{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:FFFNSU32SDGZLIZZR4FLDORVLZ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"8ac08569f10b0fa1c8f319716ab01f5bb0fc1f75dec1dde772d761a1b4ad709a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-09-20T10:06:19Z","title_canon_sha256":"6bf03965f42ed65ad9b4b710a2b7a7914f888aad0bd2182632f11376a4ee9930"},"schema_version":"1.0","source":{"id":"1609.06077","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.06077","created_at":"2026-05-17T23:44:43Z"},{"alias_kind":"arxiv_version","alias_value":"1609.06077v1","created_at":"2026-05-17T23:44:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.06077","created_at":"2026-05-17T23:44:43Z"},{"alias_kind":"pith_short_12","alias_value":"FFFNSU32SDGZ","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FFFNSU32SDGZLIZZ","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FFFNSU32","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:3469c7ae582f2e6fa0bac65b3244e272b367f34f33ed0fe8decba17aa874de26","target":"graph","created_at":"2026-05-17T23:44:43Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Andrea Lucchini, Colva M. Roney-Dougal, Peter J. Cameron","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-09-20T10:06:19Z","title":"Generating sets of finite groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.06077","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:63046cda946057b1e4cc51d87c3fd34b1bf4a1b38b60703d15d1877cd12d824d","target":"record","created_at":"2026-05-17T23:44:43Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"8ac08569f10b0fa1c8f319716ab01f5bb0fc1f75dec1dde772d761a1b4ad709a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-09-20T10:06:19Z","title_canon_sha256":"6bf03965f42ed65ad9b4b710a2b7a7914f888aad0bd2182632f11376a4ee9930"},"schema_version":"1.0","source":{"id":"1609.06077","kind":"arxiv","version":1}},"canonical_sha256":"294ad9537a90cd95a3398f0ab1ba355e60c38bc10371c9793ca32654c6b76673","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"294ad9537a90cd95a3398f0ab1ba355e60c38bc10371c9793ca32654c6b76673","first_computed_at":"2026-05-17T23:44:43.635672Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:44:43.635672Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZqAqm/t3B8HTC2i4BQsQrgQu4FOeR1xpuLAWyMHX6IVDj2WmYZiyJwiG0zJpVO8p4yyihcX7SrN1AsBW0xY8CQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:44:43.636154Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.06077","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:63046cda946057b1e4cc51d87c3fd34b1bf4a1b38b60703d15d1877cd12d824d","sha256:3469c7ae582f2e6fa0bac65b3244e272b367f34f33ed0fe8decba17aa874de26"],"state_sha256":"7ad53847ec1c11b88c3c954c62e5537effe19df598d543f83adcf02b2449b76c"}