论文标题
D（2） - 某些Metacyclic组的范围
The D(2)-Property for some metacyclic groups
论文作者
论文摘要
我们研究了与D（2）类型$ G（P，P，P-1）$的问题有关的问题，其中$ p $是一个奇怪的素数。具体而言，我们以纳迪姆的论文为基础\ cite {jamil}，这表明$ \ mathbb {z} [g（5,4）] $ - module $ \ mathbb {z} $允许对斜分辨率和第三个syzygy $ω_3（z______________} $ \ z} $ c {z} $ { $ r（2）\ oplus [y-1）$。在此结果的激励下，我们证明了$ \ mathbb {z} [g（p，p，p-1）] $ - 模块$ r（2）\ oplus [y-1）$对于任何奇数prime $ p $既饱满，又是直的。 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 $ \ mathbb {z} [g（p，p，p-1）] $上的$ω_3（\ mathbb {z}）$，我们称为条件m（p）。遵循此结果，我们证明了一个定理，该定理简化了证明条件M（p）所需的计算。最后，在$ p = 7 $并证明M（7）持有的情况下，我们进行了这些计算，这足以证明$ g（7,6）$满足D（2） - 专业。
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 $Ω_3(\mathbb{Z})$ 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 $Ω_3(\mathbb{Z})$ 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.
