A necessary and sufficient condition for a prime to be an integer group determinant of certain $p$-groups

Abstract

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 ${\mathrm{C}}_p\times H$, where ${\mathrm{C}}_p$ 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.