{
  "abstract": "Let $g_1H_1,\\ldots,g_nH_n$ be left cosets of subgroups of a group $G$ whose union $U$ is a proper subset of $G$. We prove that $G$ is the union of at most $2^n$ left translates of $A:=G\\setminus U$. For finite $G$ this yields $|A|\\ge|G|/2^n$, which settles a conjecture of Tărnăuceanu and the author. Equality $|A|=|G|/2^n$ can only hold when $A$ is a left coset of $K:=H_1\\cap\\ldots\\cap H_n$. Moreover, every subgroup occurring in an irredundant cover of an arbitrary group by $n$ cosets has index at most $2^{n-1}$. This improves a lemma of Neumann.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09052",
  "author_total": 1,
  "authors": [
    {
      "name": "Benjamin Sambale"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0dyekz8g9w034.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09052",
  "fulltext_url": null,
  "id": "0dyekz8g9w034",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09052",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7212037224",
  "pdf_url": "https://arxiv.org/pdf/2609.09052",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0dyekz8g9w034.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.09052",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09052"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the complement of a union of cosets",
  "title_text": "On the complement of a union of cosets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/0dyekz8g9w034",
  "venue": "arXiv",
  "versions": []
}