{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:A6WB3FRBUUX6X36VKKX2KNQMWT","short_pith_number":"pith:A6WB3FRB","schema_version":"1.0","canonical_sha256":"07ac1d9621a52febefd552afa5360cb4feb7ec5aceba578aa7f30ec025994347","source":{"kind":"arxiv","id":"1802.02194","version":2},"attestation_state":"computed","paper":{"title":"On the length and depth of finite groups (with an appendix by D.R. Heath-Brown)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Aner Shalev, Martin W. Liebeck, Timothy C. Burness","submitted_at":"2018-02-06T20:15:15Z","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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1802.02194","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-02-06T20:15:15Z","cross_cats_sorted":[],"title_canon_sha256":"b42ebf661e84527568d757093c00e403d3aa62a1a9fb16ff8d8ff9a7c925c631","abstract_canon_sha256":"be5bf56a6d72dfd66d5af1e374bbd3310ec464f1f366fe2ff974130752529834"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:46.854285Z","signature_b64":"0YjasZl9TPCYJG3k0Xo+RAxd/2bkxHJE1do3gQov3taFd+O5Mw6FnuMxGl7ViE+RI0YI4/OoFveMAgOOuc8bDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"07ac1d9621a52febefd552afa5360cb4feb7ec5aceba578aa7f30ec025994347","last_reissued_at":"2026-05-17T23:41:46.853740Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:46.853740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the length and depth of finite groups (with an appendix by D.R. Heath-Brown)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Aner Shalev, Martin W. Liebeck, Timothy C. Burness","submitted_at":"2018-02-06T20:15:15Z","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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.02194","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1802.02194","created_at":"2026-05-17T23:41:46.853834+00:00"},{"alias_kind":"arxiv_version","alias_value":"1802.02194v2","created_at":"2026-05-17T23:41:46.853834+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.02194","created_at":"2026-05-17T23:41:46.853834+00:00"},{"alias_kind":"pith_short_12","alias_value":"A6WB3FRBUUX6","created_at":"2026-05-18T12:32:13.499390+00:00"},{"alias_kind":"pith_short_16","alias_value":"A6WB3FRBUUX6X36V","created_at":"2026-05-18T12:32:13.499390+00:00"},{"alias_kind":"pith_short_8","alias_value":"A6WB3FRB","created_at":"2026-05-18T12:32:13.499390+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/A6WB3FRBUUX6X36VKKX2KNQMWT","json":"https://pith.science/pith/A6WB3FRBUUX6X36VKKX2KNQMWT.json","graph_json":"https://pith.science/api/pith-number/A6WB3FRBUUX6X36VKKX2KNQMWT/graph.json","events_json":"https://pith.science/api/pith-number/A6WB3FRBUUX6X36VKKX2KNQMWT/events.json","paper":"https://pith.science/paper/A6WB3FRB"},"agent_actions":{"view_html":"https://pith.science/pith/A6WB3FRBUUX6X36VKKX2KNQMWT","download_json":"https://pith.science/pith/A6WB3FRBUUX6X36VKKX2KNQMWT.json","view_paper":"https://pith.science/paper/A6WB3FRB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1802.02194&json=true","fetch_graph":"https://pith.science/api/pith-number/A6WB3FRBUUX6X36VKKX2KNQMWT/graph.json","fetch_events":"https://pith.science/api/pith-number/A6WB3FRBUUX6X36VKKX2KNQMWT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/A6WB3FRBUUX6X36VKKX2KNQMWT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/A6WB3FRBUUX6X36VKKX2KNQMWT/action/storage_attestation","attest_author":"https://pith.science/pith/A6WB3FRBUUX6X36VKKX2KNQMWT/action/author_attestation","sign_citation":"https://pith.science/pith/A6WB3FRBUUX6X36VKKX2KNQMWT/action/citation_signature","submit_replication":"https://pith.science/pith/A6WB3FRBUUX6X36VKKX2KNQMWT/action/replication_record"}},"created_at":"2026-05-17T23:41:46.853834+00:00","updated_at":"2026-05-17T23:41:46.853834+00:00"}