{
  "abstract": "Investigating the descendant tree T(Delta) of 3-groups G of maximal nilpotency-class, with root Delta=SmallGroup(27,3), we show for the first time that an entropy-value H(P_r)=sum P_r(G)|log(P_r(G))| can also be determined for the real probability-distribution P_r(G) of all terminal vertices outside of the mainline on this infinite digraph. These vertices arise as automorphism groups Gal(F(3,infinity,K)/K) of maximal unramified pro-3-extensions F(3,infinity,K) of real-quadratic number fields K=Q(d^1/2), d>0. For the groups G of another descendant tree T(N), with root N=SmallGroup(729,45) and unbounded coclass, which has been studied by L. Bartholdi and M. R. Bush, we compare the entropy H(P_r) of the real measure P_r(G) with the entropy H(P_i) of the imaginary measure P_i(G).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06166",
  "author_total": 1,
  "authors": [
    {
      "name": "Helga Boyer von Berghof"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fytf2h6cmjm1m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06166",
  "fulltext_url": null,
  "id": "fytf2h6cmjm1m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06166",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7212012045",
  "pdf_url": "https://arxiv.org/pdf/2609.06166",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fytf2h6cmjm1m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06166",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06166"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The entropy of coclass one and of SmallGroup(729,45)",
  "title_text": "The entropy of coclass one and of SmallGroup(729,45)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/fytf2h6cmjm1m",
  "venue": "arXiv",
  "versions": []
}