{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:A2TB7BLGEEQJTM6J4XEXHPH6SN","short_pith_number":"pith:A2TB7BLG","schema_version":"1.0","canonical_sha256":"06a61f8566212099b3c9e5c973bcfe935cd06a109396674f93f2799d810fb037","source":{"kind":"arxiv","id":"1305.2368","version":1},"attestation_state":"computed","paper":{"title":"A Characterization of the Prime Graphs of Solvable Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alexander Gruber, Benjamin Strasser, Keeley Naughton, Mark Lewis, Thomas Keller","submitted_at":"2013-05-10T15:32:24Z","abstract_excerpt":"Let \\pi(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G is the graph with vertex set \\pi(G) with edges {p,q} if and only if there exists an element of order pq in G. In this paper, we prove that a graph is isomorphic to the prime graph of a solvable group if and only if its complement is 3-colorable and triangle free. We then introduce the idea of a minimal prime graph. We prove that there exists an infinite class of solvable groups whose prime graphs are minimal. We prove the 3k-conjecture on prime divisors in element orders for solvable groups with "},"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":"1305.2368","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-05-10T15:32:24Z","cross_cats_sorted":[],"title_canon_sha256":"2e9c6cdf72afd12b0168fac6806a0307771aad75665ceffe422e0270e99e5e69","abstract_canon_sha256":"ebf210c768c3297c1a3be328641490e7b9891869f101bcc9a68a3bcd1cd6fb02"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:26:00.193337Z","signature_b64":"CxpiiYBEXWia78z44M0g/JvXHLrYyO4HKAX+JgnGxFKLy80e7ZDQVcvenqR4WwvBQUyLGmOPQTs54NzjgWIlAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"06a61f8566212099b3c9e5c973bcfe935cd06a109396674f93f2799d810fb037","last_reissued_at":"2026-05-18T03:26:00.192586Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:26:00.192586Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Characterization of the Prime Graphs of Solvable Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alexander Gruber, Benjamin Strasser, Keeley Naughton, Mark Lewis, Thomas Keller","submitted_at":"2013-05-10T15:32:24Z","abstract_excerpt":"Let \\pi(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G is the graph with vertex set \\pi(G) with edges {p,q} if and only if there exists an element of order pq in G. In this paper, we prove that a graph is isomorphic to the prime graph of a solvable group if and only if its complement is 3-colorable and triangle free. We then introduce the idea of a minimal prime graph. We prove that there exists an infinite class of solvable groups whose prime graphs are minimal. We prove the 3k-conjecture on prime divisors in element orders for solvable groups with "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.2368","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":"1305.2368","created_at":"2026-05-18T03:26:00.192706+00:00"},{"alias_kind":"arxiv_version","alias_value":"1305.2368v1","created_at":"2026-05-18T03:26:00.192706+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.2368","created_at":"2026-05-18T03:26:00.192706+00:00"},{"alias_kind":"pith_short_12","alias_value":"A2TB7BLGEEQJ","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_16","alias_value":"A2TB7BLGEEQJTM6J","created_at":"2026-05-18T12:27:38.830355+00:00"},{"alias_kind":"pith_short_8","alias_value":"A2TB7BLG","created_at":"2026-05-18T12:27:38.830355+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/A2TB7BLGEEQJTM6J4XEXHPH6SN","json":"https://pith.science/pith/A2TB7BLGEEQJTM6J4XEXHPH6SN.json","graph_json":"https://pith.science/api/pith-number/A2TB7BLGEEQJTM6J4XEXHPH6SN/graph.json","events_json":"https://pith.science/api/pith-number/A2TB7BLGEEQJTM6J4XEXHPH6SN/events.json","paper":"https://pith.science/paper/A2TB7BLG"},"agent_actions":{"view_html":"https://pith.science/pith/A2TB7BLGEEQJTM6J4XEXHPH6SN","download_json":"https://pith.science/pith/A2TB7BLGEEQJTM6J4XEXHPH6SN.json","view_paper":"https://pith.science/paper/A2TB7BLG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1305.2368&json=true","fetch_graph":"https://pith.science/api/pith-number/A2TB7BLGEEQJTM6J4XEXHPH6SN/graph.json","fetch_events":"https://pith.science/api/pith-number/A2TB7BLGEEQJTM6J4XEXHPH6SN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/A2TB7BLGEEQJTM6J4XEXHPH6SN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/A2TB7BLGEEQJTM6J4XEXHPH6SN/action/storage_attestation","attest_author":"https://pith.science/pith/A2TB7BLGEEQJTM6J4XEXHPH6SN/action/author_attestation","sign_citation":"https://pith.science/pith/A2TB7BLGEEQJTM6J4XEXHPH6SN/action/citation_signature","submit_replication":"https://pith.science/pith/A2TB7BLGEEQJTM6J4XEXHPH6SN/action/replication_record"}},"created_at":"2026-05-18T03:26:00.192706+00:00","updated_at":"2026-05-18T03:26:00.192706+00:00"}