Hall complement numbers

Abstract

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.