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

TL;DR

This paper characterizes when a prime number can be realized as an integer group determinant for certain abelian p-groups and relates prime group determinants to their abelianizations.

Contribution

It provides a necessary and sufficient condition for primes to be integer group determinants of specific p-groups and links prime group determinants to abelianizations.

Findings

Prime group determinants are characterized for abelian p-groups of the form C_p x H.

Under certain conditions, prime group determinants of a group G equal those of its abelianization.

The result simplifies understanding of prime group determinants for 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 ${\rm C}_{p} \times H$, 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.