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Let $N\\unlhd G$, write ${\\rm Irr}(G|N)=\\{\\chi \\in {\\rm Irr}(G)~|~N\\nsubseteq {\\rm ker}\\chi\\}$, and ${\\rm cod}(G|N)=\\{ {\\rm cod}(\\chi) ~|~\\chi\\in{\\rm Irr}(G|N)\\}.$ In this Ipaper, we prove that if $N\\unlhd G$ and every member of ${\\rm cod}(G|N')$ is not divisible by some fixed prime $p\\in \\pi(G)$, then $N$ has a normal $p$-complemen"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2102.07132","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2021-02-14T11:42:00Z","cross_cats_sorted":[],"title_canon_sha256":"a68b9975402b7aae7c4de5f34508655f955ae6351f97a4dc0dc7a2b9eef20ae8","abstract_canon_sha256":"3aaf520f9506b9b1e232a0c250da31a9ee8ceaab6f00b0d727252d933e49b545"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:32:26.660062Z","signature_b64":"HqeSvwwIC+OKGw9YrBA4Gpy5ZuOBu2mXWyXDPhvYJkSFeiiyJP0rP6vIjC3cI+7SaxPOGIZvbep52HV/b8m2Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4558912ae9b99f739bb9ff7dc701ae476d9ad13d0f68bb55ffe72f8fe8d9c5ca","last_reissued_at":"2026-07-05T02:32:26.659642Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:32:26.659642Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Normal $p$-complements and irreducible character codegrees","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Boru Zhang, Jiakuan Lu, Yu Li","submitted_at":"2021-02-14T11:42:00Z","abstract_excerpt":"Let $G$ be a finite group and $p\\in \\pi(G)$, and let Irr$(G)$ be the set of all irreducible complex characters of $G$. Let $\\chi \\in {\\rm Irr}(G)$, we write ${\\rm cod}(\\chi)=|G:{\\rm ker} \\chi|/\\chi(1)$, and called it the codegree of the irreducible character $\\chi$. Let $N\\unlhd G$, write ${\\rm Irr}(G|N)=\\{\\chi \\in {\\rm Irr}(G)~|~N\\nsubseteq {\\rm ker}\\chi\\}$, and ${\\rm cod}(G|N)=\\{ {\\rm cod}(\\chi) ~|~\\chi\\in{\\rm Irr}(G|N)\\}.$ In this Ipaper, we prove that if $N\\unlhd G$ and every member of ${\\rm cod}(G|N')$ is not divisible by some fixed prime $p\\in \\pi(G)$, then $N$ has a normal $p$-complemen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.07132","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2102.07132/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2102.07132","created_at":"2026-07-05T02:32:26.659702+00:00"},{"alias_kind":"arxiv_version","alias_value":"2102.07132v2","created_at":"2026-07-05T02:32:26.659702+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2102.07132","created_at":"2026-07-05T02:32:26.659702+00:00"},{"alias_kind":"pith_short_12","alias_value":"IVMJCKXJXGPX","created_at":"2026-07-05T02:32:26.659702+00:00"},{"alias_kind":"pith_short_16","alias_value":"IVMJCKXJXGPXHG5Z","created_at":"2026-07-05T02:32:26.659702+00:00"},{"alias_kind":"pith_short_8","alias_value":"IVMJCKXJ","created_at":"2026-07-05T02:32:26.659702+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IVMJCKXJXGPXHG5Z7564OANOI5","json":"https://pith.science/pith/IVMJCKXJXGPXHG5Z7564OANOI5.json","graph_json":"https://pith.science/api/pith-number/IVMJCKXJXGPXHG5Z7564OANOI5/graph.json","events_json":"https://pith.science/api/pith-number/IVMJCKXJXGPXHG5Z7564OANOI5/events.json","paper":"https://pith.science/paper/IVMJCKXJ"},"agent_actions":{"view_html":"https://pith.science/pith/IVMJCKXJXGPXHG5Z7564OANOI5","download_json":"https://pith.science/pith/IVMJCKXJXGPXHG5Z7564OANOI5.json","view_paper":"https://pith.science/paper/IVMJCKXJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2102.07132&json=true","fetch_graph":"https://pith.science/api/pith-number/IVMJCKXJXGPXHG5Z7564OANOI5/graph.json","fetch_events":"https://pith.science/api/pith-number/IVMJCKXJXGPXHG5Z7564OANOI5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IVMJCKXJXGPXHG5Z7564OANOI5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IVMJCKXJXGPXHG5Z7564OANOI5/action/storage_attestation","attest_author":"https://pith.science/pith/IVMJCKXJXGPXHG5Z7564OANOI5/action/author_attestation","sign_citation":"https://pith.science/pith/IVMJCKXJXGPXHG5Z7564OANOI5/action/citation_signature","submit_replication":"https://pith.science/pith/IVMJCKXJXGPXHG5Z7564OANOI5/action/replication_record"}},"created_at":"2026-07-05T02:32:26.659702+00:00","updated_at":"2026-07-05T02:32:26.659702+00:00"}