{
  "abstract": "Let $p$ be an odd prime, $G$ a finite $p$-group, and $V$ a finite faithful irreducible $G$-module. We show that if $G$ has exactly one regular orbit on $V$, then $p=2^{n}-1$ is a Mersenne prime and $(G,V)$ embeds into $(P_m,V_m)$ for some $m\\ge 1$, where $P_m$ is a Sylow $p$-subgroup of the monomial group ${\\rm GL}_1(2^{n})\\wr S_{p^{m-1}}$ and $V_m$ is its natural module, regarded over ${\\Bbb F}_2$. We determine the largest orbit size $p^{x_m}$ of $P_m$ on $V_m$ and the number of orbits of this size, and we construct subgroups of $P_m$ of order $p^{x_m}$ whose unique regular orbit is a largest orbit of $P_m$; for $p=3$ such subgroups exist for every $m$, whereas for Mersenne primes $p\\ge 7$ we prove that they exist only for $m\\ge 3$. Finally, we classify the subgroups of $P_m$ of order $p^{x_m}$ with exactly one regular orbit for $m=2$ and, with the help of a computer, for $p=3$ and $m=3$, where there are $13$ such pairs up to isomorphism; for $p=3$ and $m=4$ we show that there are exactly $4914$ conjugacy classes of such subgroups in $P_4$. In particular, such a subgroup is in general not determined by its regular orbit.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25737",
  "author_total": 2,
  "authors": [
    {
      "name": "Thomas Michael Keller"
    },
    {
      "name": "Jeffrey M. Riedl"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8m0cvrs8hebgv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25737",
  "fulltext_url": null,
  "id": "8m0cvrs8hebgv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25737",
  "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": "W7223010159",
  "pdf_url": "https://arxiv.org/pdf/2610.25737",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8m0cvrs8hebgv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25737",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25737"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Linear Group Actions with Exactly One Regular Orbit",
  "title_text": "Linear Group Actions with Exactly One Regular Orbit",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/8m0cvrs8hebgv",
  "venue": "arXiv",
  "versions": []
}