{"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. Alspach, J.-C. Bermond, and D. Sotteau, Decomposition into cycles. I. Hamilton decompositions, in Cycles and Rays, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 301, Kluwer Academic Publisher","work_id":"982b8357-d3c2-446a-9391-9e0f6d45fc89","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1974,"title":"D´ enes and A","work_id":"55cc653d-7b4b-42df-919a-f5af65271d6b","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"A. B. Evans, Applications of complete mappings and orthomorphisms of finite groups,Quasigroups Related Systems23(2015), no. 1, 5–30","work_id":"c0d5bd02-41d5-47aa-9fe3-cf603d95091c","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1958,"title":"Fuchs, Ein kombinatorisches Problem bez¨ uglich abelscher Gruppen,Math","work_id":"c69068bb-8320-4c23-b974-2e0f03e976fd","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2006,"title":"W. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: A survey,Expo. Math.24 (2006), no. 4, 337–369. 17","work_id":"25bf4214-a6d9-4458-846f-1987722b302c","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":12,"snapshot_sha256":"e5124684713c06d8a2d802d1680479bea34cb43a0d31d9ae5601bcb6b1803bdb","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}