Factoring nonabelian finite groups into two subsets

TL;DR

This paper investigates the unique factorization of finite groups into subsets, proposing a conjecture about such factorizations related to group order partitions, and proves it for groups with small nonabelian composition factors.

Contribution

It introduces a conjecture on group factorizations into subsets based on order partitions and proves it for groups with small nonabelian composition factors.

Findings

Counterexamples must be nonabelian simple groups

Conjecture holds for groups with nonabelian factors under 10,000 in order

Minimal counterexamples are nonabelian simple groups

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 $10\,000$.