Complete factorizations of finite groups

Abstract

Let $G$ be a group. The subsets $A_1,\ldots,A_k$ of $G$ form a complete factorization of group $G$ if if they are pairwise disjoint and each element $g\in G$ is uniquely represented as $g=a_1\ldots a_k$, with $a_i\in A_i$. We prove the following theorem: Let $G$ be a finite nilpotent group. If $|G|=m_1\ldots m_k$ where $m_1,\ldots,m_k$ are integers greater $1$ and $k\geq3$, then there exist subsets $A_1,\ldots,A_k$ of $G$ which form a complete factorization of group $G$ and $|A_i|=m_i$ for all $i=1,2,\ldots,k$. In addition, we give several examples of building complete factorization for some groups and formulate one open question.