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The generalized Fitting height of a finite group $G$ is the least number $h=h^*(G)$ such that $F^*_h(G)=G$, where $F^*_1(G)=F^*(G)$ is the generalized Fitting subgroup, and $F^*_{i+1}(G)$ is the inverse image of $F^*(G/F^*_{i}(G))$. It is proved that if a finite group $G=AB$ is factorized by two subgroups of coprime orders, then the nonsoluble length of $G$ is bounded in t"},"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":"1405.1899","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-05-08T12:04:32Z","cross_cats_sorted":[],"title_canon_sha256":"476cbc980da357c1b797008ef00045dd22cff11f5805be7cbf2094bb5e180ca2","abstract_canon_sha256":"bec27de0abd7777a645b70f118f9e285de893609031e60e665998a622dc072bd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:52:15.191807Z","signature_b64":"3Oylhk+5utlxs55YQvyq8V26o2HRduDvD+Rv6uIY9qRVm0ikw+cv2HbCechsBnQ9OZ8EMk1xKNvgz9BUQH4gAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"339ca6676f6f54720797c9354c136e62ef7da22a304cf69858bcc02610318b6f","last_reissued_at":"2026-05-18T02:52:15.191145Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:52:15.191145Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the length of finite factorized groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"E. I. Khukhro, P. Shumyatsky","submitted_at":"2014-05-08T12:04:32Z","abstract_excerpt":"The nonsoluble length $\\lambda (G)$ of a finite group $G$ is defined as the number of nonsoluble factors in a shortest normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group $G$ is the least number $h=h^*(G)$ such that $F^*_h(G)=G$, where $F^*_1(G)=F^*(G)$ is the generalized Fitting subgroup, and $F^*_{i+1}(G)$ is the inverse image of $F^*(G/F^*_{i}(G))$. It is proved that if a finite group $G=AB$ is factorized by two subgroups of coprime orders, then the nonsoluble length of $G$ is bounded in t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.1899","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1405.1899","created_at":"2026-05-18T02:52:15.191251+00:00"},{"alias_kind":"arxiv_version","alias_value":"1405.1899v1","created_at":"2026-05-18T02:52:15.191251+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.1899","created_at":"2026-05-18T02:52:15.191251+00:00"},{"alias_kind":"pith_short_12","alias_value":"GOOKMZ3PN5KH","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_16","alias_value":"GOOKMZ3PN5KHEB4X","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_8","alias_value":"GOOKMZ3P","created_at":"2026-05-18T12:28:30.664211+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/GOOKMZ3PN5KHEB4XZE2UYE3OML","json":"https://pith.science/pith/GOOKMZ3PN5KHEB4XZE2UYE3OML.json","graph_json":"https://pith.science/api/pith-number/GOOKMZ3PN5KHEB4XZE2UYE3OML/graph.json","events_json":"https://pith.science/api/pith-number/GOOKMZ3PN5KHEB4XZE2UYE3OML/events.json","paper":"https://pith.science/paper/GOOKMZ3P"},"agent_actions":{"view_html":"https://pith.science/pith/GOOKMZ3PN5KHEB4XZE2UYE3OML","download_json":"https://pith.science/pith/GOOKMZ3PN5KHEB4XZE2UYE3OML.json","view_paper":"https://pith.science/paper/GOOKMZ3P","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1405.1899&json=true","fetch_graph":"https://pith.science/api/pith-number/GOOKMZ3PN5KHEB4XZE2UYE3OML/graph.json","fetch_events":"https://pith.science/api/pith-number/GOOKMZ3PN5KHEB4XZE2UYE3OML/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GOOKMZ3PN5KHEB4XZE2UYE3OML/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GOOKMZ3PN5KHEB4XZE2UYE3OML/action/storage_attestation","attest_author":"https://pith.science/pith/GOOKMZ3PN5KHEB4XZE2UYE3OML/action/author_attestation","sign_citation":"https://pith.science/pith/GOOKMZ3PN5KHEB4XZE2UYE3OML/action/citation_signature","submit_replication":"https://pith.science/pith/GOOKMZ3PN5KHEB4XZE2UYE3OML/action/replication_record"}},"created_at":"2026-05-18T02:52:15.191251+00:00","updated_at":"2026-05-18T02:52:15.191251+00:00"}