{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:TMXYYTV4P7WPNGIYPJWC3L72DA","short_pith_number":"pith:TMXYYTV4","canonical_record":{"source":{"id":"1505.06762","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-05-25T21:34:05Z","cross_cats_sorted":[],"title_canon_sha256":"4e20d1cc9847bdd24b4a132d259f66535a491fe24f28c02e5cbe2ca4981036d8","abstract_canon_sha256":"7e7ac3e2bfd701d889d2ece2c08ba36987ea7499cf95174108c2b918cf329d9a"},"schema_version":"1.0"},"canonical_sha256":"9b2f8c4ebc7fecf699187a6c2daffa18043ecd595bf4d11cd20bb4d27c529732","source":{"kind":"arxiv","id":"1505.06762","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.06762","created_at":"2026-05-18T02:03:31Z"},{"alias_kind":"arxiv_version","alias_value":"1505.06762v1","created_at":"2026-05-18T02:03:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.06762","created_at":"2026-05-18T02:03:31Z"},{"alias_kind":"pith_short_12","alias_value":"TMXYYTV4P7WP","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"TMXYYTV4P7WPNGIY","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"TMXYYTV4","created_at":"2026-05-18T12:29:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:TMXYYTV4P7WPNGIYPJWC3L72DA","target":"record","payload":{"canonical_record":{"source":{"id":"1505.06762","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-05-25T21:34:05Z","cross_cats_sorted":[],"title_canon_sha256":"4e20d1cc9847bdd24b4a132d259f66535a491fe24f28c02e5cbe2ca4981036d8","abstract_canon_sha256":"7e7ac3e2bfd701d889d2ece2c08ba36987ea7499cf95174108c2b918cf329d9a"},"schema_version":"1.0"},"canonical_sha256":"9b2f8c4ebc7fecf699187a6c2daffa18043ecd595bf4d11cd20bb4d27c529732","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:03:31.914269Z","signature_b64":"cYQl7rclItguWkZQz/zJYLIOzI/iKuerWmBVASR6EerBZK3u90H3+jlH/mE0lZHl5BwP6MR4mgAHFKSBRlWHBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9b2f8c4ebc7fecf699187a6c2daffa18043ecd595bf4d11cd20bb4d27c529732","last_reissued_at":"2026-05-18T02:03:31.913665Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:03:31.913665Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1505.06762","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:03:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q84+QIa0UEZNCIOqQf/Qcbsif4yPwap87KCE8B+Ve925csB2SQy2qFZbcdImfpWfPozoYNzpoEn5ee1ysZ+ECg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T07:50:18.929557Z"},"content_sha256":"090f7c59cb88eec3ea9ebf5141f9c399405f971022d1e90faeb6e5990920039a","schema_version":"1.0","event_id":"sha256:090f7c59cb88eec3ea9ebf5141f9c399405f971022d1e90faeb6e5990920039a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:TMXYYTV4P7WPNGIYPJWC3L72DA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Variants of theorems of Baer and Hall on finite-by-hypercentral groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Carlo Casolo, Silvana Rinauro, Ulderico Dardano","submitted_at":"2015-05-25T21:34:05Z","abstract_excerpt":"We show that if a group $G$ has a finite normal subgroup $L$ such that $G/L$ is hypercentral, then the index of the hypercenter of $G$ is bounded by a function of the order of $L$. This completes recent results generalizing classical theorems by R. Baer and P. Hall. Then we apply our results to groups of automorphisms of a group $G$ acting in a restricted way on an ascending normal series of $G$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.06762","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:03:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"F177RyYlj9TwJbmfXlrPjzQwgKnC2jQ7XJboX92rYx3Mj02pANUW28uZeSeKo2FqukiPUPvhVEr4ltpfPi9jCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T07:50:18.929907Z"},"content_sha256":"7181811fcf55bde26f2d402b8c15be0d1bfc96823675fe1a628d29159d1894d2","schema_version":"1.0","event_id":"sha256:7181811fcf55bde26f2d402b8c15be0d1bfc96823675fe1a628d29159d1894d2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TMXYYTV4P7WPNGIYPJWC3L72DA/bundle.json","state_url":"https://pith.science/pith/TMXYYTV4P7WPNGIYPJWC3L72DA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TMXYYTV4P7WPNGIYPJWC3L72DA/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-02T07:50:18Z","links":{"resolver":"https://pith.science/pith/TMXYYTV4P7WPNGIYPJWC3L72DA","bundle":"https://pith.science/pith/TMXYYTV4P7WPNGIYPJWC3L72DA/bundle.json","state":"https://pith.science/pith/TMXYYTV4P7WPNGIYPJWC3L72DA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TMXYYTV4P7WPNGIYPJWC3L72DA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:TMXYYTV4P7WPNGIYPJWC3L72DA","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":"7e7ac3e2bfd701d889d2ece2c08ba36987ea7499cf95174108c2b918cf329d9a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-05-25T21:34:05Z","title_canon_sha256":"4e20d1cc9847bdd24b4a132d259f66535a491fe24f28c02e5cbe2ca4981036d8"},"schema_version":"1.0","source":{"id":"1505.06762","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.06762","created_at":"2026-05-18T02:03:31Z"},{"alias_kind":"arxiv_version","alias_value":"1505.06762v1","created_at":"2026-05-18T02:03:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.06762","created_at":"2026-05-18T02:03:31Z"},{"alias_kind":"pith_short_12","alias_value":"TMXYYTV4P7WP","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"TMXYYTV4P7WPNGIY","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"TMXYYTV4","created_at":"2026-05-18T12:29:42Z"}],"graph_snapshots":[{"event_id":"sha256:7181811fcf55bde26f2d402b8c15be0d1bfc96823675fe1a628d29159d1894d2","target":"graph","created_at":"2026-05-18T02:03:31Z","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 show that if a group $G$ has a finite normal subgroup $L$ such that $G/L$ is hypercentral, then the index of the hypercenter of $G$ is bounded by a function of the order of $L$. This completes recent results generalizing classical theorems by R. Baer and P. Hall. Then we apply our results to groups of automorphisms of a group $G$ acting in a restricted way on an ascending normal series of $G$.","authors_text":"Carlo Casolo, Silvana Rinauro, Ulderico Dardano","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-05-25T21:34:05Z","title":"Variants of theorems of Baer and Hall on finite-by-hypercentral groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.06762","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":"sha256:090f7c59cb88eec3ea9ebf5141f9c399405f971022d1e90faeb6e5990920039a","target":"record","created_at":"2026-05-18T02:03:31Z","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":"7e7ac3e2bfd701d889d2ece2c08ba36987ea7499cf95174108c2b918cf329d9a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-05-25T21:34:05Z","title_canon_sha256":"4e20d1cc9847bdd24b4a132d259f66535a491fe24f28c02e5cbe2ca4981036d8"},"schema_version":"1.0","source":{"id":"1505.06762","kind":"arxiv","version":1}},"canonical_sha256":"9b2f8c4ebc7fecf699187a6c2daffa18043ecd595bf4d11cd20bb4d27c529732","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9b2f8c4ebc7fecf699187a6c2daffa18043ecd595bf4d11cd20bb4d27c529732","first_computed_at":"2026-05-18T02:03:31.913665Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:03:31.913665Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cYQl7rclItguWkZQz/zJYLIOzI/iKuerWmBVASR6EerBZK3u90H3+jlH/mE0lZHl5BwP6MR4mgAHFKSBRlWHBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:03:31.914269Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.06762","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:090f7c59cb88eec3ea9ebf5141f9c399405f971022d1e90faeb6e5990920039a","sha256:7181811fcf55bde26f2d402b8c15be0d1bfc96823675fe1a628d29159d1894d2"],"state_sha256":"3ecbe74c14885bcef5335d7bc17a57139c4bf6cfb45d7608e4f224118f7de905"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U/MYnUa1/izFspu0C7eNVYTY0hlZxALrug6wV/sqvrAdpmrQy+PzMVKCWqjOvj80A9xOwsbiVRxQo0jx67bfDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T07:50:18.931857Z","bundle_sha256":"9ff43edfe8937f09353a5b1d478aea557739fed77e66eeedeb622a1a4922156c"}}