2WmYZiyJwiG0zJpVO8p4yyihcX7SrN1AsBW0xY8CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"294ad9537a90cd95a3398f0ab1ba355e60c38bc10371c9793ca32654c6b76673","last_reissued_at":"2026-05-17T23:44:43.635672Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:43.635672Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1609.06077","created_at":"2026-05-17T23:44:43.635746+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.06077v1","created_at":"2026-05-17T23:44:43.635746+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.06077","created_at":"2026-05-17T23:44:43.635746+00:00"},{"alias_kind":"pith_short_12","alias_value":"FFFNSU32SDGZ","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_16","alias_value":"FFFNSU32SDGZLIZZ","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_8","alias_value":"FFFNSU32","created_at":"2026-05-18T12:30:15.759754+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/FFFNSU32SDGZLIZZR4FLDORVLZ","json":"https://pith.science/pith/FFFNSU32SDGZLIZZR4FLDORVLZ.json","graph_json":"https://pith.science/api/pith-number/FFFNSU32SDGZLIZZR4FLDORVLZ/graph.json","events_json":"https://pith.science/api/pith-number/FFFNSU32SDGZLIZZR4FLDORVLZ/events.json","paper":"https://pith.science/paper/FFFNSU32"},"agent_actions":{"view_html":"https://pith.science/pith/FFFNSU32SDGZLIZZR4FLDORVLZ","download_json":"https://pith.science/pith/FFFNSU32SDGZLIZZR4FLDORVLZ.json","view_paper":"https://pith.science/paper/FFFNSU32","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.06077&json=true","fetch_graph":"https://pith.science/api/pith-number/FFFNSU32SDGZLIZZR4FLDORVLZ/graph.json","fetch_events":"https://pith.science/api/pith-number/FFFNSU32SDGZLIZZR4FLDORVLZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FFFNSU32SDGZLIZZR4FLDORVLZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FFFNSU32SDGZLIZZR4FLDORVLZ/action/storage_attestation","attest_author":"https://pith.science/pith/FFFNSU32SDGZLIZZR4FLDORVLZ/action/author_attestation","sign_citation":"https://pith.science/pith/FFFNSU32SDGZLIZZR4FLDORVLZ/action/citation_signature","submit_replication":"https://pith.science/pith/FFFNSU32SDGZLIZZR4FLDORVLZ/action/replication_record"}},"created_at":"2026-05-17T23:44:43.635746+00:00","updated_at":"2026-05-17T23:44:43.635746+00:00"}