Factorizations of groups of small order

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.