{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:GBCNJ7SIUI2TBGMRVDDRJGPVPB","short_pith_number":"pith:GBCNJ7SI","schema_version":"1.0","canonical_sha256":"3044d4fe48a235309991a8c71499f578734ae9f69b3d6e5b29c8e800a195d37c","source":{"kind":"arxiv","id":"1206.4279","version":1},"attestation_state":"computed","paper":{"title":"Normal coverings of linear groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Attila Maroti, John R. Britnell","submitted_at":"2012-06-19T17:49:11Z","abstract_excerpt":"For a non-cyclic finite group $G$, let $\\gamma(G)$ denote the smallest number of conjugacy classes of proper subgroups of $G$ needed to cover $G$. Bubboloni, Praeger and Spiga, motivated by questions in number theory, have recently established that $\\gamma(S_n)$ and $\\gamma(A_{n})$ are bounded above and below by linear functions of $n$. In this paper we show that if $G$ is in the range $\\SL_{n}(q)\\le G\\le \\GL_{n}(q)$ for $n>2$, then $n/\\pi^2 < \\gamma(G) \\le (n+1)/2$. We give various alternative bounds, and derive explicit formulas for $\\gamma(G)$ in some cases."},"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":"1206.4279","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-06-19T17:49:11Z","cross_cats_sorted":[],"title_canon_sha256":"a109394520ccd9b64721ea73d61c70663b6ccf3ef8b623479cc0461ea8f28f8b","abstract_canon_sha256":"e299167419049a6751081b4e6e0a4cbcfb91bfd798b8f3e3a208ca2fb4393dfe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:53:07.510307Z","signature_b64":"yU1nosfkeHwk5AqfuGDiBGPoh9SlXlw87E5OeVDBnYoC+6l+a+6o33LcDvPNIQWPZWzu0xZ7ljFtUBjlCmZqBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3044d4fe48a235309991a8c71499f578734ae9f69b3d6e5b29c8e800a195d37c","last_reissued_at":"2026-05-18T03:53:07.509596Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:53:07.509596Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Normal coverings of linear groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Attila Maroti, John R. Britnell","submitted_at":"2012-06-19T17:49:11Z","abstract_excerpt":"For a non-cyclic finite group $G$, let $\\gamma(G)$ denote the smallest number of conjugacy classes of proper subgroups of $G$ needed to cover $G$. Bubboloni, Praeger and Spiga, motivated by questions in number theory, have recently established that $\\gamma(S_n)$ and $\\gamma(A_{n})$ are bounded above and below by linear functions of $n$. In this paper we show that if $G$ is in the range $\\SL_{n}(q)\\le G\\le \\GL_{n}(q)$ for $n>2$, then $n/\\pi^2 < \\gamma(G) \\le (n+1)/2$. We give various alternative bounds, and derive explicit formulas for $\\gamma(G)$ in some cases."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.4279","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1206.4279","created_at":"2026-05-18T03:53:07.509701+00:00"},{"alias_kind":"arxiv_version","alias_value":"1206.4279v1","created_at":"2026-05-18T03:53:07.509701+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.4279","created_at":"2026-05-18T03:53:07.509701+00:00"},{"alias_kind":"pith_short_12","alias_value":"GBCNJ7SIUI2T","created_at":"2026-05-18T12:27:06.952714+00:00"},{"alias_kind":"pith_short_16","alias_value":"GBCNJ7SIUI2TBGMR","created_at":"2026-05-18T12:27:06.952714+00:00"},{"alias_kind":"pith_short_8","alias_value":"GBCNJ7SI","created_at":"2026-05-18T12:27:06.952714+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/GBCNJ7SIUI2TBGMRVDDRJGPVPB","json":"https://pith.science/pith/GBCNJ7SIUI2TBGMRVDDRJGPVPB.json","graph_json":"https://pith.science/api/pith-number/GBCNJ7SIUI2TBGMRVDDRJGPVPB/graph.json","events_json":"https://pith.science/api/pith-number/GBCNJ7SIUI2TBGMRVDDRJGPVPB/events.json","paper":"https://pith.science/paper/GBCNJ7SI"},"agent_actions":{"view_html":"https://pith.science/pith/GBCNJ7SIUI2TBGMRVDDRJGPVPB","download_json":"https://pith.science/pith/GBCNJ7SIUI2TBGMRVDDRJGPVPB.json","view_paper":"https://pith.science/paper/GBCNJ7SI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1206.4279&json=true","fetch_graph":"https://pith.science/api/pith-number/GBCNJ7SIUI2TBGMRVDDRJGPVPB/graph.json","fetch_events":"https://pith.science/api/pith-number/GBCNJ7SIUI2TBGMRVDDRJGPVPB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GBCNJ7SIUI2TBGMRVDDRJGPVPB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GBCNJ7SIUI2TBGMRVDDRJGPVPB/action/storage_attestation","attest_author":"https://pith.science/pith/GBCNJ7SIUI2TBGMRVDDRJGPVPB/action/author_attestation","sign_citation":"https://pith.science/pith/GBCNJ7SIUI2TBGMRVDDRJGPVPB/action/citation_signature","submit_replication":"https://pith.science/pith/GBCNJ7SIUI2TBGMRVDDRJGPVPB/action/replication_record"}},"created_at":"2026-05-18T03:53:07.509701+00:00","updated_at":"2026-05-18T03:53:07.509701+00:00"}