An observation on factorizations of finite groups

Given a finite group $G$ and a factorization of its order, $\mathrm{\mid }G\mathrm{\mid }=a_1{a}_2,$ we show that a sufficient condition for there to exist subsets $A_1,A_2\subseteq G$ such that $\mathrm{\mid }{A}_1\mathrm{\mid }=a_1,$ $\mathrm{\mid }{A}_2\mathrm{\mid }=a_2,$ and $G=A_1{A}_2,$ is that there exist a chain of subgroups $\{e\}=G_0<\cdots <G_n=G$ such that $a_1$ is the product of some subfamily of the indices $\mathrm{\mid }{G}_i:G_{i-1}\mathrm{\mid }$ (and hence $a_2$ is the product of the complementary subfamily).