{
  "abstract": "For a finite group $G$, let $λ(G)$ denote the minimum number of orbits on the elements of a finite lattice $L$ with $\\operatorname{Aut}(L)\\cong G$. Babai and Goodman conjectured that $λ(G)$ is bounded by an absolute constant. We prove that $λ(G)\\leq 50$ for every finite group $G$, thereby confirming their conjecture. Moreover, the lattice can be chosen to have a regular orbit. The main algebraic ingredient is a decomposition of a generating set of an arbitrary finite $2$-group into an elementary abelian part and two sets in which no quotient of distinct elements is an involution.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35191",
  "author_total": 4,
  "authors": [
    {
      "name": "JiaLi Du"
    },
    {
      "name": "Andrea Lucchini"
    },
    {
      "name": "Joy Morris"
    },
    {
      "name": "Pablo Spiga"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dzm56t3mtm8kr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35191",
  "fulltext_url": null,
  "id": "dzm56t3mtm8kr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35191",
  "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.CO"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7214783247",
  "pdf_url": "https://arxiv.org/pdf/2609.35191",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dzm56t3mtm8kr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.35191",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35191"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Bounded-orbit lattice representations of finite groups",
  "title_text": "Bounded-orbit lattice representations of finite groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/dzm56t3mtm8kr",
  "venue": "arXiv",
  "versions": []
}