Factoring nonabelian finite groups into two subsets

Abstract

A group $G$ is said to be factorized into subsets $A_1,A_2,\dots ,A_s\subseteq G$ if every element $g$ in $G$ can be uniquely represented as $g=g_1{g}_2\dots g_s$, where $g_i\in A_i$, $i=1,2,\dots ,s$. We consider the following conjecture: for every finite group $G$ and every factorization $n=ab$ of its order, there is a factorization $G=AB$ with $|A|=a$ and $|B|=b$. We show that a minimal counterexample to this conjecture must be a nonabelian simple group and prove the conjecture for every finite group the nonabelian composition factors of which have orders less than $10000$.