论文标题
无扭转组的功率图确定了定向的功率图
The Power Graph of a Torsion-Free Group Determines the Directed Power Graph
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。 $ \ Mathbf G $的电源图，用$ \ Mathcal G（\ Mathbf G）$表示为基础简单图。 在本文中，对于$ \ mathbf g $和$ \ mathbf H $的组，以下证明了以下内容。如果$ \ mathbf g $没有Quasicyclic子组$ \ MATHBF C_ {p^\ infty} $，它与每个环状子组$ \ MATHBF K $ of $ \ MATHBF G $都具有微不足道的相交g（\ mathbf g）\ cong \ mathcal g（\ mathbf h）$含义$ \ vec {\ mathcal g}（\ mathbf g）\ cong \ cong \ vec {\ vec {\ mathcal g}（\ mathbf h）$。因此，具有同构功率图的任何两个无扭转的组具有同构的定向功率图。
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 of $\mathbf G$, denoted with $\mathcal G(\mathbf G)$, is the underlying simple graph. In this paper, for groups $\mathbf G$ and $\mathbf H$, the following is proved. If $\mathbf G$ has no quasicyclic subgroup $\mathbf C_{p^\infty}$ which has trivial intersection with every cyclic subgroup $\mathbf K$ of $\mathbf G$ such that $\mathbf K\not\leq\mathbf C_{p^\infty}$, then $\mathcal G(\mathbf G)\cong \mathcal G(\mathbf H)$ implies $\vec{\mathcal G}(\mathbf G)\cong \vec{\mathcal G}(\mathbf H)$. Consequently, any two torsion-free groups having isomorphic power graphs have isomorphic directed power graphs.
