论文标题
所有Nilpotent组的功率图
Power graphs of all nilpotent groups
论文作者
论文摘要
组$ \ Mathbf G $的定向电源图$ \ vec {\ Mathcal G}（\ Mathbf G）$是带有顶点套装$ g $的简单挖掘物，因此，如果$ x \ rightarrow y $ y $是$ x $的电源。组$ \ mathbf g $的电源图$ \ MATHCAL G（\ MATHBF G）$是基础简单图。 在本文中，我们证明了Prüfer组是唯一一个尼尔氏群体，其功率图并未确定为同构的定向功率图。另外，我们将一个$ \ mathbf g $与准扭转子组呈现，该子组由其功率图确定为同构为同构，即$ \ Mathcal G（\ Mathbf H）\ Mathbf H）\ cong \ Mathcal g（\ Mathcal G（\ Mathbf G）$表示$ \ \ MATHBF H \ MATHBF H \ MATHB $ MATHBF G $ $ $ $ $ $ $ $ $ $。
The directed power graph $\vec{\mathcal G}(\mathbf G)$ of a group $\mathbf G$ is the simple digraph with vertex set $G$ such that $x\rightarrow y$ if $y$ is a power of $x$. The power graph $\mathcal G(\mathbf G)$ of the group $\mathbf G$ is the underlying simple graph. In this paper, we prove that Prüfer group is the only nilpotent group whose power graph does not determine the directed power graph up to isomorphism. Also, we present a group $\mathbf G$ with quasicyclic torsion subgroup that is determined by its power graph up to isomorphism, i.e. such that $\mathcal G(\mathbf H)\cong\mathcal G(\mathbf G)$ implies $\mathbf H\cong \mathbf G$ for any group $\mathbf H$.
