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/ may be updated more frequently than arXiv copy. I've cleaned up the wording and some arguments from previous version. As noted in 2nd paragraph of section 2, I've decided not to publish this note, since the main result, Theorem 1.1, is easily seen to hold using the method of proof of [2, Proposition 2]