{"paper":{"title":"A note on partitions of groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Igor Protasov, Sergii Slobodianiuk","submitted_at":"2014-08-24T13:57:27Z","abstract_excerpt":"Every infinite group $G$ of regular cardinality can be partitioned $G=A_1\\cup A_2$ so that $G\\neq FA_1$, $G\\neq FA_2$ for every subset $F\\subset G$ of cardinality $|F|<|G|$. The first author asked whether the same is true for each group $G$ of singular cardinality. We show that an answer depends on the algebraic structure of $G$. In particular, this is so for each free group but the statement does not hold for every Abelian group $G$ of singular cardinality. As an application, we prove that every Abelian group of singular cardinality k admits maximal translation invariant k-bounded topology th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.5607","kind":"arxiv","version":1},"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"}