{"paper":{"title":"On a permutation problem for finite abelian groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.GR"],"primary_cat":"math.NT","authors_text":"Fan Ge, Zhi-Wei Sun","submitted_at":"2016-01-19T16:51:21Z","abstract_excerpt":"Let $G$ be a finite additive abelian group with exponent $n>1$, and let $a_1,\\ldots,a_{n-1}\\in G$. We show that there is a permutation $\\sigma\\in S_{n-1}$ such that all the elements $sa_{\\sigma(s)}\\ (s=1,\\ldots,n-1)$ are nonzero if and only if $$\\left|\\left\\{1\\le s<n:\\ \\frac{n}{d}a_s\\ne 0\\right\\}\\right|\\ge d-1\\ \\ \\textrm{ for every positive divisor }\\ d\\ \\textrm{ of }\\ n.$$ When $G$ is the cyclic group $\\mathbb Z/n\\mathbb Z$, this confirms a conjecture of Z.-W. Sun."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.04988","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","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"}