{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:PBTZL2TY7VAI2GQQEKXHVAM2IK","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":"2824581b5c650ac36711919798e4ef9b9b51e64fa71a37a8f75dad29dac391d0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-08-05T11:44:34Z","title_canon_sha256":"e918c814ab0d3da8ed2ecd0ad854fa9d13f582803e8e5b2cd1b297366f990f08"},"schema_version":"1.0","source":{"id":"1308.0948","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.0948","created_at":"2026-05-18T01:35:57Z"},{"alias_kind":"arxiv_version","alias_value":"1308.0948v2","created_at":"2026-05-18T01:35:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.0948","created_at":"2026-05-18T01:35:57Z"},{"alias_kind":"pith_short_12","alias_value":"PBTZL2TY7VAI","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"PBTZL2TY7VAI2GQQ","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"PBTZL2TY","created_at":"2026-05-18T12:27:54Z"}],"graph_snapshots":[{"event_id":"sha256:a95f2aed30fc17e2bb6fe019e8f03ac53e27a64f39168e96617655a98d9bb0e8","target":"graph","created_at":"2026-05-18T01:35:57Z","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 $\\mathfrak{H}$ be a Fitting class and $\\mathfrak{F}$ a formation. We call a subgroup $\\mathcal{N}_{\\mathfrak{H},\\mathfrak{F}}(G)$ of a finite group $G$ the $\\mathfrak{H}$-$\\mathfrak{F}$-norm of $G$ if $\\mathcal{N}_{\\mathfrak{H},\\mathfrak{F}}(G)$ is the intersection of the normalizers of the products of the $\\mathfrak{F}$-residuals of all subgroups of $G$ and the $\\mathfrak{H}$-radical of $G$. Let $\\pi$ denote a set of primes and let $\\mathfrak{G}_\\pi$ denote the class of all finite $\\pi$-groups. We call the subgroup $\\mathcal{N}_{\\mathfrak{G}_\\pi,\\mathfrak{F}}(G)$ of $G$ the $\\pi\\mathfrak{","authors_text":"Wenbin Guo, Xiaoyu Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-08-05T11:44:34Z","title":"On the $\\pi$$\\mathfrak{F}$-norm and the $\\mathfrak{H}$-$\\mathfrak{F}$-norm of a finite group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.0948","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:add8fee157aee177c3d74d9015aa2cd37588a8d1ec060e2b319a42b3227232cd","target":"record","created_at":"2026-05-18T01:35:57Z","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":"2824581b5c650ac36711919798e4ef9b9b51e64fa71a37a8f75dad29dac391d0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-08-05T11:44:34Z","title_canon_sha256":"e918c814ab0d3da8ed2ecd0ad854fa9d13f582803e8e5b2cd1b297366f990f08"},"schema_version":"1.0","source":{"id":"1308.0948","kind":"arxiv","version":2}},"canonical_sha256":"786795ea78fd408d1a1022ae7a819a42b86e2cdac84ef4b656610852f0ad913e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"786795ea78fd408d1a1022ae7a819a42b86e2cdac84ef4b656610852f0ad913e","first_computed_at":"2026-05-18T01:35:57.540561Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:35:57.540561Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"o/Kj9hQhKtgEKodSB8auNSIj47qMg/O2D8u9K8irq0kc8I3LFi0+8NtaG6E42HBnrA1LkP2QPI4qC/FlUa74DA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:35:57.541229Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.0948","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:add8fee157aee177c3d74d9015aa2cd37588a8d1ec060e2b319a42b3227232cd","sha256:a95f2aed30fc17e2bb6fe019e8f03ac53e27a64f39168e96617655a98d9bb0e8"],"state_sha256":"2a972e6975704bc2d0b13ce59d30d378e1ea5113ff16a9b2ea34a2a1787439d1"}