{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:7M3UKS7DBBMWEILTBPRFH6JIEF","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":"130cf27fc2b87829375675287025fc3da37589bc1746e35069a360392a984533","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-03-27T16:53:45Z","title_canon_sha256":"4faa1dc5dc9e1b93272270399074edcb7030b53269a2713738821481b2c144b4"},"schema_version":"1.0","source":{"id":"1803.10179","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.10179","created_at":"2026-05-18T00:07:27Z"},{"alias_kind":"arxiv_version","alias_value":"1803.10179v3","created_at":"2026-05-18T00:07:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.10179","created_at":"2026-05-18T00:07:27Z"},{"alias_kind":"pith_short_12","alias_value":"7M3UKS7DBBMW","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"7M3UKS7DBBMWEILT","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"7M3UKS7D","created_at":"2026-05-18T12:32:11Z"}],"graph_snapshots":[{"event_id":"sha256:ac95ac7fa450145be759f0a9b7b6bc2a938f024fc69c8a509f9d1e50bcb2e69f","target":"graph","created_at":"2026-05-18T00:07:27Z","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":"An $integral$ of a group $G$ is a group $H$ whose derived group (commutator subgroup) is isomorphic to $G$. This paper discusses integrals of groups, and in particular questions about which groups have integrals and how big or small those integrals can be. Our main results are:\n  (1) If a finite group has an integral, then it has a finite integral.\n  (2) A precise characterization of the set of natural numbers $n$ for which every group of order $n$ is integrable: these are the cubefree numbers $n$ which do not have prime divisors $p$ and $q$ with $q\\mid p-1$.\n  (3) An abelian group of order $n","authors_text":"Carlo Casolo, Francesco Matucci, Jo\\~ao Ara\\'ujo, Peter J. Cameron","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-03-27T16:53:45Z","title":"Integrals of groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.10179","kind":"arxiv","version":3},"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:b04b89efcce6ab30ed0b2247b221aea0bb13558020f14698193c151cbd4f6a47","target":"record","created_at":"2026-05-18T00:07:27Z","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":"130cf27fc2b87829375675287025fc3da37589bc1746e35069a360392a984533","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-03-27T16:53:45Z","title_canon_sha256":"4faa1dc5dc9e1b93272270399074edcb7030b53269a2713738821481b2c144b4"},"schema_version":"1.0","source":{"id":"1803.10179","kind":"arxiv","version":3}},"canonical_sha256":"fb37454be308596221730be253f9282148c2beb13e3f8e953f5fccb7279054b2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fb37454be308596221730be253f9282148c2beb13e3f8e953f5fccb7279054b2","first_computed_at":"2026-05-18T00:07:27.801341Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:07:27.801341Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Il6/njFgn7fmuoGM2ZxeDnAlnpAvqSjhNccTnYHo/oj0Yk61M6QKo3U2Ha1CGaTRvL/E6t2dvY80NNjpQ9BUBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:07:27.801966Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.10179","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b04b89efcce6ab30ed0b2247b221aea0bb13558020f14698193c151cbd4f6a47","sha256:ac95ac7fa450145be759f0a9b7b6bc2a938f024fc69c8a509f9d1e50bcb2e69f"],"state_sha256":"ecb95a9875fe3965bfc6bda2710b4d304688e541ee17d64bccec198fd9b69bbe"}