The D(2)-Property for some metacyclic groups

TL;DR

This paper investigates the D(2)-Property for certain metacyclic groups, establishing conditions under which these groups satisfy the property, and confirms the property for specific cases like G(7,6).

Contribution

It introduces the condition M(p) for the D(2)-Property and proves its validity for p=7, advancing understanding of the property for metacyclic groups.

Findings

G(5,4) satisfies the D(2)-property

Condition M(p) is sufficient for the D(2)-property

M(7) holds, confirming G(7,6) satisfies the D(2)-property

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.