{
  "abstract": "Let $G$ be a finite group, and let $p$ be the smallest prime divisor of $|G|$. The Gao constant $\\mathsf{E}(G)$ is the least integer $\\ell$ such that every sequence over $G$ of length $\\ell$ contains a subsequence of length $|G|$ whose terms can be ordered to have product equal to the identity. In 1961, Erdős, Ginzburg and Ziv proved that every sequence of $2n-1$ integers has an $n$-term subsequence whose sum is divisible by $n$; equivalently, $\\mathsf{E}(C_n)\\leq 2n-1$, where $C_n$ denotes the cyclic group of order $n$. In this paper, we prove that $\\mathsf{E}(G)\\leq |G|+|G|/p-1$ whenever $G$ contains no cyclic subgroup of index $p$. Consequently, every finite non-cyclic group satisfies $\\mathsf{E}(G)\\leq |G|+|G|/p+p-2$, with equality if and only if $G$ contains a cyclic subgroup of index $p$. This establishes the conjectured upper bound for the Gao constant and provides a complete characterization of the equality case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26122",
  "author_total": 1,
  "authors": [
    {
      "name": "Yongke Qu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d9chhhrwr1cbt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26122",
  "fulltext_url": null,
  "id": "d9chhhrwr1cbt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26122",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223011498",
  "pdf_url": "https://arxiv.org/pdf/2610.26122",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d9chhhrwr1cbt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.26122",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26122"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Erdős--Ginzburg--Ziv theorem for finite groups",
  "title_text": "The Erdős--Ginzburg--Ziv theorem for finite groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/d9chhhrwr1cbt",
  "venue": "arXiv",
  "versions": []
}