Skip to content
DraconiticResearch
Search Sources API
194,226 studies31 sourcesUpdated 12 min ago
PreprintCC BY 4.0arXiv

Bounded-orbit lattice representations of finite groups

  • JiaLi Du
  • Andrea Lucchini
  • Joy Morris
  • Pablo Spiga
Open at sourcePDFDOICite

Abstract

For a finite group G, let λ(G) denote the minimum number of orbits on the elements of a finite lattice L with Aut(L)≅ G. Babai and Goodman conjectured that λ(G) is bounded by an absolute constant. We prove that λ(G)≤ 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.

Related

  1. Tate cohomology of unipotent cuspidal representations of finite groups of Lie type
    Sabyasachi Dhar
    arXivPreprintCC0Full textAlgebra & number theoryarXiv
  2. Absence of bounded-orbit wandering domains
    Nikolai Prochorov, Lasse Rempe, James Waterman
    arXivPreprintarXiv licenceFull textAnalysis & PDEsDynamical systemsarXiv
  3. Bounded-orbit wandering domains do not exist
    Kostiantyn Drach, Leticia Pardo-Simón, Beno Učakar
    arXivPreprintCC BY 4.0Full textDynamical systemsarXiv
  4. The Complexity of Conjugacy, Orbit Equivalence, and von Neumann Equivalence of Actions of Nonamenable Groups
    Eusebio Gardella, Martino Lupini
    Memoirs of the American Mathematical SocietyPeer-reviewedLicence not statedFull text6 citedMathematical physicsOpenAlex
  5. Abelian quotients and orbit sizes for subgroups of linear groups
    Yong Yang
    arXivPreprintCC BY-NC-ND 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{du2026boundedorbit,
  title = {{Bounded-orbit lattice representations of finite groups}},
  author = {JiaLi Du and Andrea Lucchini and Joy Morris and Pablo Spiga},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.35191},
  eprint = {2609.35191},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.35191},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Bounded-orbit lattice representations of finite groups
AU  - JiaLi Du
AU  - Andrea Lucchini
AU  - Joy Morris
AU  - Pablo Spiga
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.35191
UR  - https://arxiv.org/abs/2609.35191
AB  - For a finite group G, let λ(G) denote the minimum number of orbits on the elements of a finite lattice L with Aut(L)≅ G. Babai and Goodman conjectured that λ(G) is bounded by an absolute constant. We prove that λ(G)≤ 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.
ER  -