{"paper":{"title":"The Herzog-Sch\\\"onheim Conjecture for small groups and harmonic subgroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Leo Margolis, Ofir Schnabel","submitted_at":"2018-03-09T15:39:43Z","abstract_excerpt":"We prove that the Herzog-Sch\\\"onheim Conjecture holds for any group $G$ of order smaller than $1440$. In other words we show that in any non-trivial coset partition $\\{g_i U_i\\}_{i=1}^n $ of $G$ there exist distinct $1 \\leq i, j \\leq n$ such that $[G:U_i]=[G:U_j]$.\n  We also study interaction between the indices of subgroups having cosets with pairwise trivial intersection and harmonic integers. We prove that if $U_1$,...,$U_n$ are subgroups of $G$ which have pairwise trivially intersecting cosets and $n \\leq 4$ then $[G:U_1]$,...,$[G:U_n]$ are harmonic integers."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.03569","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"}