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

TL;DR

The paper addresses the problem of determining all integer values taken by the group determinant for semidirect products $G=\mathbb Z_p \rtimes \mathbb Z_n$ with $p$ prime and $n \mid p-1$, focusing on the affine groups GA$(1,p)$ and related cases. It derives a unified factorization $D = A B^n$, with $A$ coming from a $\mathbb Z_n$-determinant and $B$ as a product of determinants across irreducible representations; in the GA$(1,p)$ case ($n=p-1$) $B(\omega)$ is integral, yielding $D = A B^{p-1}$. It proves sharp divisibility and congruence constraints and provides explicit classifications in several instances (notably GA$(1,5)$ and certain half-derivative and safe-prime cases), expressing results in norm-like forms such as $D = m \, N\bigl(m+\sum \beta_i(\alpha_i-n)\bigr)^n$ or as norms from quadratic, cubic, or quartic fields. The work also presents constructive descriptions of attainable values and ends with speculations and open questions about extending these patterns to broader families and to cases with Sophie Germain primes.

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 $\mathbb Z_7\rtimes \mathbb Z_3,$ $\mathbb Z_{11}\rtimes \mathbb Z_5$ and $\mathbb Z_{13}\rtimes \mathbb Z_6,$ 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, $\mathbb Z_{13}\rtimes_5 \mathbb Z_4$ and $\mathbb Z_{13}\rtimes \mathbb Z_3$, (working in an appropriate cubic and quartic field).

Theorems & Definitions (22)

• Theorem 3.1
• proof
• Lemma 4.1
• proof : Proof of Lemma \ref{['ex']}
• Theorem 5.1
• proof : Proof of Theorem \ref{['GA']}
• Lemma 5.1
• proof
• Theorem 5.2
• proof