{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:C42PKWDYSPB67GAHP5SZGL6AJG","short_pith_number":"pith:C42PKWDY","schema_version":"1.0","canonical_sha256":"1734f5587893c3ef98077f65932fc04993d9f3e6a04c5fdcbfc7d8db3420ecc7","source":{"kind":"arxiv","id":"1810.03508","version":1},"attestation_state":"computed","paper":{"title":"The generating graph of the abelian groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Andrea Lucchini, Cristina Acciarri","submitted_at":"2018-10-08T14:52:23Z","abstract_excerpt":"For a group $G,$ let $\\Gamma(G)$ denote the graph defined on the elements of $G$ in such a way that two distinct vertices are connected by an edge if and only if they generate $G$. Moreover let $\\Gamma^*(G)$ be the subgraph of $\\Gamma(G)$ that is induced by all the vertices of $\\Gamma(G)$ that are not isolated. We prove that if $G$ is a 2-generated non-cyclic abelian group then $\\Gamma^*(G)$ is connected. Moreover $\\mathrm{diam}(\\Gamma^*(G))=2$ if the torsion subgroup of $G$ is non-trivial and $\\mathrm{diam}(\\Gamma^*(G))=\\infty$ otherwise. If $F$ is the free group of rank 2, then $\\Gamma^*(F)$"},"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":"1810.03508","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-10-08T14:52:23Z","cross_cats_sorted":[],"title_canon_sha256":"9df262740247d8503c54db9b95d5b539812804c51d798840bd1bf7d2bb854f58","abstract_canon_sha256":"76e29f6e411fd1860348a611cca88a2d5b9595f2c4024a0fbe253f25628cb257"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:51.082788Z","signature_b64":"Z9v4kcwPJ+pNlTvjEe3nI41XOtCjLwpiLNbaM6Qn3aLT5iL/dg2NpdlBUVv6IRQAWxf0yDbI8qL2DZObeUxIAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1734f5587893c3ef98077f65932fc04993d9f3e6a04c5fdcbfc7d8db3420ecc7","last_reissued_at":"2026-05-18T00:03:51.082184Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:51.082184Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The generating graph of the abelian groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Andrea Lucchini, Cristina Acciarri","submitted_at":"2018-10-08T14:52:23Z","abstract_excerpt":"For a group $G,$ let $\\Gamma(G)$ denote the graph defined on the elements of $G$ in such a way that two distinct vertices are connected by an edge if and only if they generate $G$. Moreover let $\\Gamma^*(G)$ be the subgraph of $\\Gamma(G)$ that is induced by all the vertices of $\\Gamma(G)$ that are not isolated. We prove that if $G$ is a 2-generated non-cyclic abelian group then $\\Gamma^*(G)$ is connected. Moreover $\\mathrm{diam}(\\Gamma^*(G))=2$ if the torsion subgroup of $G$ is non-trivial and $\\mathrm{diam}(\\Gamma^*(G))=\\infty$ otherwise. If $F$ is the free group of rank 2, then $\\Gamma^*(F)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.03508","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":"1810.03508","created_at":"2026-05-18T00:03:51.082275+00:00"},{"alias_kind":"arxiv_version","alias_value":"1810.03508v1","created_at":"2026-05-18T00:03:51.082275+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.03508","created_at":"2026-05-18T00:03:51.082275+00:00"},{"alias_kind":"pith_short_12","alias_value":"C42PKWDYSPB6","created_at":"2026-05-18T12:32:16.446611+00:00"},{"alias_kind":"pith_short_16","alias_value":"C42PKWDYSPB67GAH","created_at":"2026-05-18T12:32:16.446611+00:00"},{"alias_kind":"pith_short_8","alias_value":"C42PKWDY","created_at":"2026-05-18T12:32:16.446611+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/C42PKWDYSPB67GAHP5SZGL6AJG","json":"https://pith.science/pith/C42PKWDYSPB67GAHP5SZGL6AJG.json","graph_json":"https://pith.science/api/pith-number/C42PKWDYSPB67GAHP5SZGL6AJG/graph.json","events_json":"https://pith.science/api/pith-number/C42PKWDYSPB67GAHP5SZGL6AJG/events.json","paper":"https://pith.science/paper/C42PKWDY"},"agent_actions":{"view_html":"https://pith.science/pith/C42PKWDYSPB67GAHP5SZGL6AJG","download_json":"https://pith.science/pith/C42PKWDYSPB67GAHP5SZGL6AJG.json","view_paper":"https://pith.science/paper/C42PKWDY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1810.03508&json=true","fetch_graph":"https://pith.science/api/pith-number/C42PKWDYSPB67GAHP5SZGL6AJG/graph.json","fetch_events":"https://pith.science/api/pith-number/C42PKWDYSPB67GAHP5SZGL6AJG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/C42PKWDYSPB67GAHP5SZGL6AJG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/C42PKWDYSPB67GAHP5SZGL6AJG/action/storage_attestation","attest_author":"https://pith.science/pith/C42PKWDYSPB67GAHP5SZGL6AJG/action/author_attestation","sign_citation":"https://pith.science/pith/C42PKWDYSPB67GAHP5SZGL6AJG/action/citation_signature","submit_replication":"https://pith.science/pith/C42PKWDYSPB67GAHP5SZGL6AJG/action/replication_record"}},"created_at":"2026-05-18T00:03:51.082275+00:00","updated_at":"2026-05-18T00:03:51.082275+00:00"}