Factoring nonabelian finite groups into two subsets

Abstract

A group G is said to be factorized into subsets $A_1, A_2,$ $\ldots, A_s\subseteq G$ if every element g in G can be uniquely represented as $g=g_1g_2\ldots g_s$, where $g_i\in A_i$, $i=1,2,\ldots,s$. 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 $10\,000$.