{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:522K3WBPO5AO2EMCW3XIM7BJKE","short_pith_number":"pith:522K3WBP","canonical_record":{"source":{"id":"1206.6340","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-06-27T17:14:50Z","cross_cats_sorted":["math.CO","math.RT"],"title_canon_sha256":"855724928acc1520094eeb17247786d07403386f0d950ac0d72d075930df55d9","abstract_canon_sha256":"83f7e19e0b0e9436e7946f51e7f3a59a2c448a1474ea5e41fbe36cd6046c31c3"},"schema_version":"1.0"},"canonical_sha256":"eeb4add82f7740ed1182b6ee867c29513bb5ea54771322b00540b0f9443ec06b","source":{"kind":"arxiv","id":"1206.6340","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.6340","created_at":"2026-05-18T03:52:26Z"},{"alias_kind":"arxiv_version","alias_value":"1206.6340v1","created_at":"2026-05-18T03:52:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.6340","created_at":"2026-05-18T03:52:26Z"},{"alias_kind":"pith_short_12","alias_value":"522K3WBPO5AO","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"522K3WBPO5AO2EMC","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"522K3WBP","created_at":"2026-05-18T12:26:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:522K3WBPO5AO2EMCW3XIM7BJKE","target":"record","payload":{"canonical_record":{"source":{"id":"1206.6340","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-06-27T17:14:50Z","cross_cats_sorted":["math.CO","math.RT"],"title_canon_sha256":"855724928acc1520094eeb17247786d07403386f0d950ac0d72d075930df55d9","abstract_canon_sha256":"83f7e19e0b0e9436e7946f51e7f3a59a2c448a1474ea5e41fbe36cd6046c31c3"},"schema_version":"1.0"},"canonical_sha256":"eeb4add82f7740ed1182b6ee867c29513bb5ea54771322b00540b0f9443ec06b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:52:26.934592Z","signature_b64":"2SgCmdXAc/UjWPYkMjShRx4NpNtS2GpMFLBX66pMxbIYVYNt1J7x374EwFcRJ96tGecGFf1OSPqEdXnyR/itBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eeb4add82f7740ed1182b6ee867c29513bb5ea54771322b00540b0f9443ec06b","last_reissued_at":"2026-05-18T03:52:26.933937Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:52:26.933937Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.6340","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:52:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+05gibYALGTcfqW3nYR3iBjQm6IASii63g5KPPwhzWfRHBs/NpVFk8G9lZwwtCJuBghDYqpqp2w6pMJhTJ7CDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T07:07:48.621563Z"},"content_sha256":"dee6ec98af617cd54f6274a7d747c684c465733f99ac64a3259bba26a54776ed","schema_version":"1.0","event_id":"sha256:dee6ec98af617cd54f6274a7d747c684c465733f99ac64a3259bba26a54776ed"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:522K3WBPO5AO2EMCW3XIM7BJKE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On extendability of permutations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.RT"],"primary_cat":"math.GR","authors_text":"Mark Pankov","submitted_at":"2012-06-27T17:14:50Z","abstract_excerpt":"Let $V$ be a left vector space over a division ring and let ${\\mathcal P}(V)$ be the associated projective space. We describe all finite subsets $X\\subset V$ such that every permutation on $X$ can be extended to a linear automorphism of $V$ and all finite subsets ${\\mathcal X}\\subset {\\mathcal P}(V)$ such that every permutation on ${\\mathcal X}$ can be extended to an element of ${\\rm PGL}(V)$. Also, we reformulate the results in terms of linear and projective representations of symmetric groups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.6340","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:52:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lLf6oYxETGuOS/MBXewU09qx8y6UGYnyNwg+Xf1JBD8M0a35zNLw43rjiYGPH2CqOxKi5MpeuuYgbtPAJnBUCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T07:07:48.621906Z"},"content_sha256":"bd01e853d64b6781eb2fbd4f6b90062f1325d21015ebc5e69bc8b2dda18317cf","schema_version":"1.0","event_id":"sha256:bd01e853d64b6781eb2fbd4f6b90062f1325d21015ebc5e69bc8b2dda18317cf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/522K3WBPO5AO2EMCW3XIM7BJKE/bundle.json","state_url":"https://pith.science/pith/522K3WBPO5AO2EMCW3XIM7BJKE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/522K3WBPO5AO2EMCW3XIM7BJKE/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-30T07:07:48Z","links":{"resolver":"https://pith.science/pith/522K3WBPO5AO2EMCW3XIM7BJKE","bundle":"https://pith.science/pith/522K3WBPO5AO2EMCW3XIM7BJKE/bundle.json","state":"https://pith.science/pith/522K3WBPO5AO2EMCW3XIM7BJKE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/522K3WBPO5AO2EMCW3XIM7BJKE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:522K3WBPO5AO2EMCW3XIM7BJKE","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":"83f7e19e0b0e9436e7946f51e7f3a59a2c448a1474ea5e41fbe36cd6046c31c3","cross_cats_sorted":["math.CO","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-06-27T17:14:50Z","title_canon_sha256":"855724928acc1520094eeb17247786d07403386f0d950ac0d72d075930df55d9"},"schema_version":"1.0","source":{"id":"1206.6340","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.6340","created_at":"2026-05-18T03:52:26Z"},{"alias_kind":"arxiv_version","alias_value":"1206.6340v1","created_at":"2026-05-18T03:52:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.6340","created_at":"2026-05-18T03:52:26Z"},{"alias_kind":"pith_short_12","alias_value":"522K3WBPO5AO","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"522K3WBPO5AO2EMC","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"522K3WBP","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:bd01e853d64b6781eb2fbd4f6b90062f1325d21015ebc5e69bc8b2dda18317cf","target":"graph","created_at":"2026-05-18T03:52:26Z","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":"Let $V$ be a left vector space over a division ring and let ${\\mathcal P}(V)$ be the associated projective space. We describe all finite subsets $X\\subset V$ such that every permutation on $X$ can be extended to a linear automorphism of $V$ and all finite subsets ${\\mathcal X}\\subset {\\mathcal P}(V)$ such that every permutation on ${\\mathcal X}$ can be extended to an element of ${\\rm PGL}(V)$. Also, we reformulate the results in terms of linear and projective representations of symmetric groups.","authors_text":"Mark Pankov","cross_cats":["math.CO","math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-06-27T17:14:50Z","title":"On extendability of permutations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.6340","kind":"arxiv","version":1},"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:dee6ec98af617cd54f6274a7d747c684c465733f99ac64a3259bba26a54776ed","target":"record","created_at":"2026-05-18T03:52:26Z","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":"83f7e19e0b0e9436e7946f51e7f3a59a2c448a1474ea5e41fbe36cd6046c31c3","cross_cats_sorted":["math.CO","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-06-27T17:14:50Z","title_canon_sha256":"855724928acc1520094eeb17247786d07403386f0d950ac0d72d075930df55d9"},"schema_version":"1.0","source":{"id":"1206.6340","kind":"arxiv","version":1}},"canonical_sha256":"eeb4add82f7740ed1182b6ee867c29513bb5ea54771322b00540b0f9443ec06b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eeb4add82f7740ed1182b6ee867c29513bb5ea54771322b00540b0f9443ec06b","first_computed_at":"2026-05-18T03:52:26.933937Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:52:26.933937Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2SgCmdXAc/UjWPYkMjShRx4NpNtS2GpMFLBX66pMxbIYVYNt1J7x374EwFcRJ96tGecGFf1OSPqEdXnyR/itBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:52:26.934592Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.6340","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dee6ec98af617cd54f6274a7d747c684c465733f99ac64a3259bba26a54776ed","sha256:bd01e853d64b6781eb2fbd4f6b90062f1325d21015ebc5e69bc8b2dda18317cf"],"state_sha256":"c9554392ad49b27497ca8762557de39927ef239e54e02cc4f28819b731420bcc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5aeX+KaCJKmpa9XLr6/g87LRuMYb80ifkmYANdbki/rgbEjd0tkE+QHa0fJ2TwdHhkit9oPMw3bel2UgUX1HCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T07:07:48.623937Z","bundle_sha256":"b7d2360a368c839d46e95b2004107c70cf57d8744b4226a3470c64bf9c14fb21"}}