A note on factorizations of finite groups

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 goup 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 a related open question.