Some answers regarding factorizations of finite groups

Ryan McCulloch

paper

Some answers regarding factorizations of finite groups

arXiv:2607.20569

Abstract

In this note we answer some questions of G. M. Bergman concerning factorizations of finite groups. Such questions originated when M. H. Hooshmand asked whether, given a finite group $G$ and a factorization $∣ G ∣ = n_{1} \dots n_{k}$ , one can always find subsets $A_{1}, \dots, A_{k}$ of $G$ with $∣ A_{i} ∣ = n_{i}$ such that $G = A_{1} \dots A_{k}$ . This was Question 19.35 of the Kourovka Notebook. G. M. Bergman provided a counterexample for $k = 3$ , and Kabenyuk provided counterexamples for all $k \geq 3$ . We present these counterexamples with direct proofs that mirror Bergman's original $k = 3$ argument, and we answer another recent question of Bergman.