  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 $100$.
