Solubility from a disconnected common-divisor graph on p-regular conjugacy-class sizes

Abstract

Let G be a finite group and let p be a prime. We prove that if the common-divisor graph on the nontrivial conjugacy-class sizes of p-regular elements of G is disconnected, then G is soluble. This settles the remaining case identified by Camina and collaborators. Their structure results provide an abelian normal p'-subgroup M. We choose a noncentral r-element x∈ M, with r≠ p, such that [G:C_G(x)] is coprime to pr, and show that C_G(x)/M is a {p,r}-group. Thus C_G(x)/M is a Hall {p,r}-subgroup of G/M. The separation between the two graph components then implies that every p-regular conjugacy-class size in G/M has all its prime divisors in {p,r}. A theorem of Dolfi and Lucido, together with Burnside's p^a q^b-theorem, yields the solubility of G/M and hence of G.

On the number of ℓ-regular conjugacy classes
Zhicheng Feng
arXiv26 Sep 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Computing the Greatest Common Divisor of Binomial Coefficients binommnmk
Chai Wah Wu
arXiv4 Aug 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Havel--Hakimi Residues of Common-Divisor Graphs: Complete Asymptotics and Prime-Counting Structure
Randy Davila
arXiv31 Aug 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
Higher localised Â-genera: a note on disconnected fixed-point sets
Hao Guo
arXiv13 Oct 2026PreprintarXiv licenceFull textAlgebra & number theory Geometry & topologyarXiv
On the existence of F-irregular graphs for disconnected graphs F
Tatiana Dovzhenok
arXiv13 Oct 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv

Cite

BibTeX RIS

BibTeX

@misc{zhang2026solubility,
  title = {{Solubility from a disconnected common-divisor graph on $p$-regular conjugacy-class sizes}},
  author = {Zhi-Lin Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.27960},
  eprint = {2609.27960},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27960},
}

RIS

TY  - UNPB
TI  - Solubility from a disconnected common-divisor graph on p-regular conjugacy-class sizes
AU  - Zhi-Lin Zhang
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.27960
UR  - https://arxiv.org/abs/2609.27960
AB  - Let G be a finite group and let p be a prime. We prove that if the common-divisor graph on the nontrivial conjugacy-class sizes of p-regular elements of G is disconnected, then G is soluble. This settles the remaining case identified by Camina and collaborators. Their structure results provide an abelian normal p'-subgroup M. We choose a noncentral r-element x∈ M, with r≠ p, such that [G:C_G(x)] is coprime to pr, and show that C_G(x)/M is a {p,r}-group. Thus C_G(x)/M is a Hall {p,r}-subgroup of G/M. The separation between the two graph components then implies that every p-regular conjugacy-class size in G/M has all its prime divisors in {p,r}. A theorem of Dolfi and Lucido, together with Burnside's p^a q^b-theorem, yields the solubility of G/M and hence of G.
ER  -

Abstract

Related

Cite