The integer group determinants for $GA(1,q)$

TL;DR

This paper characterizes the structure of integer group determinants for the affine group of degree one over prime power fields, revealing a specific form and conditions for certain prime powers.

Contribution

It generalizes the form of integer group determinants for $GA(1,q)$ to prime powers and establishes necessary and sufficient conditions for specific cases like Mersenne primes.

Findings

Determinants take the form $D=AB^{q-1}$ with $A$ and $B$ related modulo $q$

For $q=2^k$, the condition involving Mersenne primes is necessary and sufficient

Explicit results for $GA(1,9)$ and $GA(1,27)$ cases

Abstract

We show that the integer group determinants for the general affine group of degree one, $GA(1,q)$ with $q=p^k$ a prime power, take the form $D=AB^{q-1},$ where $A$ is a $\mathbb Z_{q-1}$ integer group determinant and $B\equiv A \bmod q$. This generalizes the result for $k=1$. When $2^k-1$ is a Mersenne prime we show that this condition is both necessary and sufficient for $GA(1,2^k).$ The same is true for $GA(1,9)$ and $GA(1,27)$.