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

TL;DR

The paper proves that integer group determinants for the general affine group $GA(1,q)$ with $q=p^k$ factorize as $D=AB^{q-1}$, where $A$ is a $\mathbb{Z}_{q-1}$ determinant and $B\equiv A\pmod q$, extending the known $k=1$ case. Using a group presentation and Frobenius' determinant formula, it identifies the two determinant factors and proves the key congruence $B\equiv A\pmod q$. It then establishes achievability results: (i) when $\gcd(A,q-1)=1$; (ii) when $(q-1)^2|A$; and, in the special cases $q=2^k$ with $q-1$ prime (Mersenne primes) and $q=9,27$, provides complete or near-complete classifications of attainable $D$. For $GA(1,9)$ and $GA(1,27)$ the paper develops constructive procedures and reports explicit computational instances that realize the predicted structures, illustrating the practicality of the method for larger $q$.

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 $GA(1,27)$.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 5.1
• proof
• Theorem 5.2
• proof
• Theorem 6.1
• Theorem 7.1
• proof
• Theorem 8.1
• proof