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We show that the number of z-classes in I(V, B) is finite when F is perfect and has the property that it has only finitely many field extensions of degree at most n."},"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":"0906.0563","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2009-06-02T19:30:16Z","cross_cats_sorted":[],"title_canon_sha256":"dce7de41d1f5ccaec4f6516921910bf28eb39afebe31c43a01818d2f53f5569c","abstract_canon_sha256":"78d5863e722bb1b05f5d133642b9cceae132a205a1a266729f70e9de95d96e05"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:55.557270Z","signature_b64":"/603RIi3EbaB3xmcKqG3M0ofQ6j/BKfswTrvVDZ/AfEI18b7YlcKENE7hE1kefPHrIfQ8S76nA/FnrfkmR1XBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1d77f3e76e0c31f6b4706e2d4f54d8e81cdfa049b6054f61e10d07d966c3bd68","last_reissued_at":"2026-05-18T02:28:55.556752Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:55.556752Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The z-Classes of Isometries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Krishnendu Gongopadhyay, Ravi S. 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