  We give a necessary and sufficient condition for a prime to be an integer
group determinant for an arbitrary abelian $p$-group of the form ${\rm C}_{p}
\times H$, where ${\rm C}_{p}$ is the cyclic group of order $p$. Also, we show
that under certain conditions, the integer group determinant of a finite group
$G$ that is prime is the integer group determinant of the abelianization of
$G$. As a result, we know that the integer group determinant of a $p$-group
that is prime is the integer group determinant of its abelianization.
