Factorizations of groups of small order

Publication date: 15 March 2021

Abstract: Let $G$ be a finite group and let $A_1,ldots,A_k$ be a collection of subsets of $G$ such that $G=A_1ldots A_k$ is the product of all the $A_i$'s with $|G|=|A_1|ldots|A_k|$. We write $G=A_1cdotldotscdot A_k$ 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 $kgeq2$ be a fixed integer. If $G$ has an $(a_1,ldots,a_k)$-factorization, whenever $|G|=a_1ldots a_k$ 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 $kgeq2$. 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: $A_4$, $(C_2 imes C_2) times C_9$, $A_4 imes C_3$, $(C_2 imes C_2 imes C_2) times C_7$, $A_5$, $A_4 imes C_5$. Some related open questions are presented.

Mathematics Subject Classification ID

Arithmetic and combinatorial problems involving abstract finite groups (20D60)

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