{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:UHCRIGM63CEBBNBOZS4FHX3XUY","short_pith_number":"pith:UHCRIGM6","schema_version":"1.0","canonical_sha256":"a1c514199ed88810b42eccb853df77a607a04058db05ec45dab2bcf419a0c860","source":{"kind":"arxiv","id":"1505.03462","version":1},"attestation_state":"computed","paper":{"title":"Classification of finite groups with toroidal or projective-planar permutability graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Andrei Gagarin, P. Devi, R. Rajkumar","submitted_at":"2015-05-13T17:24:46Z","abstract_excerpt":"Let $G$ be a group. The permutability graph of subgroups of $G$, denoted by $\\Gamma(G)$, is a graph having all the proper subgroups of $G$ as its vertices, and two subgroups are adjacent in $\\Gamma(G)$ if and only if they permute. In this paper, we classify the finite groups whose permutability graphs are toroidal or projective-planar. In addition, we classify the finite groups whose permutability graph does not contain one of $K_{3,3}$, $K_{1,5}$, $C_6$, $P_5$, or $P_6$ as a subgraph."},"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":"1505.03462","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-05-13T17:24:46Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d2cc376316648402406b8b8eb21036c92ab04b13df84696301adb6f2424aadf9","abstract_canon_sha256":"b9f5d27bb772d7091b9f0a5a8fd3f6c465d242fc250ff4e4fa10253c6098bd67"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:59.770118Z","signature_b64":"mFCW/xV+ou2eNv+oDnwMA/pTBTZ1NafnRp0yCvUdQbTQo3ofBa78v075NWzAr58RHMYoF9uAgN8zWgWKd6HhBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a1c514199ed88810b42eccb853df77a607a04058db05ec45dab2bcf419a0c860","last_reissued_at":"2026-05-18T01:12:59.769758Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:59.769758Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Classification of finite groups with toroidal or projective-planar permutability graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Andrei Gagarin, P. Devi, R. Rajkumar","submitted_at":"2015-05-13T17:24:46Z","abstract_excerpt":"Let $G$ be a group. The permutability graph of subgroups of $G$, denoted by $\\Gamma(G)$, is a graph having all the proper subgroups of $G$ as its vertices, and two subgroups are adjacent in $\\Gamma(G)$ if and only if they permute. In this paper, we classify the finite groups whose permutability graphs are toroidal or projective-planar. In addition, we classify the finite groups whose permutability graph does not contain one of $K_{3,3}$, $K_{1,5}$, $C_6$, $P_5$, or $P_6$ as a subgraph."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.03462","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":"1505.03462","created_at":"2026-05-18T01:12:59.769813+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.03462v1","created_at":"2026-05-18T01:12:59.769813+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.03462","created_at":"2026-05-18T01:12:59.769813+00:00"},{"alias_kind":"pith_short_12","alias_value":"UHCRIGM63CEB","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"UHCRIGM63CEBBNBO","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"UHCRIGM6","created_at":"2026-05-18T12:29:44.643036+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/UHCRIGM63CEBBNBOZS4FHX3XUY","json":"https://pith.science/pith/UHCRIGM63CEBBNBOZS4FHX3XUY.json","graph_json":"https://pith.science/api/pith-number/UHCRIGM63CEBBNBOZS4FHX3XUY/graph.json","events_json":"https://pith.science/api/pith-number/UHCRIGM63CEBBNBOZS4FHX3XUY/events.json","paper":"https://pith.science/paper/UHCRIGM6"},"agent_actions":{"view_html":"https://pith.science/pith/UHCRIGM63CEBBNBOZS4FHX3XUY","download_json":"https://pith.science/pith/UHCRIGM63CEBBNBOZS4FHX3XUY.json","view_paper":"https://pith.science/paper/UHCRIGM6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.03462&json=true","fetch_graph":"https://pith.science/api/pith-number/UHCRIGM63CEBBNBOZS4FHX3XUY/graph.json","fetch_events":"https://pith.science/api/pith-number/UHCRIGM63CEBBNBOZS4FHX3XUY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UHCRIGM63CEBBNBOZS4FHX3XUY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UHCRIGM63CEBBNBOZS4FHX3XUY/action/storage_attestation","attest_author":"https://pith.science/pith/UHCRIGM63CEBBNBOZS4FHX3XUY/action/author_attestation","sign_citation":"https://pith.science/pith/UHCRIGM63CEBBNBOZS4FHX3XUY/action/citation_signature","submit_replication":"https://pith.science/pith/UHCRIGM63CEBBNBOZS4FHX3XUY/action/replication_record"}},"created_at":"2026-05-18T01:12:59.769813+00:00","updated_at":"2026-05-18T01:12:59.769813+00:00"}