{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:GBCNJ7SIUI2TBGMRVDDRJGPVPB","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":"e299167419049a6751081b4e6e0a4cbcfb91bfd798b8f3e3a208ca2fb4393dfe","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-06-19T17:49:11Z","title_canon_sha256":"a109394520ccd9b64721ea73d61c70663b6ccf3ef8b623479cc0461ea8f28f8b"},"schema_version":"1.0","source":{"id":"1206.4279","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.4279","created_at":"2026-05-18T03:53:07Z"},{"alias_kind":"arxiv_version","alias_value":"1206.4279v1","created_at":"2026-05-18T03:53:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.4279","created_at":"2026-05-18T03:53:07Z"},{"alias_kind":"pith_short_12","alias_value":"GBCNJ7SIUI2T","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"GBCNJ7SIUI2TBGMR","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"GBCNJ7SI","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:d5c95c021f071e83deb7aaf4c7e53087fe49e735fc6f24412ece27dde17b881d","target":"graph","created_at":"2026-05-18T03:53:07Z","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":"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.","authors_text":"Attila Maroti, John R. Britnell","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-06-19T17:49:11Z","title":"Normal coverings of linear groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.4279","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":"sha256:34b9e0f2224f2bbafc8e583a7659e3621bccaf55a3f5b1df56c1edebd1bc6c0d","target":"record","created_at":"2026-05-18T03:53:07Z","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":"e299167419049a6751081b4e6e0a4cbcfb91bfd798b8f3e3a208ca2fb4393dfe","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-06-19T17:49:11Z","title_canon_sha256":"a109394520ccd9b64721ea73d61c70663b6ccf3ef8b623479cc0461ea8f28f8b"},"schema_version":"1.0","source":{"id":"1206.4279","kind":"arxiv","version":1}},"canonical_sha256":"3044d4fe48a235309991a8c71499f578734ae9f69b3d6e5b29c8e800a195d37c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3044d4fe48a235309991a8c71499f578734ae9f69b3d6e5b29c8e800a195d37c","first_computed_at":"2026-05-18T03:53:07.509596Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:53:07.509596Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yU1nosfkeHwk5AqfuGDiBGPoh9SlXlw87E5OeVDBnYoC+6l+a+6o33LcDvPNIQWPZWzu0xZ7ljFtUBjlCmZqBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:53:07.510307Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.4279","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:34b9e0f2224f2bbafc8e583a7659e3621bccaf55a3f5b1df56c1edebd1bc6c0d","sha256:d5c95c021f071e83deb7aaf4c7e53087fe49e735fc6f24412ece27dde17b881d"],"state_sha256":"4d3c878d611b35aa57b2e2335b26cc6b5750fe86575b6531a4be336d026ed094"}