{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:6NCPVD7MDB7KURBX5NLZQJLXNF","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":"ec3e8f606f3f0ca09ef22c06c51b6b4f31ec2756d52d21081d6f9ee6aed91674","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-10-21T15:14:22Z","title_canon_sha256":"61ca1018202ffa99c5fcd117ca83bb4e94e958c268d3ef8781df8f487f22dcac"},"schema_version":"1.0","source":{"id":"1510.06293","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.06293","created_at":"2026-05-18T01:26:09Z"},{"alias_kind":"arxiv_version","alias_value":"1510.06293v2","created_at":"2026-05-18T01:26:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.06293","created_at":"2026-05-18T01:26:09Z"},{"alias_kind":"pith_short_12","alias_value":"6NCPVD7MDB7K","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6NCPVD7MDB7KURBX","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6NCPVD7M","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:39dbec9ed1b60aa9030247bc132a9a2ffd803bcebf055b28951dfb01eae3005b","target":"graph","created_at":"2026-05-18T01:26:09Z","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":"Let $G$ be a Camina $p$-group of nilpotence class $3$. We prove that if $G' < C_G (G')$, then $|Z(G)| \\le |G':G_3|^{1/2}$. We also prove that if $G/G_3$ has only one or two abelian subgroups of order $|G:G'|$, then $G' < C_G (G')$. If $G/G_3$ has $p^a + 1$ abelian subgroups of order $|G:G'|$, then either $G' < C_G (G')$ or $|Z(G)| \\le p^{2a}$.","authors_text":"Mark L. Lewis","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-10-21T15:14:22Z","title":"Centralizers of Camina $p$-groups of nilpotence class $3$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.06293","kind":"arxiv","version":2},"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":"sha256:830f6c5bfb478b26a62dc26633ad576dbcff95ac9fdd95abf4d9152af87aad59","target":"record","created_at":"2026-05-18T01:26:09Z","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":"ec3e8f606f3f0ca09ef22c06c51b6b4f31ec2756d52d21081d6f9ee6aed91674","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-10-21T15:14:22Z","title_canon_sha256":"61ca1018202ffa99c5fcd117ca83bb4e94e958c268d3ef8781df8f487f22dcac"},"schema_version":"1.0","source":{"id":"1510.06293","kind":"arxiv","version":2}},"canonical_sha256":"f344fa8fec187eaa4437eb57982577694b22c2dc4623dab4ef6eb9bd3aba9766","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f344fa8fec187eaa4437eb57982577694b22c2dc4623dab4ef6eb9bd3aba9766","first_computed_at":"2026-05-18T01:26:09.325116Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:26:09.325116Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mwAMP0wdc3hbthGfJ10AcwB97xkIPuoWxVxIFBbJdVXjO+t+/DA+J8+A/6BQRmGLhh4dEzHS4o7De4CeOcRZDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:26:09.325782Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.06293","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:830f6c5bfb478b26a62dc26633ad576dbcff95ac9fdd95abf4d9152af87aad59","sha256:39dbec9ed1b60aa9030247bc132a9a2ffd803bcebf055b28951dfb01eae3005b"],"state_sha256":"dbb6d820f4cafccd2943cc9e9466c3f66389183b1185e25b079a97d51710b6fc"}