Factorizations of groups of small order

TL;DR

This paper classifies small finite groups based on their ability to be factored into subsets with specific product properties, identifying exactly six groups that do not have this property among those of order up to 60.

Contribution

It provides a complete classification of non-multifold-factorizable groups of order at most 60, introducing the concept of $k$-factorizability and analyzing its implications.

Findings

Identifies exactly 6 non-multifold-factorizable groups up to order 60.

Defines and explores the concept of $k$-factorizability in finite groups.

Provides open questions related to group factorizations.

Abstract

Let $G$ be a finite group and let $A_1,\ldots,A_k$ be a collection of subsets of $G$ such that $G=A_1\ldots A_k$ is the product of all the $A_i$'s with $|G|=|A_1|\ldots|A_k|$. We write $G=A_1\cdot\ldots\cdot A_k$ and call this a $k$-fold factorization of $G$ of the form $(|A_1|,\ldots,|A_k|)$ or more briefly an $(|A_1|,\ldots,|A_k|)$-factorization of $G$. Let $k\geq2$ be a fixed integer. If $G$ has an $(a_1,\ldots,a_k)$-factorization, whenever $|G|=a_1\ldots a_k$ with $a_i>1$, $i=1,\ldots,k$, we say that $G$ is $k$-factorizable. We say that $G$ is multifold-factorizable if $G$ is $k$-factorizable for any possible integer $k\geq2$. In this paper we prove that there are exactly $6$ non-multifold-factorizable groups among the groups of order at most $60$. Here is their complete list: $A_4$, $(C_2\times C_2)\rtimes C_9$, $A_4\times C_3$, $(C_2\times C_2\times C_2)\rtimes C_7$, $A_5$, $A_4\times C_5$. Some related open questions are presented.