AI 中文总结
受张纪平对霍尔数分类问题启发，钱国华提出霍尔补数分类问题。本文证明每个霍尔补数要么是\(1\)，要么是\(4k + 2\)（\(k\)为非负整数），解决了钱国华的问题。
AI 中文摘要
一个正整数\(m\)，若每个阶数恰好能被\(m\)整除的有限群\(G\)都有一个阶为\(m\)的霍尔子群，则称\(m\)为一个“霍尔数”。为寻求有限可解群的西罗定理和霍尔定理的推广，张纪平提出对霍尔数进行完全分类，该问题最近由郭、胡和李解决。受张问题启发，钱国华提出了关于霍尔补数完全分类的类似问题。若每个满足\(m\)恰好整除\(|G|\)的有限群\(G\)都有一个阶为\(|G|/m\)的霍尔子群，则正整数\(m\)称为“霍尔补数”。本文证明每个霍尔补数要么是\(1\)，要么是\(4k + 2\)的形式（\(k\)为非负整数），从而回答了钱的问题。
英文摘要
A positive integer $m$ is termed a \emph{Hall number} if every finite group $G$ whose order is precisely divisible by $m$ possesses a Hall subgroup of order $m$. Seeking generalizations of Sylow's theorem and Hall's theorem for finite solvable groups, Jiping Zhang asked for a full classification of Hall numbers, a problem recently solved by Guo, Hu and Li. Inspired by Zhang's problem, Guohua Qian put forward an analogous problem on a full classification of Hall complement numbers. Recall that a positive integer $m$ is called a \emph{Hall complement number} provided that every finite group $G$ with $m$ precisely dividing $|G|$ admits a Hall subgroup of order $|G|/m$. In the present paper, we prove that every Hall complement number is either $1$ or of the form $4k+2$ for some non-negative integer $k$, thus answering Qian's problem.
