A note on factorizations of finite groups

Abstract

In Question 19.35 of the Kourovka Notebook, M. H. Hooshmand asks whether, given a finite group G and a factorization card(G)= n1… nk, one can always find subsets A1,…,Ak of G with card(Ai)=ni such that G=A1… Ak; equivalently, such that the group multiplication map A1×…× Ak G is a bijection. We show that for G the alternating group on 4 elements, k=3, and (n1,n2,n3) = (2,3,2), the answer is negative. We then generalize some of the tools used in our proof, and note an open question.