{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:ONR7NVDY6V7EW6VETRHMNTL45Q","short_pith_number":"pith:ONR7NVDY","schema_version":"1.0","canonical_sha256":"7363f6d478f57e4b7aa49c4ec6cd7cec0800ca7901c0397fe664272cef893766","source":{"kind":"arxiv","id":"1305.3748","version":3},"attestation_state":"computed","paper":{"title":"Nilpotent covers and non-nilpotent subsets of finite groups of Lie type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Azizollah Azad, John R. Britnell, Nick Gill","submitted_at":"2013-05-16T10:22:16Z","abstract_excerpt":"Let $G$ be a finite group, and $c$ an element of $\\mathbb{Z}\\cup \\{\\infty\\}$. A subgroup $H$ of $G$ is said to be {\\it $c$-nilpotent} if it is nilpotent, and has nilpotency class at most $c$. A subset $X$ of $G$ is said to be {\\it non-$c$-nilpotent} if it contains no two elements $x$ and $y$ such that the subgroup $< x,y>$ is $c$-nilpotent. In this paper we study the quantity $\\omega_c(G)$, defined to be the size of the largest non-$c$-nilpotent subset of $L$.\n  In the case that $L$ is a finite group of Lie type, we identify covers of $L$ by $c$-nilpotent subgroups, and we use these covers to "},"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":"1305.3748","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-05-16T10:22:16Z","cross_cats_sorted":[],"title_canon_sha256":"1f8c1b165f526961b1ef7bfa0ad595165fad45f89c4e58ed3e01b8475ca7f5de","abstract_canon_sha256":"ff8752f299ce78622c8aa7f244c2c13a408730037bb0bd71ac8d0789444e544e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:45:32.041564Z","signature_b64":"iRy9WlnLlDHRAeDgsGxrfMs6oC9REYmWCrXr7+logSItY4BzzmaXj5lHjgpR7rqlQGjs1zLUWt6I/xVUwFJ/Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7363f6d478f57e4b7aa49c4ec6cd7cec0800ca7901c0397fe664272cef893766","last_reissued_at":"2026-05-18T02:45:32.040844Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:45:32.040844Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nilpotent covers and non-nilpotent subsets of finite groups of Lie type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Azizollah Azad, John R. Britnell, Nick Gill","submitted_at":"2013-05-16T10:22:16Z","abstract_excerpt":"Let $G$ be a finite group, and $c$ an element of $\\mathbb{Z}\\cup \\{\\infty\\}$. A subgroup $H$ of $G$ is said to be {\\it $c$-nilpotent} if it is nilpotent, and has nilpotency class at most $c$. A subset $X$ of $G$ is said to be {\\it non-$c$-nilpotent} if it contains no two elements $x$ and $y$ such that the subgroup $< x,y>$ is $c$-nilpotent. In this paper we study the quantity $\\omega_c(G)$, defined to be the size of the largest non-$c$-nilpotent subset of $L$.\n  In the case that $L$ is a finite group of Lie type, we identify covers of $L$ by $c$-nilpotent subgroups, and we use these covers to "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.3748","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1305.3748","created_at":"2026-05-18T02:45:32.040971+00:00"},{"alias_kind":"arxiv_version","alias_value":"1305.3748v3","created_at":"2026-05-18T02:45:32.040971+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.3748","created_at":"2026-05-18T02:45:32.040971+00:00"},{"alias_kind":"pith_short_12","alias_value":"ONR7NVDY6V7E","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_16","alias_value":"ONR7NVDY6V7EW6VE","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_8","alias_value":"ONR7NVDY","created_at":"2026-05-18T12:27:54.935989+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/ONR7NVDY6V7EW6VETRHMNTL45Q","json":"https://pith.science/pith/ONR7NVDY6V7EW6VETRHMNTL45Q.json","graph_json":"https://pith.science/api/pith-number/ONR7NVDY6V7EW6VETRHMNTL45Q/graph.json","events_json":"https://pith.science/api/pith-number/ONR7NVDY6V7EW6VETRHMNTL45Q/events.json","paper":"https://pith.science/paper/ONR7NVDY"},"agent_actions":{"view_html":"https://pith.science/pith/ONR7NVDY6V7EW6VETRHMNTL45Q","download_json":"https://pith.science/pith/ONR7NVDY6V7EW6VETRHMNTL45Q.json","view_paper":"https://pith.science/paper/ONR7NVDY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1305.3748&json=true","fetch_graph":"https://pith.science/api/pith-number/ONR7NVDY6V7EW6VETRHMNTL45Q/graph.json","fetch_events":"https://pith.science/api/pith-number/ONR7NVDY6V7EW6VETRHMNTL45Q/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ONR7NVDY6V7EW6VETRHMNTL45Q/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ONR7NVDY6V7EW6VETRHMNTL45Q/action/storage_attestation","attest_author":"https://pith.science/pith/ONR7NVDY6V7EW6VETRHMNTL45Q/action/author_attestation","sign_citation":"https://pith.science/pith/ONR7NVDY6V7EW6VETRHMNTL45Q/action/citation_signature","submit_replication":"https://pith.science/pith/ONR7NVDY6V7EW6VETRHMNTL45Q/action/replication_record"}},"created_at":"2026-05-18T02:45:32.040971+00:00","updated_at":"2026-05-18T02:45:32.040971+00:00"}