{
  "abstract": "In the binary diameter problem, the input consists of explicitly given permutations generating a finite group $G$ and a binary-encoded number $k$, and the question is whether every element of $G$ is a product of at most $k$ of the given generators. In the binary length problem, the input additionally contains a permutation $g \\in G$, and the question is whether $g$ is such a product. We show that the binary diameter problem is PSPACE-complete, thereby confirming a conjecture of Lohrey and Rosowski (2023). Moreover, we show that PSPACE-hardness already holds for solvable groups of derived length at most 3, whereas the problem is $Π_2^{\\mathrm{P}}$-complete for 2-step nilpotent groups. For the binary length problem, we establish a sharp complexity threshold: the problem is NP-complete for 2-step nilpotent groups, which extends a result of Lohrey and Rosowski for abelian groups, and PSPACE-complete for groups that are simultaneously 3-step nilpotent and metabelian, which strengthens the main result of Jerrum (1985). Finally, we show that the unary diameter problem, where $k$ is encoded in unary, is $Π_2^{\\mathrm{P}}$-complete for generating sets of the full symmetric group, confirming another conjecture of Lohrey and Rosowski.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26307",
  "author_total": 2,
  "authors": [
    {
      "name": "Markus Lohrey"
    },
    {
      "name": "Alexander Thumm"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ajt5yenc663gz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26307",
  "fulltext_url": null,
  "id": "ajt5yenc663gz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26307",
  "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",
      "cs.CC"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7214117224",
  "pdf_url": "https://arxiv.org/pdf/2609.26307",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ajt5yenc663gz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26307",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26307"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Word Length and Diameter in Permutation Groups",
  "title_text": "Word Length and Diameter in Permutation Groups",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/ajt5yenc663gz",
  "venue": "arXiv",
  "versions": []
}