无挠的群的幂图决定有向幂图
The Power Graph of a Torsion-Free Group Determines the Directed Power Graph
摘要：
群$\mathbf G$的有向幂图$\vec{\mathcal G}(\mathbf G)$是顶点集为$G$的简单有向图，使得如果$y$是$x$的幂，则有$x\rightarrow y$.用$\mathcal G(\mathbf G)$表示的$\mathbf G$的幂图是底层的简单图。本文对群$\mathbf G$和$\mathbf H$证明了如下定理。如果$\mathbf G$没有与$\mathbf G$的每个循环子群$\mathbf K$有平凡交集使得$\mathbf K\not\leq\mathbf C_{p^\infty}$的拟循环子群$\mathbf C_{p^\infty}$，则$\mathcal G(\mathbf G)\cong \mathcal G(\mathbf H)$蕴涵$\vec{\mathcal G}(\mathbf G)\cong \vec{\mathcal G}(\mathbf H)$。因此，任何两个具有同构幂图的无挠无挠的群都具有同构的有向幂图。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
