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PreprintCC BY 4.0arXiv

Arithmetic properties of solubilizers

  • Luca Sabatini
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Abstract

Let G be a finite group and x ∈ G. In 2013, Hai-Reuven proved that the cardinality of the centralizer of x divides the cardinality of the solubilizer of x. Settling a conjecture of Mousavi, Poozesh and Zamani, we show that the cardinality of the normalizer of ⟨ x ⟩ is also a divisor. We actually give direct short proofs of more general results.

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Cite

BibTeXRIS
BibTeX
@misc{sabatini2026arithmetic,
  title = {{Arithmetic properties of solubilizers}},
  author = {Luca Sabatini},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03151},
  eprint = {2610.03151},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03151},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Arithmetic properties of solubilizers
AU  - Luca Sabatini
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.03151
UR  - https://arxiv.org/abs/2610.03151
AB  - Let G be a finite group and x ∈ G. In 2013, Hai-Reuven proved that the cardinality of the centralizer of x divides the cardinality of the solubilizer of x. Settling a conjecture of Mousavi, Poozesh and Zamani, we show that the cardinality of the normalizer of ⟨ x ⟩ is also a divisor. We actually give direct short proofs of more general results.
ER  -