{"paper":{"title":"Set-Direct Factorizations of Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Attila Mar\\'oti, Dan Levy","submitted_at":"2017-07-14T21:19:21Z","abstract_excerpt":"We consider factorizations $G=XY$ where $G$ is a general group, $X$ and $Y$ are normal subsets of $G$ and any $g\\in G$ has a unique representation $g=xy$ with $x\\in X$ and $y\\in Y$. This definition coincides with the customary and extensively studied definition of a direct product decomposition by subsets of a finite abelian group. Our main result states that a group $G$ has such a factorization if and only if $G$ is a central product of $\\left\\langle X\\right\\rangle $ and $\\left\\langle Y\\right\\rangle $ and the central subgroup $\\left\\langle X\\right\\rangle \\cap \\left\\langle Y\\right\\rangle $ sat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.04643","kind":"arxiv","version":2},"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"}