The integer group determinants for GA(1,p) and related semidirect products

Abstract

We consider the integer group determinants for groups that are semidirect products of ${\mathbb{Z}}_p$ and ${\mathbb{Z}}_n$ with $p$ prime and $n\mid p-1$. We give a complete description of the integer group determinants for the general affine groups of degree one GA(1,$p$) when $p=5,7,11$ and $23$, and for ${\mathbb{Z}}_7⋊{\mathbb{Z}}_3,$ ${\mathbb{Z}}_{11}⋊{\mathbb{Z}}_5$ and ${\mathbb{Z}}_{13}⋊{\mathbb{Z}}_6,$ showing that the obvious divisibility and congruence conditions arising from the form of the group determinant when $n=p-1$ or $\frac{1}{2}(p-1)$, can be sufficient as well as necessary for these types of groups (although in the latter case we must work with norms of integers in a quadratic field). For $p=13$ this also happens for the remaining groups of this type, ${\mathbb{Z}}_{13}{⋊}_5{\mathbb{Z}}_4$ and ${\mathbb{Z}}_{13}⋊{\mathbb{Z}}_3$, (working in an appropriate cubic and quartic field).

arXiv commentCorrects typo in the statement of Theorem 5.2. Simplifies computations needed for Theorem 5.3