An observation on factorizations of finite groups

Abstract

Given a finite group $G$ and a factorization of its order, $|G|=a_1{a}_2,$ we show that a sufficient condition for there to exist subsets $A_1,A_2\subseteq G$ such that $|A_1|=a_1,$ $|A_2|=a_2,$ 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 $|G_i:G_{i-1}|$ (and hence $a_2$ is the product of the complementary subfamily).

3 pages. Copy at https://math.berkeley.edu/~gbergman/papers/unpub/, or if I decide to publish it, https://math.berkeley.edu/~gbergman/papers/, may be updated more frequently than arXiv copy