Some answers regarding factorizations of finite groups

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 $|G| = n_1 \cdots n_k$, one can always find subsets $A_1, \dots , A_k$ of $G$ with $|A_i| = n_i$ such that $G = A_1 \cdots A_k$. 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.