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When $\\Gamma$ and $G$ are finite we show that such universal factorization exists: $\\Gamma\\to\\Gamma_{\\infty}\\to G,$ where $\\Gamma_{\\infty}$ is a hypercentral extension of the subnormal closure $\\mathcal{C}$ of $\\varphi(\\Gamma)$ in $G$ (i.e.~"},"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":"1405.0090","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-05-01T04:24:03Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"37e93104af438e26726d11bb98c2eb9b77fa57f90c25cadba50c580a2757617e","abstract_canon_sha256":"fc3d585c20af6fac284abb727a62d484bf42476c8d1a2c9a6e850a95ae1d09de"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:52:49.013709Z","signature_b64":"EHkXxCM3OTXCMEJbvX+QR0FxyQyF8fSvoExxaDMe0GRkmobFSkDWapMdE6r10A4rrWkVwXmp14aLe/CW57lHAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4bf61adef6ee455416507ba6b028795923f6fc327135597792d4dd81736f972d","last_reissued_at":"2026-05-18T02:52:49.013247Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:52:49.013247Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Subnormal closure of a homomorphism","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.GR","authors_text":"Emmanuel D. 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