{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:G73GWJCLRYPD6KUQIRMW4MFNBZ","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":"9eb774b34db09c16b7fb0f54e429ef310cd2efd3ecacbd384a7f89ed379ddebc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-07-04T16:38:11Z","title_canon_sha256":"0f9bee2ccf0ccd6e7d0a9cf64f2a74de0df9b24c69ae508c881c41aff18accee"},"schema_version":"1.0","source":{"id":"1007.0568","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.0568","created_at":"2026-05-17T23:53:07Z"},{"alias_kind":"arxiv_version","alias_value":"1007.0568v2","created_at":"2026-05-17T23:53:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.0568","created_at":"2026-05-17T23:53:07Z"},{"alias_kind":"pith_short_12","alias_value":"G73GWJCLRYPD","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"G73GWJCLRYPD6KUQ","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"G73GWJCL","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:5b925d3ba50cf32814e7722b60b192e7392838d71bf4af90bf874abcdf1702a7","target":"graph","created_at":"2026-05-17T23:53:07Z","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":"If $G$ is a finite group and $x\\in G$ then the set of all elements of $G$ having the same order as $x$ is called {\\em an order subset of $G$ determined by $x$} (see [2]). We say that $G$ is a {\\em group with perfect order subsets} or briefly, $G$ is a {\\em $POS$-group} if the number of elements in each order subset of $G$ is a divisor of $|G|$. In this paper we prove that for any $n\\geq 4$, the symmetric group $S_n$ is not $POS$-group. This gives the positive answer to one of two questions rising from Conjecture 5.2 in [3].","authors_text":"Bui Xuan Hai, Nguyen Trong Tuan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-07-04T16:38:11Z","title":"On perfect order subsets in finite groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.0568","kind":"arxiv","version":2},"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":"sha256:2a845a295fc58e76f6095ebb0bd853923e5806bd4f299a956963d8d9df1ae85e","target":"record","created_at":"2026-05-17T23:53:07Z","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":"9eb774b34db09c16b7fb0f54e429ef310cd2efd3ecacbd384a7f89ed379ddebc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-07-04T16:38:11Z","title_canon_sha256":"0f9bee2ccf0ccd6e7d0a9cf64f2a74de0df9b24c69ae508c881c41aff18accee"},"schema_version":"1.0","source":{"id":"1007.0568","kind":"arxiv","version":2}},"canonical_sha256":"37f66b244b8e1e3f2a9044596e30ad0e43474e1027c18714ce25f875a064ae8a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"37f66b244b8e1e3f2a9044596e30ad0e43474e1027c18714ce25f875a064ae8a","first_computed_at":"2026-05-17T23:53:07.075132Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:53:07.075132Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TVgCNS7H1A9p7lZ4Ct2GfD024R9PhUjC+KIUIi2KUJl56OE4lZRm4brI0Wi9euX0GAc/E9GaF3y5QMrg0NBjAw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:53:07.075618Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.0568","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2a845a295fc58e76f6095ebb0bd853923e5806bd4f299a956963d8d9df1ae85e","sha256:5b925d3ba50cf32814e7722b60b192e7392838d71bf4af90bf874abcdf1702a7"],"state_sha256":"d047a424b062efd8643a000cf96c6c78eca3c1e2241a0b58b34a8d7cc5be9a4f"}