{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:6C7ZTJKLWKQKB3RNCPF6FJ6CJ3","short_pith_number":"pith:6C7ZTJKL","schema_version":"1.0","canonical_sha256":"f0bf99a54bb2a0a0ee2d13cbe2a7c24ec692de3a6a8678bd8ef7b288aa9d9467","source":{"kind":"arxiv","id":"1303.4966","version":1},"attestation_state":"computed","paper":{"title":"Automorphisms of groups and converse of Schur's theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Deepak Gumber, Hemant Kalra, Sandeep Singh","submitted_at":"2013-03-20T15:29:48Z","abstract_excerpt":"An automorphism of a group G is called an IA-automorphism if it induces the identity automorphism on the abelianized group G/G'. Let IA(G) denote the group of all IA-automorphisms of G. We classify all finitely generated nilpotent groups G of class 2 for which IA(G) is isomorphic to Inn(G). In particular, we classify all finite nilpotent groups of class 2 for which each IA-automorphism is inner. As consequences, we give surprisingly very easy proofs of converse of Schur's theorem and also prove some other related results."},"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":"1303.4966","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-03-20T15:29:48Z","cross_cats_sorted":[],"title_canon_sha256":"ba2294bfe5e351d565e440f12c40526c838d3248e45a191124f199ca72d701cc","abstract_canon_sha256":"77c04cb5549c53cb2dc0ff01def3355b0f0b1768ef8a855efc471b98d47fdaee"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:30:15.532417Z","signature_b64":"lkrlOF5t+xF4wYa/MxSiBWKG5xUNuYFFG0LclXIhqBq8A2dtvWJSBdl5kSizGdlhdi66D+k1cVHDSsnm/Q/IAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0bf99a54bb2a0a0ee2d13cbe2a7c24ec692de3a6a8678bd8ef7b288aa9d9467","last_reissued_at":"2026-05-18T03:30:15.531652Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:30:15.531652Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Automorphisms of groups and converse of Schur's theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Deepak Gumber, Hemant Kalra, Sandeep Singh","submitted_at":"2013-03-20T15:29:48Z","abstract_excerpt":"An automorphism of a group G is called an IA-automorphism if it induces the identity automorphism on the abelianized group G/G'. Let IA(G) denote the group of all IA-automorphisms of G. We classify all finitely generated nilpotent groups G of class 2 for which IA(G) is isomorphic to Inn(G). In particular, we classify all finite nilpotent groups of class 2 for which each IA-automorphism is inner. As consequences, we give surprisingly very easy proofs of converse of Schur's theorem and also prove some other related results."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.4966","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":"1303.4966","created_at":"2026-05-18T03:30:15.531770+00:00"},{"alias_kind":"arxiv_version","alias_value":"1303.4966v1","created_at":"2026-05-18T03:30:15.531770+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.4966","created_at":"2026-05-18T03:30:15.531770+00:00"},{"alias_kind":"pith_short_12","alias_value":"6C7ZTJKLWKQK","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_16","alias_value":"6C7ZTJKLWKQKB3RN","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_8","alias_value":"6C7ZTJKL","created_at":"2026-05-18T12:27:36.564083+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/6C7ZTJKLWKQKB3RNCPF6FJ6CJ3","json":"https://pith.science/pith/6C7ZTJKLWKQKB3RNCPF6FJ6CJ3.json","graph_json":"https://pith.science/api/pith-number/6C7ZTJKLWKQKB3RNCPF6FJ6CJ3/graph.json","events_json":"https://pith.science/api/pith-number/6C7ZTJKLWKQKB3RNCPF6FJ6CJ3/events.json","paper":"https://pith.science/paper/6C7ZTJKL"},"agent_actions":{"view_html":"https://pith.science/pith/6C7ZTJKLWKQKB3RNCPF6FJ6CJ3","download_json":"https://pith.science/pith/6C7ZTJKLWKQKB3RNCPF6FJ6CJ3.json","view_paper":"https://pith.science/paper/6C7ZTJKL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1303.4966&json=true","fetch_graph":"https://pith.science/api/pith-number/6C7ZTJKLWKQKB3RNCPF6FJ6CJ3/graph.json","fetch_events":"https://pith.science/api/pith-number/6C7ZTJKLWKQKB3RNCPF6FJ6CJ3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6C7ZTJKLWKQKB3RNCPF6FJ6CJ3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6C7ZTJKLWKQKB3RNCPF6FJ6CJ3/action/storage_attestation","attest_author":"https://pith.science/pith/6C7ZTJKLWKQKB3RNCPF6FJ6CJ3/action/author_attestation","sign_citation":"https://pith.science/pith/6C7ZTJKLWKQKB3RNCPF6FJ6CJ3/action/citation_signature","submit_replication":"https://pith.science/pith/6C7ZTJKLWKQKB3RNCPF6FJ6CJ3/action/replication_record"}},"created_at":"2026-05-18T03:30:15.531770+00:00","updated_at":"2026-05-18T03:30:15.531770+00:00"}