{"paper":{"title":"Difference sets disjoint from a subgroup","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Courtney Hoagland, Seth Poulsen, Stephen P. Humphries","submitted_at":"2017-03-20T21:46:24Z","abstract_excerpt":"We study finite groups $G$ having a subgroup $H$ and $D \\subset G \\setminus H$ such that the multiset $\\{ xy^{-1}:x,y \\in D\\}$ has every non-identity element occur the same number of times (such a $D$ is called a {\\it difference set}). We show that $H$ has to be normal, that $|G|=|H|^2$, and that $|D \\cap Hg|=|H|/2$ for all $g \\notin H$. We show that $H$ is contained in every normal subgroup of prime index, and other properties. We give a $2$-parameter family of examples of such groups. We show that such groups have Schur rings with four principal sets."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.06979","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"}