2026/07/16 by Yu Zeng, Hangyang Meng
#math.GR
A positive integer m is termed a 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 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.