Factorizations of finite groups

Abstract: A finite group $G$ is called $k$-factorizable if for any factorization $|G|=a_1\cdots a_k$ with $a_i>1$ there exist subsets $A_i$ of $G$ with $|A_i|=a_i$ such that $G=A_1\cdots A_k$. The main results are as follows. 1. For every integer $k\geq3$ there exist a finite group $G$ such that $G$ is not $k$-factorizable. 2. If a Sylow $2$-subgroup of a finite group $G$ is elementary abelian, all involutions of $G$ are conjugate, and the centralizer of every involution is the direct product of a Sylow $2$-subgroup of $G$ and a group of odd order. Then $G$ is not $3$-factorizable. 3. We give a complete list of not $k$-factorizable groups for some $k$ of order at most $100$.