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The subgroup $H$ is called observable if $H=\\hat H$ and epimorphic if $G=\\hat H$. In this work I show that under some natural restrictions $H$ is observable if and only if some orbit of some group contains 0 in the closure and $H$ is epimorphic if and only the same orbit is closed."},"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":"1007.1348","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-07-08T11:08:40Z","cross_cats_sorted":[],"title_canon_sha256":"4b3fe37a49de2e38fdd20b5439d57d21dbcae6f3997627283c74ffe3b2fd4d65","abstract_canon_sha256":"c5692721e7b8ab7ee72eb846910dffb1fa7a53b528e4c6e1d25d13e4f07acf9a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:06:23.928842Z","signature_b64":"JCYHjIPQlYlefN5IYlJbC/TdrsXrwalh07qdizYkZq1dbYaAVNSJNtj4XzYxGa5GiH4xPPsmKkoXVzJ94oLRAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aad2948447ade7309b73bf96f48e646042d280d5852e1d42bca7632c9ef3a769","last_reissued_at":"2026-05-18T02:06:23.928340Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:06:23.928340Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Geometric Description of Epimorphic Subgroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alexey Petukhov","submitted_at":"2010-07-08T11:08:40Z","abstract_excerpt":"Let $G$ be an affine algebraic group over an algrebraically closed field $\\mathbb K$ of characteristic 0 and $H$ be a subgroup of $G$. 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