{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:6525CTUXBD66E2XYQDBOCJT4Q7","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":"d0a9a2dc6767081b75cd87f4dbbb8bbf67a76d4b2647a11a20c1bba437537f2a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-12-17T16:32:38Z","title_canon_sha256":"883b9e1471db30eef91a128a166b725fcc13653d4f9231d3e331adf644448a2e"},"schema_version":"1.0","source":{"id":"1412.5469","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.5469","created_at":"2026-05-18T02:31:02Z"},{"alias_kind":"arxiv_version","alias_value":"1412.5469v1","created_at":"2026-05-18T02:31:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.5469","created_at":"2026-05-18T02:31:02Z"},{"alias_kind":"pith_short_12","alias_value":"6525CTUXBD66","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"6525CTUXBD66E2XY","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"6525CTUX","created_at":"2026-05-18T12:28:16Z"}],"graph_snapshots":[{"event_id":"sha256:2f63593c5516eb5d11246b34e2e01297a8fe732a244cede4e58347fed1ad4c84","target":"graph","created_at":"2026-05-18T02:31:02Z","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":"A proper subgroup $H$ of a group $G$ is said to be: $\\Bbb{P}$-subnormal in $G$ if there exists a chain of subgroups $H=H_0 < H_1< ... < H_{n}=G$ such that $|H_{i}:H_{i-1}|$ is a prime for $i=1,...,n$; $\\Bbb{P}$-abnormal in $G$ if for every two subgroups $K\\leq L$ of $G$, where $H\\leq K$, $|L:K|$ is not a prime.\n In this paper we describe finite groups in which every non-identity subgroup is either $\\Bbb{P}$-subnormal or $\\Bbb{P}$-abnormal.","authors_text":"Alexander N. Skiba, Vladimir N. Semenchuk","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-12-17T16:32:38Z","title":"On one generalization of finite $\\frak U$-critical groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.5469","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:da0fe3b2bc0c2a440795d2f771108ade89cb7537be01d90193b529c789898ad6","target":"record","created_at":"2026-05-18T02:31:02Z","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":"d0a9a2dc6767081b75cd87f4dbbb8bbf67a76d4b2647a11a20c1bba437537f2a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-12-17T16:32:38Z","title_canon_sha256":"883b9e1471db30eef91a128a166b725fcc13653d4f9231d3e331adf644448a2e"},"schema_version":"1.0","source":{"id":"1412.5469","kind":"arxiv","version":1}},"canonical_sha256":"f775d14e9708fde26af880c2e1267c87d6218d0c318b641d8a083bfc250a9e21","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f775d14e9708fde26af880c2e1267c87d6218d0c318b641d8a083bfc250a9e21","first_computed_at":"2026-05-18T02:31:02.908475Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:31:02.908475Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wbBvsdyV8JLvln8+GDNoMCQXPy0NR3nFxpFr7VatpC2CRQ0kG9LnMtFTfBFthPB8kJ/l1vx8zvODqOCopWR8Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:31:02.908883Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.5469","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:da0fe3b2bc0c2a440795d2f771108ade89cb7537be01d90193b529c789898ad6","sha256:2f63593c5516eb5d11246b34e2e01297a8fe732a244cede4e58347fed1ad4c84"],"state_sha256":"2afa577edb3f0ef513bd35ec2c52d84ffac3e89bda7bcedd62cdce881e7fb9fe"}