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In this paper we show that if $\\Cal A$ covers all the elements of G the same times and $G_1,...,G_k$ are subnormal subgroups of G not all equal to G, then $M=\\max_{1\\le j\\le k}|\\{1\\le i\\le k:n_i=n_j\\}|$ is not less than the smallest prime divisor of $n_1... n_k$, moreover $\\min_{1\\ls i\\ls k}\\log n_i=O(M\\log^2 M)$ where the O-constant is absolute."},"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":"math/0306099","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.GR","submitted_at":"2003-06-05T16:38:18Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"d31db4ab5bd9b5ee840aa0f3927e04afc4df66718372a0aad085442b3ef889f3","abstract_canon_sha256":"b933561657e9ad14a66a39d9e5e75691dc6298e110b4049e4b370fe0e1d0eeda"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:37:22.956215Z","signature_b64":"JXH+4mnwA1I6Pf3n5Iepm10P3SgopmiNxVG1Q408Wa0V3l9smuAwsh2tGdvq6qcGLn/zaTzG39QjWTup6n7aAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c5b33f611af8ad730d4aa9781997a7106be0b3d18d433ac5218008228323c45c","last_reissued_at":"2026-07-04T14:37:22.955828Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:37:22.955828Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Herzog-Sch\\\"onheim conjecture for uniform covers of groups","license":"","headline":"","cross_cats":["math.NT"],"primary_cat":"math.GR","authors_text":"Zhi-Wei Sun","submitted_at":"2003-06-05T16:38:18Z","abstract_excerpt":"Let G be any group and $a_1G_1,...,a_kG_k (k>1)$ be left cosets in G. In 1974 Herzog and Sch\\\"onheim conjectured that if $\\Cal A=\\{a_iG_i\\}_{i=1}^k$ is a partition of G then the (finite) indices $n_1=[G:G_1],...,n_k=[G:G_k]$ cannot be distinct. In this paper we show that if $\\Cal A$ covers all the elements of G the same times and $G_1,...,G_k$ are subnormal subgroups of G not all equal to G, then $M=\\max_{1\\le j\\le k}|\\{1\\le i\\le k:n_i=n_j\\}|$ is not less than the smallest prime divisor of $n_1... n_k$, moreover $\\min_{1\\ls i\\ls k}\\log n_i=O(M\\log^2 M)$ where the O-constant is absolute."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0306099","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/math/0306099/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":"math/0306099","created_at":"2026-07-04T14:37:22.955885+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0306099v2","created_at":"2026-07-04T14:37:22.955885+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0306099","created_at":"2026-07-04T14:37:22.955885+00:00"},{"alias_kind":"pith_short_12","alias_value":"YWZT6YI27CWX","created_at":"2026-07-04T14:37:22.955885+00:00"},{"alias_kind":"pith_short_16","alias_value":"YWZT6YI27CWXGDKK","created_at":"2026-07-04T14:37:22.955885+00:00"},{"alias_kind":"pith_short_8","alias_value":"YWZT6YI2","created_at":"2026-07-04T14:37:22.955885+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/YWZT6YI27CWXGDKKVF4BTF5HCB","json":"https://pith.science/pith/YWZT6YI27CWXGDKKVF4BTF5HCB.json","graph_json":"https://pith.science/api/pith-number/YWZT6YI27CWXGDKKVF4BTF5HCB/graph.json","events_json":"https://pith.science/api/pith-number/YWZT6YI27CWXGDKKVF4BTF5HCB/events.json","paper":"https://pith.science/paper/YWZT6YI2"},"agent_actions":{"view_html":"https://pith.science/pith/YWZT6YI27CWXGDKKVF4BTF5HCB","download_json":"https://pith.science/pith/YWZT6YI27CWXGDKKVF4BTF5HCB.json","view_paper":"https://pith.science/paper/YWZT6YI2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0306099&json=true","fetch_graph":"https://pith.science/api/pith-number/YWZT6YI27CWXGDKKVF4BTF5HCB/graph.json","fetch_events":"https://pith.science/api/pith-number/YWZT6YI27CWXGDKKVF4BTF5HCB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YWZT6YI27CWXGDKKVF4BTF5HCB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YWZT6YI27CWXGDKKVF4BTF5HCB/action/storage_attestation","attest_author":"https://pith.science/pith/YWZT6YI27CWXGDKKVF4BTF5HCB/action/author_attestation","sign_citation":"https://pith.science/pith/YWZT6YI27CWXGDKKVF4BTF5HCB/action/citation_signature","submit_replication":"https://pith.science/pith/YWZT6YI27CWXGDKKVF4BTF5HCB/action/replication_record"}},"created_at":"2026-07-04T14:37:22.955885+00:00","updated_at":"2026-07-04T14:37:22.955885+00:00"}