{
  "abstract": "Let $G$ be a finite nonabelian group. The small Davenport constant $\\mathsf d(G)$ of $G$ is the largest integer $\\ell$ such that there exists a product-one free sequence over $G$ of length $\\ell$, while the Gao constant $E(G)$ of $G$ is the least integer $\\ell$ such that every sequence over $G$ of length at least $\\ell$ contains a product-one subsequence of length exactly $|G|$. A long-standing conjecture of Gao and Zhuang \\cite{ZG2005} asserts that $E(G)=\\mathsf d(G)+|G|$ for every finite nonabelian group $G$. Let $p$ be an odd prime and let $H_{p^3}=\\operatorname{UT}_3(\\mathbb F_p)$ be the finite Heisenberg group over $\\mathbb F_p$. Godara and Sarkar proved the Gao-Zhuang equality for $H_{27}=\\operatorname{UT}_3(\\mathbb F_3)$ and asked whether the same equality holds for $H_{p^3}$ for every odd prime $p$. Recently, Volkmann proved that $\\mathsf d(H_{p^3})=3p-3$. In this paper, we determine the Gao constant of $H_{p^3}$ and prove that $E(H_{p^3})=\\mathsf d(H_{p^3})+|H_{p^3}|=p^3+3p-3$. Together with the known abelian and cyclic-index cases, this completes the verification of the Gao-Zhuang equality for all groups of order $p^3$, for every prime $p$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.23319",
  "author_total": 2,
  "authors": [
    {
      "name": "Yongke Qu"
    },
    {
      "name": "Guoqing Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f4we6fscvfemr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.23319",
  "fulltext_url": null,
  "id": "f4we6fscvfemr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.23319",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7204155431",
  "pdf_url": "https://arxiv.org/pdf/2608.23319",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f4we6fscvfemr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2608.23319",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.23319"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Gao-Zhuang conjecture for the Heisenberg group over $\\mathbb{F}_p$",
  "title_text": "The Gao-Zhuang conjecture for the Heisenberg group over 𝔽_p",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/f4we6fscvfemr",
  "venue": "arXiv",
  "versions": []
}