{
  "abstract": "A Steiner quadruple system of order $v$, denoted by $\\operatorname{SQS}(v)$, is called cyclic if it admits $\\mathbb Z_v$ as a group of automorphisms acting regularly on its point set. A cyclic system is called symmetric cyclic, or S-cyclic, if the multiplier $-1$ preserves each block orbit individually. Grannell and Griggs conjectured in 1983 that such a system of order $v$ exists if and only if $v\\equiv2,4,10,20\\pmod{24}$ and every odd prime divisor $p$ of $v$ satisfies $p\\equiv1$ or $5\\pmod{12}$. This paper proves the conjecture. The main step is to construct an $S$-cyclic Steiner quadruple system of order $2p$ for every prime $p\\equiv1\\pmod4$. Together with existing reduction theorems, these constructions determine which finite abelian groups admit a symmetric group-invariant Steiner quadruple system.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25951",
  "author_total": 2,
  "authors": [
    {
      "name": "Hengrui Liu"
    },
    {
      "name": "Tao Feng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fd5jbww72aqj0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25951",
  "fulltext_url": null,
  "id": "fd5jbww72aqj0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25951",
  "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.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7222996893",
  "pdf_url": "https://arxiv.org/pdf/2610.25951",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fd5jbww72aqj0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25951",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25951"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The existence of symmetric cyclic Steiner quadruple systems",
  "title_text": "The existence of symmetric cyclic Steiner quadruple systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/fd5jbww72aqj0",
  "venue": "arXiv",
  "versions": []
}