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The necessary and sufficient condition is the zero-sum condition $$ a_1+\\cdots+a_{|G|}=0. $$ This paper studies the corresponding problem for finite nonabelian groups, with differences replaced by quotients. Thus we ask when a multiset $A$ of cardinality $|G|$ can be represented as $$ A=\\{b(i)c(i)^{-1}:1\\le i\\le |G|\\}, $$ where $b$ an"},"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":true,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.16478","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-05-15T16:15:43Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"fa0b4642c61fb2f0c243a154b1ff334fe09a6acf549a4c5b23e590a5a29a0456","abstract_canon_sha256":"ae3e4b40066f03e36b93843083d33084ffdef90d5521e6f8053f35dcd3957ee2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:02:24.017767Z","signature_b64":"+IIS4A0SzpMz4PH+Ek6il3RcxTHaMWQp7hfq7s1bo6UYoTBb8Vk//fZEYRE7x5l/j8FdL3ChFiSPjqXKO9SWBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"494d210d5d8aa1555e101703d0c646232c23eff2e977f3075ecdb246a08aa6a6","last_reissued_at":"2026-05-20T00:02:24.017239Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:02:24.017239Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A nonabelian twist on differences of bijections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Quotient-realizability in nonabelian groups requires a cycle-tiling decomposition of partial products beyond the abelianization condition.","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Mohsen Aliabadi","submitted_at":"2026-05-15T16:15:43Z","abstract_excerpt":"Hall's theorem on differences of bijections characterizes the multisets $$ \\{a_1,\\ldots,a_{|G|}\\} $$ in a finite abelian group $G$ that can be written in the form $$ a_i=b_i-c_i, $$ where both $b_1,\\ldots,b_{|G|}$ and $c_1,\\ldots,c_{|G|}$ are enumerations of $G$. The necessary and sufficient condition is the zero-sum condition $$ a_1+\\cdots+a_{|G|}=0. $$ This paper studies the corresponding problem for finite nonabelian groups, with differences replaced by quotients. Thus we ask when a multiset $A$ of cardinality $|G|$ can be represented as $$ A=\\{b(i)c(i)^{-1}:1\\le i\\le |G|\\}, $$ where $b$ an"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"quotient-realizability is equivalent to a decomposition of A into product-one words whose partial-product sets tile G by right translates","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the standard use of permutation cycles and the translation of quotient-realizability into an exact tiling condition on partial-product sets is sufficient to capture all obstructions without further group-specific invariants beyond the abelianization product (as described in the main structural result).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A cycle-tiling criterion characterizes when a multiset A in a finite nonabelian group G can be realized as quotients from two bijections, with the abelianization product condition shown insufficient even when the product in G is the identity, via a counterexample in S3.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Quotient-realizability in nonabelian groups requires a cycle-tiling decomposition of partial products beyond the abelianization condition.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"8046b55d3101d68853ee92010787985ff983638e1974854ef1bed4d2c4a96a28"},"source":{"id":"2605.16478","kind":"arxiv","version":1},"verdict":{"id":"cc0cc229-3dbc-468f-b952-eda29253b5ee","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T21:34:26.851803Z","strongest_claim":"quotient-realizability is equivalent to a decomposition of A into product-one words whose partial-product sets tile G by right translates","one_line_summary":"A cycle-tiling criterion characterizes when a multiset A in a finite nonabelian group G can be realized as quotients from two bijections, with the abelianization product condition shown insufficient even when the product in G is the identity, via a counterexample in S3.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the standard use of permutation cycles and the translation of quotient-realizability into an exact tiling condition on partial-product sets is sufficient to capture all obstructions without further group-specific invariants beyond the abelianization product (as described in the main structural result).","pith_extraction_headline":"Quotient-realizability in nonabelian groups requires a cycle-tiling decomposition of partial products beyond the abelianization condition."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16478/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T22:01:23.239787Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T21:41:22.832904Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T19:33:23.112348Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T19:21:57.038580Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"71d8bbe2f8d5c393d9b47dcaabffe81c4f880ad24fd109ef2847d63b9d9173e6"},"references":{"count":12,"sample":[{"doi":"","year":1990,"title":"B. 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