{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:6RFHU4Q4M6JMUQ43J3KPRRSP7E","short_pith_number":"pith:6RFHU4Q4","canonical_record":{"source":{"id":"1508.00725","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-08-04T10:22:17Z","cross_cats_sorted":[],"title_canon_sha256":"8b6c5eff5e6c8690bb9ff92847b4cd880d00f619a7ccc7a12385091377252c20","abstract_canon_sha256":"6ce23414c42757ee4f7221c9b9d705d1ff5d7dd8736bda31f2f7eeb2757b9b0d"},"schema_version":"1.0"},"canonical_sha256":"f44a7a721c6792ca439b4ed4f8c64ff912a11ce49a21804b2e36de3db4c06ba8","source":{"kind":"arxiv","id":"1508.00725","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.00725","created_at":"2026-05-18T01:29:20Z"},{"alias_kind":"arxiv_version","alias_value":"1508.00725v2","created_at":"2026-05-18T01:29:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.00725","created_at":"2026-05-18T01:29:20Z"},{"alias_kind":"pith_short_12","alias_value":"6RFHU4Q4M6JM","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6RFHU4Q4M6JMUQ43","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6RFHU4Q4","created_at":"2026-05-18T12:29:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:6RFHU4Q4M6JMUQ43J3KPRRSP7E","target":"record","payload":{"canonical_record":{"source":{"id":"1508.00725","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-08-04T10:22:17Z","cross_cats_sorted":[],"title_canon_sha256":"8b6c5eff5e6c8690bb9ff92847b4cd880d00f619a7ccc7a12385091377252c20","abstract_canon_sha256":"6ce23414c42757ee4f7221c9b9d705d1ff5d7dd8736bda31f2f7eeb2757b9b0d"},"schema_version":"1.0"},"canonical_sha256":"f44a7a721c6792ca439b4ed4f8c64ff912a11ce49a21804b2e36de3db4c06ba8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:29:20.562286Z","signature_b64":"ytRhaRZq0M5R1MkxMGkXpG6ujG24BzntUFpwOz6ce8VMDtKyJv5jJd9pXK6LbcXjaUBlm8+nxfM9FeCRi1S9Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f44a7a721c6792ca439b4ed4f8c64ff912a11ce49a21804b2e36de3db4c06ba8","last_reissued_at":"2026-05-18T01:29:20.561573Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:29:20.561573Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.00725","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:29:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2JxoJA2fXEnbHX0hWgBhh/UAyUY93EQj2rtAmPvEOFXOUtcC4ZI/0oYtb2VQdsgBGLllxEhHbeRlj/0LzMgFBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T15:38:41.559997Z"},"content_sha256":"0264d8bc4f4eff94acb1b845fe3811830215016f48d58609dcc6b21eb08d08b8","schema_version":"1.0","event_id":"sha256:0264d8bc4f4eff94acb1b845fe3811830215016f48d58609dcc6b21eb08d08b8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:6RFHU4Q4M6JMUQ43J3KPRRSP7E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A note on automorphisms of finite $p$-groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Anitha Thillaisundaram, Gustavo A. Fern\\'andez-Alcober","submitted_at":"2015-08-04T10:22:17Z","abstract_excerpt":"Let $G$ be a finite non-cyclic $p$-group of order at least $p^3$. If $G$ has an abelian maximal subgroup, or if $G$ has an elementary abelian centre with $C_G(Z(\\Phi(G))) \\ne \\Phi(G)$, then $|G|$ divides $|\\text{Aut}(G)|$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.00725","kind":"arxiv","version":2},"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":"2026-05-18T01:29:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zrYXlrw6tg2P08bj9RhqYoIngWYjmyw61zNdwRkFaw374wLlWznvPuPpzycHUj5WN7o476p49T3odZ2AEOafDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T15:38:41.560733Z"},"content_sha256":"8a7bdab2f696a98c9ddfdf62d821c16675802222e2becf003478365d97a7da10","schema_version":"1.0","event_id":"sha256:8a7bdab2f696a98c9ddfdf62d821c16675802222e2becf003478365d97a7da10"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6RFHU4Q4M6JMUQ43J3KPRRSP7E/bundle.json","state_url":"https://pith.science/pith/6RFHU4Q4M6JMUQ43J3KPRRSP7E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6RFHU4Q4M6JMUQ43J3KPRRSP7E/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":"2026-06-08T15:38:41Z","links":{"resolver":"https://pith.science/pith/6RFHU4Q4M6JMUQ43J3KPRRSP7E","bundle":"https://pith.science/pith/6RFHU4Q4M6JMUQ43J3KPRRSP7E/bundle.json","state":"https://pith.science/pith/6RFHU4Q4M6JMUQ43J3KPRRSP7E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6RFHU4Q4M6JMUQ43J3KPRRSP7E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:6RFHU4Q4M6JMUQ43J3KPRRSP7E","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":"6ce23414c42757ee4f7221c9b9d705d1ff5d7dd8736bda31f2f7eeb2757b9b0d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-08-04T10:22:17Z","title_canon_sha256":"8b6c5eff5e6c8690bb9ff92847b4cd880d00f619a7ccc7a12385091377252c20"},"schema_version":"1.0","source":{"id":"1508.00725","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.00725","created_at":"2026-05-18T01:29:20Z"},{"alias_kind":"arxiv_version","alias_value":"1508.00725v2","created_at":"2026-05-18T01:29:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.00725","created_at":"2026-05-18T01:29:20Z"},{"alias_kind":"pith_short_12","alias_value":"6RFHU4Q4M6JM","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6RFHU4Q4M6JMUQ43","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6RFHU4Q4","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:8a7bdab2f696a98c9ddfdf62d821c16675802222e2becf003478365d97a7da10","target":"graph","created_at":"2026-05-18T01:29:20Z","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 $G$ be a finite non-cyclic $p$-group of order at least $p^3$. If $G$ has an abelian maximal subgroup, or if $G$ has an elementary abelian centre with $C_G(Z(\\Phi(G))) \\ne \\Phi(G)$, then $|G|$ divides $|\\text{Aut}(G)|$.","authors_text":"Anitha Thillaisundaram, Gustavo A. Fern\\'andez-Alcober","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-08-04T10:22:17Z","title":"A note on automorphisms of finite $p$-groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.00725","kind":"arxiv","version":2},"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":"sha256:0264d8bc4f4eff94acb1b845fe3811830215016f48d58609dcc6b21eb08d08b8","target":"record","created_at":"2026-05-18T01:29:20Z","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":"6ce23414c42757ee4f7221c9b9d705d1ff5d7dd8736bda31f2f7eeb2757b9b0d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-08-04T10:22:17Z","title_canon_sha256":"8b6c5eff5e6c8690bb9ff92847b4cd880d00f619a7ccc7a12385091377252c20"},"schema_version":"1.0","source":{"id":"1508.00725","kind":"arxiv","version":2}},"canonical_sha256":"f44a7a721c6792ca439b4ed4f8c64ff912a11ce49a21804b2e36de3db4c06ba8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f44a7a721c6792ca439b4ed4f8c64ff912a11ce49a21804b2e36de3db4c06ba8","first_computed_at":"2026-05-18T01:29:20.561573Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:29:20.561573Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ytRhaRZq0M5R1MkxMGkXpG6ujG24BzntUFpwOz6ce8VMDtKyJv5jJd9pXK6LbcXjaUBlm8+nxfM9FeCRi1S9Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:29:20.562286Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.00725","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0264d8bc4f4eff94acb1b845fe3811830215016f48d58609dcc6b21eb08d08b8","sha256:8a7bdab2f696a98c9ddfdf62d821c16675802222e2becf003478365d97a7da10"],"state_sha256":"0fe1c63ab47db80367f531d4dce0930aa84121a19936c83beec7f80edec5a9ea"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5qwqSDiQndkDfy3FWHZLBSXa/W5ekPwVHciqVjqyt+/9HApi6Qix4aSB6W6tKVd+axTjXYAlAXS0lhexwfp7Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T15:38:41.564730Z","bundle_sha256":"cbb67189b18117c5c55b4df9bbdc6530090630565bfaf01e0aa728f04274a4af"}}