Skip to content
DraconiticResearch
Search Sources API
188,932 studies31 sourcesUpdated 29 min ago
PreprintarXiv licencearXiv

On the number of conjugacy classes of finite solvable groups of order divisible by p2

  • Thomas Michael Keller
Open at sourcePDFDOICite

Abstract

In 2003, Héthelyi and Külshammer proved that if G is a finite solvable group and p is a prime such that p2 divides |G|, then k(G)≥ (49p+1)/60, and they constructed an infinite family of groups for which this bound was claimed to be attained. We point out that the groups in this family are not Frobenius groups, so that their class numbers are considerably larger than stated, and that in fact the bound is never attained. We then determine the best possible bound of this shape: we show that k(G)≥ (49p+7)/60 for all primes p, which is sharp for p=17, and that k(G)≥ (49p+451)/60 for all primes p>2719, which is sharp for infinitely many primes p. Thus the slope 49/60 found by Héthelyi and Külshammer is best possible, but the correct additive constant is 451/60 rather than 1/60.

Related

  1. On the number of ℓ-regular conjugacy classes
    Zhicheng Feng
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  2. On the order of a finite group with trivial centre and bounded conjugacy classes
    Ilya Gorshkov
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  3. Finite groups with conjugacy classes of size 2pn
    Antonio Beltran, Changguo shao
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  4. The flag major index on conjugacy classes of wreath products
    Harel Rozenfeld
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  5. Categorical and Floer-theoretic entropies: comparison and counting conjugacy classes in symplectic mapping class groups
    Sangjin Lee
    arXivPreprintarXiv licenceFull textDynamical systemsGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{keller2026number,
  title = {{On the number of conjugacy classes of finite solvable groups of order divisible by $p^2$}},
  author = {Thomas Michael Keller},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12962},
  eprint = {2610.12962},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12962},
}
RIS
TY  - UNPB
TI  - On the number of conjugacy classes of finite solvable groups of order divisible by p^2
AU  - Thomas Michael Keller
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.12962
UR  - https://arxiv.org/abs/2610.12962
AB  - In 2003, Héthelyi and Külshammer proved that if G is a finite solvable group and p is a prime such that p^2 divides |G|, then k(G)≥ (49p+1)/60, and they constructed an infinite family of groups for which this bound was claimed to be attained. We point out that the groups in this family are not Frobenius groups, so that their class numbers are considerably larger than stated, and that in fact the bound is never attained. We then determine the best possible bound of this shape: we show that k(G)≥ (49p+7)/60 for all primes p, which is sharp for p=17, and that k(G)≥ (49p+451)/60 for all primes p>2719, which is sharp for infinitely many primes p. Thus the slope 49/60 found by Héthelyi and Külshammer is best possible, but the correct additive constant is 451/60 rather than 1/60.
ER  -