The integer group determinants for the abelian groups of order 18

TL;DR

This paper fully characterizes the integer group determinants for the two abelian groups of order 18, completing the classification for all groups of order less than 20.

Contribution

It provides a complete description of the integer group determinants for the specific abelian groups of order 18, filling a gap in the classification of small group determinants.

Findings

Complete description of determinants for Z_18 and Z_3 x Z_6

Classification of group determinants for all groups of order less than 20

Explicit formulas for circulant determinants with integer entries

Abstract

We obtain a complete description of the integer group determinants for $\mathbb Z_{18}$ (these are the $18\times18$ circulant determinants with integer entries) and $\mathbb Z_3 \times \mathbb Z_6$, the two abelian groups of order 18. This completes the groups of order less than 20.