{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:A6WB3FRBUUX6X36VKKX2KNQMWT","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":"be5bf56a6d72dfd66d5af1e374bbd3310ec464f1f366fe2ff974130752529834","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-02-06T20:15:15Z","title_canon_sha256":"b42ebf661e84527568d757093c00e403d3aa62a1a9fb16ff8d8ff9a7c925c631"},"schema_version":"1.0","source":{"id":"1802.02194","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.02194","created_at":"2026-05-17T23:41:46Z"},{"alias_kind":"arxiv_version","alias_value":"1802.02194v2","created_at":"2026-05-17T23:41:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.02194","created_at":"2026-05-17T23:41:46Z"},{"alias_kind":"pith_short_12","alias_value":"A6WB3FRBUUX6","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_16","alias_value":"A6WB3FRBUUX6X36V","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_8","alias_value":"A6WB3FRB","created_at":"2026-05-18T12:32:13Z"}],"graph_snapshots":[{"event_id":"sha256:018f6af0c46bfb05d31142e6ec83829d99a2f04578fb2ce98e229b82ff469b91","target":"graph","created_at":"2026-05-17T23:41:46Z","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":"An unrefinable chain of a finite group $G$ is a chain of subgroups $G = G_0 > G_1 > \\cdots > G_t = 1$, where each $G_i$ is a maximal subgroup of $G_{i-1}$. The length (respectively, depth) of $G$ is the maximal (respectively, minimal) length of such a chain. We studied the depth of finite simple groups in a previous paper, which included a classification of the simple groups of depth $3$. Here we go much further by determining the finite groups of depth $3$ and $4$. We also obtain several new results on the lengths of finite groups. For example, we classify the simple groups of length at most ","authors_text":"Aner Shalev, Martin W. Liebeck, Timothy C. Burness","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-02-06T20:15:15Z","title":"On the length and depth of finite groups (with an appendix by D.R. Heath-Brown)"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.02194","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:caeec7de26b52a8932c7e0bfb738c190f6167c25df4d0fa95482472ab37d13f1","target":"record","created_at":"2026-05-17T23:41:46Z","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":"be5bf56a6d72dfd66d5af1e374bbd3310ec464f1f366fe2ff974130752529834","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-02-06T20:15:15Z","title_canon_sha256":"b42ebf661e84527568d757093c00e403d3aa62a1a9fb16ff8d8ff9a7c925c631"},"schema_version":"1.0","source":{"id":"1802.02194","kind":"arxiv","version":2}},"canonical_sha256":"07ac1d9621a52febefd552afa5360cb4feb7ec5aceba578aa7f30ec025994347","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"07ac1d9621a52febefd552afa5360cb4feb7ec5aceba578aa7f30ec025994347","first_computed_at":"2026-05-17T23:41:46.853740Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:41:46.853740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0YjasZl9TPCYJG3k0Xo+RAxd/2bkxHJE1do3gQov3taFd+O5Mw6FnuMxGl7ViE+RI0YI4/OoFveMAgOOuc8bDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:41:46.854285Z","signed_message":"canonical_sha256_bytes"},"source_id":"1802.02194","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:caeec7de26b52a8932c7e0bfb738c190f6167c25df4d0fa95482472ab37d13f1","sha256:018f6af0c46bfb05d31142e6ec83829d99a2f04578fb2ce98e229b82ff469b91"],"state_sha256":"15435ede74a3bc931c2151c3f928f2563cda93162649970c8e910dd7cd391c26"}