  A finite group of order $n$ is said to have the distinct divisor property
(DDP) if there exists a permutation $g_1,\ldots, g_n$ of its elements such that
$g_i^{-1}g_{i+1} \neq g_j^{-1}g_{j+1}$ for all $1\leq i<j<n$. We show that an
abelian group is DDP if and only if it has a unique element of order 2. We also
describe a construction of DDP groups via group extensions by abelian groups
and show that there exist infinitely many non abelian DDP groups.
