The integer group determinants for the abelian groups of order 18

Abstract

We obtain a complete description of the integer group determinants for ${\mathbb{Z}}_{18}$ (these are the $18\times 18$ 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.