{
  "abstract": "We prove that almost all finite groups are $2$-groups of nilpotency class two, when groups of order at most $x$ are counted up to isomorphism and $x\\to\\infty$. More precisely, uniformly for $2^m\\leq x<2^{m+1}$, all but an $O(2^{-cm^2})$ proportion have order $2^m$ and class two, for some absolute constant $c>0$. For each fixed prime $p$, we give a formula for the number of groups of order $p^m$, with relative error $O(p^{-m/3})$ as $m\\to\\infty$. This refines the classical logarithmic estimates for the number of $p$-groups. For every fixed prime $p$ and sufficiently large $m$, we also prove that all but a $p^{-m^2/300}$ proportion of groups of order $p^m$ have central elementary abelian Frattini subgroup. For fixed $p$, the distribution of the minimum number of generators among groups of order $p^m$ is asymptotically supported on one or two adjacent values, with explicit probabilities depending on $m$ modulo three.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11092",
  "author_total": 2,
  "authors": [
    {
      "name": "Ning Su"
    },
    {
      "name": "Binzhou Xia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a2aadxhm8dem7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11092",
  "fulltext_url": null,
  "id": "a2aadxhm8dem7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11092",
  "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": "W7221072665",
  "pdf_url": "https://arxiv.org/pdf/2610.11092",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a2aadxhm8dem7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11092",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11092"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Almost all finite groups are $2$-groups of class two",
  "title_text": "Almost all finite groups are $2$-groups of class two",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/a2aadxhm8dem7",
  "venue": "arXiv",
  "versions": []
}