Skip to content
DraconiticResearch
Search Sources API
197,768 studies31 sourcesUpdated 21 min ago
PreprintarXiv licencearXiv

A Huppert-type ρ*--σ* theorem for sub-class sizes

  • Yong Yang
Open at sourcePDFDOICite

Abstract

For a finite group G, let ρ*(G) be the set of primes dividing some sub-class size of G, and let σ*(G) be the maximum number of distinct prime divisors of a sub-class size. Motivated by the conjugacy-class version of Huppert's ρ--σ problem, we prove that |ρ*(G)|<2σ*(G) for every finite nonnilpotent group G. This answers \cite[Remark~5.3]{QY} and extends to arbitrary finite nonnilpotent groups the coefficient-$2$ bound proved in \cite[Theorem~1.7]{QY} for solvable nonnilpotent groups. The coefficient $2$ is best possible. The proof does not use the classification of finite simple groups.

Related

  1. An Ito-Michler type theorem and the prime class graph of weakly integral braided fusion categories
    Sebastian Burciu
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  2. Solubility from a disconnected common-divisor graph on p-regular conjugacy-class sizes
    Zhi-Lin Zhang
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. Cramér-type moderate deviations and precise large deviations for the elephant random walk with random step sizes
    Xiequan Fan, Shuyuan Liang, Dongxue Ma
    arXivPreprintarXiv licenceFull textProbabilityarXiv
  4. Precise large deviations for the elephant random walk with Weibull-type and lognormal-type random step sizes
    Shuyuan Liang
    arXivPreprintCC BY-NC-ND 4.0Full textProbabilityarXiv
  5. Constant Step Sizes for Muon-Type Spectral Optimizers on Strongly Convex Quadratics
    Pratik Jawanpuria, Bhabatosh Garai, Bamdev Mishra
    arXivPreprintCC BY 4.0Full textOptimisation & controlarXiv

Cite

BibTeXRIS
BibTeX
@misc{yang2026hupperttype,
  title = {{A Huppert-type $ρ^*$--$σ^*$ theorem for sub-class sizes}},
  author = {Yong Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29662},
  eprint = {2609.29662},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29662},
}
RIS
TY  - UNPB
TI  - A Huppert-type ρ^*--σ^* theorem for sub-class sizes
AU  - Yong Yang
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.29662
UR  - https://arxiv.org/abs/2609.29662
AB  - For a finite group G, let ρ^*(G) be the set of primes dividing some sub-class size of G, and let σ^*(G) be the maximum number of distinct prime divisors of a sub-class size. Motivated by the conjugacy-class version of Huppert's ρ--σ problem, we prove that |ρ^*(G)|<2σ^*(G) for every finite nonnilpotent group G. This answers \cite[Remark~5.3]{QY} and extends to arbitrary finite nonnilpotent groups the coefficient-$2$ bound proved in \cite[Theorem~1.7]{QY} for solvable nonnilpotent groups. The coefficient $2$ is best possible. The proof does not use the classification of finite simple groups.
ER  -