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Original Article

The Exceptional Hall Numbers

Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China
Department of Mathematics, SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen 518055, China
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Abstract

A positive integer m is called a Hall number if any finite group of order precisely divisible by m has a Hall subgroup of order m. We prove that, except for the obvious examples, the three integers 12, 24 and 60 are the only Hall numbers, solving a problem proposed by Jiping Zhang.

Keywords

Hall subgroupsHall numbersFinite groups

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Peking Mathematical Journal
Volume 9 Issue 3,
September 2026
Pages 497-506
DOI: 10.1007/s42543-025-00104-3

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Cite this article:
Guo, Z., Hu, Y. & Li, CH. The Exceptional Hall Numbers. Peking Math J 9, 497-506 (2026). https://doi.org/10.1007/s42543-025-00104-3

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Received: 07 August 2024
Revised: 04 February 2025
Accepted: 29 April 2025
Published: 30 May 2025
© Peking University 2025