The exceptional Hall numbers

TL;DR

This paper characterizes Hall numbers, showing that only 12, 24, and 60 are Hall numbers aside from trivial cases, resolving a problem posed by Jiping Zhang.

Contribution

It proves that 12, 24, and 60 are the only non-trivial Hall numbers, providing a complete classification.

Findings

12, 24, and 60 are the only Hall numbers besides trivial cases

Resolved a problem proposed by Jiping Zhang

Established criteria for Hall numbers in finite groups

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.