{
  "abstract": "Let H be a subgroup of a finite group G, and put n = [G:H] > 1. If p is the least prime divisor of n, we prove that the number of subgroups K with H <= K <= G is less than c(p) n^((log_p n)/4). Here c(p) is the product of (1 - p^(-j))^(-1) over all positive integers j, multiplied by the sum of p^(-z^2) over all integers z. The exponent and the constant are simultaneously optimal, already for elementary abelian p-groups of even rank. In particular, for arbitrary n > 1, the number of intermediate subgroups is less than 7.371968802 n^((log_2 n)/4). This removes the polynomial factor from the best previously known relative bound and extends the sharp absolute subgroup-counting theorem to arbitrary subgroup intervals. The group-theoretic argument is elementary. Its main ingredient is a weighted packing inequality for relative generating tuples, obtained from an escape tree on the coset space G/H.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10343",
  "author_total": 2,
  "authors": [
    {
      "name": "Sebastien Palcoux"
    },
    {
      "name": "Pablo Spiga"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0hqn00k6t8sy2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10343",
  "fulltext_url": null,
  "id": "0hqn00k6t8sy2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10343",
  "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": "W7212180127",
  "pdf_url": "https://arxiv.org/pdf/2609.10343",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0hqn00k6t8sy2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.10343",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10343"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Sharp bounds for intervals in finite subgroup lattices",
  "title_text": "Sharp bounds for intervals in finite subgroup lattices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/0hqn00k6t8sy2",
  "venue": "arXiv",
  "versions": []
}