{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:PABB7GWGO4CGP5FPSRVV4HSIBX","short_pith_number":"pith:PABB7GWG","canonical_record":{"source":{"id":"1008.3815","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-08-23T13:07:18Z","cross_cats_sorted":[],"title_canon_sha256":"3a15cb93d3682c962a59c7d0fb2f58440544cd7e4b28c247a18a0c145e5d5c36","abstract_canon_sha256":"c6330efa89df0cfe4e361f4c9bb3d538be9063b907888a3a5596b84d092ef6ae"},"schema_version":"1.0"},"canonical_sha256":"78021f9ac6770467f4af946b5e1e480dd055a848168554129b971cafadbceb40","source":{"kind":"arxiv","id":"1008.3815","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.3815","created_at":"2026-05-18T04:11:39Z"},{"alias_kind":"arxiv_version","alias_value":"1008.3815v2","created_at":"2026-05-18T04:11:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.3815","created_at":"2026-05-18T04:11:39Z"},{"alias_kind":"pith_short_12","alias_value":"PABB7GWGO4CG","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"PABB7GWGO4CGP5FP","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"PABB7GWG","created_at":"2026-05-18T12:26:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:PABB7GWGO4CGP5FPSRVV4HSIBX","target":"record","payload":{"canonical_record":{"source":{"id":"1008.3815","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-08-23T13:07:18Z","cross_cats_sorted":[],"title_canon_sha256":"3a15cb93d3682c962a59c7d0fb2f58440544cd7e4b28c247a18a0c145e5d5c36","abstract_canon_sha256":"c6330efa89df0cfe4e361f4c9bb3d538be9063b907888a3a5596b84d092ef6ae"},"schema_version":"1.0"},"canonical_sha256":"78021f9ac6770467f4af946b5e1e480dd055a848168554129b971cafadbceb40","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:39.713651Z","signature_b64":"jf9CZBt1vJJGdkejqCnjmpSPKwaylCfevDWEl8fiSiG0nEPJmN2i1ngTZiHJE62a591mvYQOFgFFyQszcCdlCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"78021f9ac6770467f4af946b5e1e480dd055a848168554129b971cafadbceb40","last_reissued_at":"2026-05-18T04:11:39.712984Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:39.712984Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1008.3815","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":"2026-05-18T04:11:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lQyIKqk4AqkhDxYhpn2eHxk5RHppk4CbmTUeKNkdx65rkFEDB2RkrGrlENWck5jFDJlsegYe2aCITmiGqjAIDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T16:55:47.968433Z"},"content_sha256":"58a0eaee9dc134ff30acee61c251e8356556707e31555a18c63c838fe90c6ee9","schema_version":"1.0","event_id":"sha256:58a0eaee9dc134ff30acee61c251e8356556707e31555a18c63c838fe90c6ee9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:PABB7GWGO4CGP5FPSRVV4HSIBX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Saturated fusion systems with parabolic families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Silvia Onofrei","submitted_at":"2010-08-23T13:07:18Z","abstract_excerpt":"Let G be group; a finite p-subgroup S of G is a Sylow p-subgroup if every finite p-subgroup of G is conjugate to a subgroup of S. In this paper, we examine the relations between the fusion system over S which is given by conjugation in G and a certain chamber system C, on which G acts chamber transitively with chamber stabilizer N_G(S). Next, we introduce the notion of a fusion system with a parabolic family and we show that a chamber system can be associated to such a fusion system. We determine some conditions the chamber system has to fulfill in order to assure the saturation of the underly"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.3815","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":"2026-05-18T04:11:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VWu9B5Zbp38hMiRBciFh/DXkja2ekbTsG3b3vdVo/Oxrh422zsZdlTksLYgQP5yCDW+7+SDpGoS9h5N8LGdzDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T16:55:47.969166Z"},"content_sha256":"be0cb0ed28b9db956e0b4fc501849ed0539e8a06574fab664d2d26f5cdb74127","schema_version":"1.0","event_id":"sha256:be0cb0ed28b9db956e0b4fc501849ed0539e8a06574fab664d2d26f5cdb74127"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PABB7GWGO4CGP5FPSRVV4HSIBX/bundle.json","state_url":"https://pith.science/pith/PABB7GWGO4CGP5FPSRVV4HSIBX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PABB7GWGO4CGP5FPSRVV4HSIBX/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":"2026-06-10T16:55:47Z","links":{"resolver":"https://pith.science/pith/PABB7GWGO4CGP5FPSRVV4HSIBX","bundle":"https://pith.science/pith/PABB7GWGO4CGP5FPSRVV4HSIBX/bundle.json","state":"https://pith.science/pith/PABB7GWGO4CGP5FPSRVV4HSIBX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PABB7GWGO4CGP5FPSRVV4HSIBX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:PABB7GWGO4CGP5FPSRVV4HSIBX","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":"c6330efa89df0cfe4e361f4c9bb3d538be9063b907888a3a5596b84d092ef6ae","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-08-23T13:07:18Z","title_canon_sha256":"3a15cb93d3682c962a59c7d0fb2f58440544cd7e4b28c247a18a0c145e5d5c36"},"schema_version":"1.0","source":{"id":"1008.3815","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.3815","created_at":"2026-05-18T04:11:39Z"},{"alias_kind":"arxiv_version","alias_value":"1008.3815v2","created_at":"2026-05-18T04:11:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.3815","created_at":"2026-05-18T04:11:39Z"},{"alias_kind":"pith_short_12","alias_value":"PABB7GWGO4CG","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"PABB7GWGO4CGP5FP","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"PABB7GWG","created_at":"2026-05-18T12:26:12Z"}],"graph_snapshots":[{"event_id":"sha256:be0cb0ed28b9db956e0b4fc501849ed0539e8a06574fab664d2d26f5cdb74127","target":"graph","created_at":"2026-05-18T04:11:39Z","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 group; a finite p-subgroup S of G is a Sylow p-subgroup if every finite p-subgroup of G is conjugate to a subgroup of S. In this paper, we examine the relations between the fusion system over S which is given by conjugation in G and a certain chamber system C, on which G acts chamber transitively with chamber stabilizer N_G(S). Next, we introduce the notion of a fusion system with a parabolic family and we show that a chamber system can be associated to such a fusion system. We determine some conditions the chamber system has to fulfill in order to assure the saturation of the underly","authors_text":"Silvia Onofrei","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-08-23T13:07:18Z","title":"Saturated fusion systems with parabolic families"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.3815","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":"sha256:58a0eaee9dc134ff30acee61c251e8356556707e31555a18c63c838fe90c6ee9","target":"record","created_at":"2026-05-18T04:11:39Z","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":"c6330efa89df0cfe4e361f4c9bb3d538be9063b907888a3a5596b84d092ef6ae","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-08-23T13:07:18Z","title_canon_sha256":"3a15cb93d3682c962a59c7d0fb2f58440544cd7e4b28c247a18a0c145e5d5c36"},"schema_version":"1.0","source":{"id":"1008.3815","kind":"arxiv","version":2}},"canonical_sha256":"78021f9ac6770467f4af946b5e1e480dd055a848168554129b971cafadbceb40","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"78021f9ac6770467f4af946b5e1e480dd055a848168554129b971cafadbceb40","first_computed_at":"2026-05-18T04:11:39.712984Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:11:39.712984Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jf9CZBt1vJJGdkejqCnjmpSPKwaylCfevDWEl8fiSiG0nEPJmN2i1ngTZiHJE62a591mvYQOFgFFyQszcCdlCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:11:39.713651Z","signed_message":"canonical_sha256_bytes"},"source_id":"1008.3815","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:58a0eaee9dc134ff30acee61c251e8356556707e31555a18c63c838fe90c6ee9","sha256:be0cb0ed28b9db956e0b4fc501849ed0539e8a06574fab664d2d26f5cdb74127"],"state_sha256":"e31ba326d5de3f2023d6ba11f78810b3d4b05aa16988f7aee5eac873520ed77f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JaARMV7bWEkVGndiEADsFGOpqO7LhKwzgvEUkPJUBmXXpuOW+VlbzHdxI5JAcXIAR+Ecq+/IZkY/WRaHL3tXAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T16:55:47.973024Z","bundle_sha256":"6a6a593af6186036ca0896d456d5f348f737320522a97efd187257f3c9e7bdc9"}}