{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:UN5Z22LI5PLV37OUMTVWRTMRYZ","short_pith_number":"pith:UN5Z22LI","canonical_record":{"source":{"id":"1509.09087","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-09-30T09:11:02Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"2ee89795024f081aa638bd32cd534469c70711acc719e250a138ae4e814dd424","abstract_canon_sha256":"04e9e9e2972e08864ec4a2860aca2a3721c28e34e2dbb5c922f57ec4e2d4ba7f"},"schema_version":"1.0"},"canonical_sha256":"a37b9d6968ebd75dfdd464eb68cd91c67c7f25210d52f87f09b4a96812462726","source":{"kind":"arxiv","id":"1509.09087","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.09087","created_at":"2026-05-18T01:31:29Z"},{"alias_kind":"arxiv_version","alias_value":"1509.09087v1","created_at":"2026-05-18T01:31:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.09087","created_at":"2026-05-18T01:31:29Z"},{"alias_kind":"pith_short_12","alias_value":"UN5Z22LI5PLV","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"UN5Z22LI5PLV37OU","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"UN5Z22LI","created_at":"2026-05-18T12:29:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:UN5Z22LI5PLV37OUMTVWRTMRYZ","target":"record","payload":{"canonical_record":{"source":{"id":"1509.09087","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-09-30T09:11:02Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"2ee89795024f081aa638bd32cd534469c70711acc719e250a138ae4e814dd424","abstract_canon_sha256":"04e9e9e2972e08864ec4a2860aca2a3721c28e34e2dbb5c922f57ec4e2d4ba7f"},"schema_version":"1.0"},"canonical_sha256":"a37b9d6968ebd75dfdd464eb68cd91c67c7f25210d52f87f09b4a96812462726","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:29.108441Z","signature_b64":"kk8eoK0QvK+Leha9R7D8M6ujT5a9VuY1uDM8kBwCFNPMh187ihYzpDloMdu9tsm3kblMrB93b7ljTGHyIP77Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a37b9d6968ebd75dfdd464eb68cd91c67c7f25210d52f87f09b4a96812462726","last_reissued_at":"2026-05-18T01:31:29.107778Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:29.107778Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.09087","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":"2026-05-18T01:31:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Vo4GGtyG6oVAOgin1VWSAn+QMl3t0XA+c/IYEXopy9TuY94lVPMt1ZGtqazGjtY9u5HisD4BUBEQ81kRshcVDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T05:04:53.374801Z"},"content_sha256":"22c4540c1aadd4a78d761d4dbb817110d96b2056c4165ff869aeb70ee63966fc","schema_version":"1.0","event_id":"sha256:22c4540c1aadd4a78d761d4dbb817110d96b2056c4165ff869aeb70ee63966fc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:UN5Z22LI5PLV37OUMTVWRTMRYZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Almost group theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.LO","authors_text":"Nadja Hempel (ICJ)","submitted_at":"2015-09-30T09:11:02Z","abstract_excerpt":"The notion of almost centralizer and almost commutator are introduced and basic properties are established. They are used to study $\\widetilde{\\mathfrak M}\\_c$-groups, i. e.groups for which every descending chain of centralizers each having infinite index in its predecessor stabilizes after finitely many steps. The Fitting subgroup of such groups is shown to be nilpotent and a theorem of Hall for nilpotent groups is generalized to ind-definable almost nilpotent subgroups of $\\widetilde{\\mathfrak M}\\_c$-groups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.09087","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":"2026-05-18T01:31:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+GmW+Cl6JZ5PHusXTVio2NnfnKQHNYEkEVpzXkh0OqQYHVsudV2Qcp5m3ZnjPEQJQD5S+3vIy+ocPsa38zMWDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T05:04:53.375463Z"},"content_sha256":"f9e2445f0b9c97ff1efcda311459ee1acf12b0bf39051b016edea10aed7fc69a","schema_version":"1.0","event_id":"sha256:f9e2445f0b9c97ff1efcda311459ee1acf12b0bf39051b016edea10aed7fc69a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UN5Z22LI5PLV37OUMTVWRTMRYZ/bundle.json","state_url":"https://pith.science/pith/UN5Z22LI5PLV37OUMTVWRTMRYZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UN5Z22LI5PLV37OUMTVWRTMRYZ/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":"2026-05-27T05:04:53Z","links":{"resolver":"https://pith.science/pith/UN5Z22LI5PLV37OUMTVWRTMRYZ","bundle":"https://pith.science/pith/UN5Z22LI5PLV37OUMTVWRTMRYZ/bundle.json","state":"https://pith.science/pith/UN5Z22LI5PLV37OUMTVWRTMRYZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UN5Z22LI5PLV37OUMTVWRTMRYZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:UN5Z22LI5PLV37OUMTVWRTMRYZ","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":"04e9e9e2972e08864ec4a2860aca2a3721c28e34e2dbb5c922f57ec4e2d4ba7f","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-09-30T09:11:02Z","title_canon_sha256":"2ee89795024f081aa638bd32cd534469c70711acc719e250a138ae4e814dd424"},"schema_version":"1.0","source":{"id":"1509.09087","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.09087","created_at":"2026-05-18T01:31:29Z"},{"alias_kind":"arxiv_version","alias_value":"1509.09087v1","created_at":"2026-05-18T01:31:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.09087","created_at":"2026-05-18T01:31:29Z"},{"alias_kind":"pith_short_12","alias_value":"UN5Z22LI5PLV","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"UN5Z22LI5PLV37OU","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"UN5Z22LI","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:f9e2445f0b9c97ff1efcda311459ee1acf12b0bf39051b016edea10aed7fc69a","target":"graph","created_at":"2026-05-18T01:31:29Z","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":"The notion of almost centralizer and almost commutator are introduced and basic properties are established. They are used to study $\\widetilde{\\mathfrak M}\\_c$-groups, i. e.groups for which every descending chain of centralizers each having infinite index in its predecessor stabilizes after finitely many steps. The Fitting subgroup of such groups is shown to be nilpotent and a theorem of Hall for nilpotent groups is generalized to ind-definable almost nilpotent subgroups of $\\widetilde{\\mathfrak M}\\_c$-groups.","authors_text":"Nadja Hempel (ICJ)","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-09-30T09:11:02Z","title":"Almost group theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.09087","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":"sha256:22c4540c1aadd4a78d761d4dbb817110d96b2056c4165ff869aeb70ee63966fc","target":"record","created_at":"2026-05-18T01:31:29Z","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":"04e9e9e2972e08864ec4a2860aca2a3721c28e34e2dbb5c922f57ec4e2d4ba7f","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-09-30T09:11:02Z","title_canon_sha256":"2ee89795024f081aa638bd32cd534469c70711acc719e250a138ae4e814dd424"},"schema_version":"1.0","source":{"id":"1509.09087","kind":"arxiv","version":1}},"canonical_sha256":"a37b9d6968ebd75dfdd464eb68cd91c67c7f25210d52f87f09b4a96812462726","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a37b9d6968ebd75dfdd464eb68cd91c67c7f25210d52f87f09b4a96812462726","first_computed_at":"2026-05-18T01:31:29.107778Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:31:29.107778Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kk8eoK0QvK+Leha9R7D8M6ujT5a9VuY1uDM8kBwCFNPMh187ihYzpDloMdu9tsm3kblMrB93b7ljTGHyIP77Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:31:29.108441Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.09087","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:22c4540c1aadd4a78d761d4dbb817110d96b2056c4165ff869aeb70ee63966fc","sha256:f9e2445f0b9c97ff1efcda311459ee1acf12b0bf39051b016edea10aed7fc69a"],"state_sha256":"697e311bccd4e2f05fc4ba83a91fcd1414543a2aaa85f275ed42cde1b4a895e4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LYWb8D97T33pDeCrpe5ifT1uLG+5cjSDARmUz7ggye1ABG8Ke0FF5MnaG+jgQWVncAmXS4ZvG7e8Ne2hAyJtAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T05:04:53.379436Z","bundle_sha256":"e7d76e717197a76290b4a301a171c7a3cff013b7a88bad7a849a1ab2fb545e27"}}