The group determinants for $\mathbb Z_n \times H$

TL;DR

This paper provides a method to express the group determinant of a product group involving a cyclic group and another group, and applies it to fully describe the integer group determinants for specific groups like d7_2 d7 D_8 and d7_2 d7 Q_8.

Contribution

It introduces a simple way to relate the group determinant of d7_n imes H to that of H, enabling complete descriptions for certain groups.

Findings

Derived explicit formulas for group determinants of d7_2 d7 D_8 and d7_2 d7 Q_8.

Established a general relation between group determinants of d7_n imes H and H.

Provided a complete characterization of integer group determinants for the specific groups studied.

Abstract

Let $\mathbb Z_n$ denote the cyclic group of order $n$. We show how the group determinant for $G= \mathbb Z_n \times H$ can be simply written in terms of the group determinant for $H$. We use this to get a complete description of the integer group determinants for $\mathbb Z_2 \times D_8$ where $D_8$ is the dihedral group of order 8, and $\mathbb Z_2 \times Q_8$ where $Q_8$ is the quaternion group of order 8.