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In order to obtain small loops of Cs\\\"org\\H{o} type, we expand our programme from `Explicit constructions of loops with commuting inner mappings', European J. Combin. 29 (2008), 1662-1681, and analyze the following setup in groups:\n  Let $G$ be a group, $Z\\le Z(G)$, and suppose that $\\delta:G/Z\\times G/Z\\to Z$ satisfies $\\delta(x,x)=1$, $\\delta(x,y)=\\delta(y,x)^{-1}$, $z^{yx}\\delta([z,y],x) = z^"},"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":"1509.05723","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-09-18T17:36:30Z","cross_cats_sorted":[],"title_canon_sha256":"ba32618d456ee563f450bf44c38ee179bea39c5c9ce35673c8b6a7deb773829c","abstract_canon_sha256":"b587e5448b9eb9a9bd3927bcca18e6a241e8cac71304e44e21b181f4ca3d12d6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:42.507754Z","signature_b64":"G6ehNdf345DXXE4ydYyI4YevWrVWLC8BB2RmMCuKn9Wrh1SqyfwGBAgmgqQasoNwRngV/2nyJQhBy6hjfEcuBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f70c3980e61d488f87851194a61e89de8658c451ad76709c19d7c3b8f9e468a7","last_reissued_at":"2026-05-18T01:32:42.507362Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:42.507362Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Small loops of nilpotency class three with commutative inner mapping groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Ale\\v{s} Dr\\'apal, Petr Vojt\\v{e}chovsk\\'y","submitted_at":"2015-09-18T17:36:30Z","abstract_excerpt":"Groups with commuting inner mappings are of nilpotency class at most two, but there exist loops with commuting inner mappings and of nilpotency class higher than two, called loops of Cs\\\"org\\H{o} type. In order to obtain small loops of Cs\\\"org\\H{o} type, we expand our programme from `Explicit constructions of loops with commuting inner mappings', European J. 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