{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:ZT3ID2TKJEOJRG7XYYVEUOK2WF","short_pith_number":"pith:ZT3ID2TK","canonical_record":{"source":{"id":"1602.07829","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-02-25T07:35:56Z","cross_cats_sorted":[],"title_canon_sha256":"865aa8b84e47cee3588c1bb1a59896ebb56f6243d163c6760116c0fe933ac302","abstract_canon_sha256":"2c115866c8db9b5d4147e7fb6ba816f56faa82f14090c2eb1c7be577098d0581"},"schema_version":"1.0"},"canonical_sha256":"ccf681ea6a491c989bf7c62a4a395ab17a040526e5cabda895ea6bb3f715b0ab","source":{"kind":"arxiv","id":"1602.07829","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.07829","created_at":"2026-05-18T00:38:21Z"},{"alias_kind":"arxiv_version","alias_value":"1602.07829v4","created_at":"2026-05-18T00:38:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.07829","created_at":"2026-05-18T00:38:21Z"},{"alias_kind":"pith_short_12","alias_value":"ZT3ID2TKJEOJ","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"ZT3ID2TKJEOJRG7X","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"ZT3ID2TK","created_at":"2026-05-18T12:30:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:ZT3ID2TKJEOJRG7XYYVEUOK2WF","target":"record","payload":{"canonical_record":{"source":{"id":"1602.07829","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-02-25T07:35:56Z","cross_cats_sorted":[],"title_canon_sha256":"865aa8b84e47cee3588c1bb1a59896ebb56f6243d163c6760116c0fe933ac302","abstract_canon_sha256":"2c115866c8db9b5d4147e7fb6ba816f56faa82f14090c2eb1c7be577098d0581"},"schema_version":"1.0"},"canonical_sha256":"ccf681ea6a491c989bf7c62a4a395ab17a040526e5cabda895ea6bb3f715b0ab","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:21.419553Z","signature_b64":"PbmD6feKhfsnjid25JkgKqofEn5E8DF0pFqT00HPtBCsdaB20Rj7bgh3NozGh+PWoM4XB6rVT9FEpe/lEv7SBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ccf681ea6a491c989bf7c62a4a395ab17a040526e5cabda895ea6bb3f715b0ab","last_reissued_at":"2026-05-18T00:38:21.418873Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:21.418873Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.07829","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:38:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zMCavfjNo2+KG62Y7yz2Z1h+5BXnq581PE6y086m9rBGdUsHKmeI8E4WIQIVySH0ThF6ceT5ZstGE+kknVxkBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T16:24:41.984481Z"},"content_sha256":"357ea4233657bb24a07193654a4a24eb380a91bd6d2fde31f64178089f4ebb4e","schema_version":"1.0","event_id":"sha256:357ea4233657bb24a07193654a4a24eb380a91bd6d2fde31f64178089f4ebb4e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:ZT3ID2TKJEOJRG7XYYVEUOK2WF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The number of composition factors of order $p$ in completely reducible groups of characteristic $p$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Cai Heng Li, Gabriel Verret, Michael Giudici, S. P. Glasby","submitted_at":"2016-02-25T07:35:56Z","abstract_excerpt":"Let $q$ be a power of a prime $p$ and let $G$ be a completely reducible subgroup of $\\mathrm{GL}(d,q)$. We prove that the number of composition factors of $G$ that have prime order $p$ is at most $(\\varepsilon_q d-1)/(p-1)$, where $\\varepsilon_q$ is a function of $q$ satisfying $1\\leqslant\\varepsilon_q\\leqslant 3/2$. For every $q$, we give examples showing this bound is sharp infinitely often."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.07829","kind":"arxiv","version":4},"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":"2026-05-18T00:38:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rOgOPrvr3j5VSROVEGTr4fmJXSxgmwqSVBVLSQMc272WE2rleIhwi7TNIwhCts557raGuKH7e4lJynmIOlBHCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T16:24:41.985113Z"},"content_sha256":"e6ab9f6863b183d3ca64cdc142bf7fbeb61e61d63696e9e4984e7ba23df72495","schema_version":"1.0","event_id":"sha256:e6ab9f6863b183d3ca64cdc142bf7fbeb61e61d63696e9e4984e7ba23df72495"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZT3ID2TKJEOJRG7XYYVEUOK2WF/bundle.json","state_url":"https://pith.science/pith/ZT3ID2TKJEOJRG7XYYVEUOK2WF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZT3ID2TKJEOJRG7XYYVEUOK2WF/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":"2026-06-06T16:24:41Z","links":{"resolver":"https://pith.science/pith/ZT3ID2TKJEOJRG7XYYVEUOK2WF","bundle":"https://pith.science/pith/ZT3ID2TKJEOJRG7XYYVEUOK2WF/bundle.json","state":"https://pith.science/pith/ZT3ID2TKJEOJRG7XYYVEUOK2WF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZT3ID2TKJEOJRG7XYYVEUOK2WF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:ZT3ID2TKJEOJRG7XYYVEUOK2WF","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":"2c115866c8db9b5d4147e7fb6ba816f56faa82f14090c2eb1c7be577098d0581","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-02-25T07:35:56Z","title_canon_sha256":"865aa8b84e47cee3588c1bb1a59896ebb56f6243d163c6760116c0fe933ac302"},"schema_version":"1.0","source":{"id":"1602.07829","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.07829","created_at":"2026-05-18T00:38:21Z"},{"alias_kind":"arxiv_version","alias_value":"1602.07829v4","created_at":"2026-05-18T00:38:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.07829","created_at":"2026-05-18T00:38:21Z"},{"alias_kind":"pith_short_12","alias_value":"ZT3ID2TKJEOJ","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"ZT3ID2TKJEOJRG7X","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"ZT3ID2TK","created_at":"2026-05-18T12:30:55Z"}],"graph_snapshots":[{"event_id":"sha256:e6ab9f6863b183d3ca64cdc142bf7fbeb61e61d63696e9e4984e7ba23df72495","target":"graph","created_at":"2026-05-18T00:38:21Z","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 $q$ be a power of a prime $p$ and let $G$ be a completely reducible subgroup of $\\mathrm{GL}(d,q)$. We prove that the number of composition factors of $G$ that have prime order $p$ is at most $(\\varepsilon_q d-1)/(p-1)$, where $\\varepsilon_q$ is a function of $q$ satisfying $1\\leqslant\\varepsilon_q\\leqslant 3/2$. For every $q$, we give examples showing this bound is sharp infinitely often.","authors_text":"Cai Heng Li, Gabriel Verret, Michael Giudici, S. P. Glasby","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-02-25T07:35:56Z","title":"The number of composition factors of order $p$ in completely reducible groups of characteristic $p$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.07829","kind":"arxiv","version":4},"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":"sha256:357ea4233657bb24a07193654a4a24eb380a91bd6d2fde31f64178089f4ebb4e","target":"record","created_at":"2026-05-18T00:38:21Z","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":"2c115866c8db9b5d4147e7fb6ba816f56faa82f14090c2eb1c7be577098d0581","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-02-25T07:35:56Z","title_canon_sha256":"865aa8b84e47cee3588c1bb1a59896ebb56f6243d163c6760116c0fe933ac302"},"schema_version":"1.0","source":{"id":"1602.07829","kind":"arxiv","version":4}},"canonical_sha256":"ccf681ea6a491c989bf7c62a4a395ab17a040526e5cabda895ea6bb3f715b0ab","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ccf681ea6a491c989bf7c62a4a395ab17a040526e5cabda895ea6bb3f715b0ab","first_computed_at":"2026-05-18T00:38:21.418873Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:38:21.418873Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PbmD6feKhfsnjid25JkgKqofEn5E8DF0pFqT00HPtBCsdaB20Rj7bgh3NozGh+PWoM4XB6rVT9FEpe/lEv7SBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:38:21.419553Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.07829","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:357ea4233657bb24a07193654a4a24eb380a91bd6d2fde31f64178089f4ebb4e","sha256:e6ab9f6863b183d3ca64cdc142bf7fbeb61e61d63696e9e4984e7ba23df72495"],"state_sha256":"9a3891e5da30ad3f2d261b88cf116159b9724c60ebba10acab4b01b2d3706f52"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"212wBvgvlD1MvizYPcJ0IWgweqOJvSV7vnrigPhjz+Q8Uz5MxcC3RXauxtTg3flk5aXKVkUkAZB0+QwReBJCDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T16:24:42.002529Z","bundle_sha256":"b4ab2e1fcdd7c3010a234827637de90d4ac3cb2650dd0d321e7c4db67f1a3e52"}}