{"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"}