{
  "abstract": "Let $k/\\mathbb{Q}$ be a finitely generated field extension. We characterize the subgroups $G\\leq GL_n(k)$ for which the power images $P_m(G)=\\{g^m:g\\in G\\}$ form a finite family: this holds precisely when $G=N\\rtimes F$, where $F$ is finite and $N$ is a uniquely divisible subgroup consisting of unipotent matrices. The subgroup $N$ is canonical and fully invariant.For $e=\\exp(G/N)$, we obtain $P_m(G)=P_{\\gcd(m,e)}(G)$, exactly $τ(e)$ distinct power images, and least positive period $e$, where $τ(e)$ is the number of positive divisors of $e$.Over $\\mathbb{Q}$, the group coincides with the full group of rational points of its Zariski closure. Since finite automorphism-orbit count implies finiteness of the power-image family, an affirmative answer to Problem~21.40 of the Kourovka Notebook\\cite{Kourovka} follows as a corollary. We further prove that a rational linear group $G$ admits a decomposition $G\\cong(\\mathbb{Q}^d,+)\\rtimes F$, with $d\\geq0$ finite and $F$ finite, whenever a virtually nilpotent subgroup of $\\operatorname{Aut}(G)$ has finitely many orbits on $G$.The latter argument uses nonconstant invariant rational functions arising from nonzero Lie brackets to obstruct dense orbits. A metabelian action with five orbits on the rational Heisenberg group shows that solubility of the acting group is insufficient.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23373",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhexin Jiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7807zc83b08pa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23373",
  "fulltext_url": null,
  "id": "7807zc83b08pa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23373",
  "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.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7222968332",
  "pdf_url": "https://arxiv.org/pdf/2610.23373",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7807zc83b08pa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23373",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23373"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Finite power image families and automorphism actions in linear groups",
  "title_text": "Finite power image families and automorphism actions in linear groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/7807zc83b08pa",
  "venue": "arXiv",
  "versions": []
}