The D(2)-Property for some metacyclic groups

Abstract

We study problems relating to the D(2)-Problem for metacyclic groups of type $G(p,p-1)$ where $p$ is an odd prime. Specifically we build on Nadim's thesis \cite{Jamil}, which showed that the $\mathbb{Z}[G(5,4)]$-module $\mathbb{Z}$ admits a diagonal resolution and a minimal representative for the third syzygy $\Omega_3(\mathbb{Z})$ is $R(2)\oplus[y-1)$. Motivated by this result, we show that the $\mathbb{Z}[G(p,p-1)]$-module $R(2)\oplus[y-1)$ is both full and straight for any odd prime $p$. Given Johnson's work on the D(2)-Problem \cite{D2}, this leads to the conclusion that $G(5,4)$ satisfies the D(2)-property, as well as providing a sufficient condition for the D(2)-property to hold for $G(p,p-1)$, namely the condition that $R(2)\oplus[y-1)$ is a minimal representative for $\Omega_3(\mathbb{Z})$ over $\mathbb{Z}[G(p,p-1)]$, which we refer to as the condition M(p). Following this result, we prove a theorem which simplifies the calculations required to show that the condition M(p) holds. Finally, we carry out these calculations in the case where $p=7$ and prove that the condition M(7) holds, which is sufficient to show that $G(7,6)$ satisfies the D(2)-property.