The aim of this paper is the characterization of algebraic properties of Leavitt path algebra of the directed power graph $\mathcal{𝒫}(G)$ and also of the directed punctured power graph ${\mathcal{𝒫}}^{\ast }(G)$ of a finite group $G$. We show that Leavitt path algebra of the power graph $\mathcal{𝒫}(G)$ of finite group $G$ over a field $K$ is simple if and only if $G$ is a direct sum of finitely many cyclic groups of order 2. Finally, we prove that the Leavitt path algebra $L_K({\mathcal{𝒫}}^{\ast }(G))$ is a prime ring if and only if $G$ is either cyclic $p$-group or generalized quaternion $2$-group.

中文翻译：

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有限群幂图的 Leavitt 路径代数

本文的目的是表征有向幂图的Leavitt路径代数的代数性质$\mathcal{𝒫}(G)$以及有向穿刺功率图${\mathcal{𝒫}}^{*}(G)$有限群的$G$. 我们展示了幂图的 Leavitt 路径代数$\mathcal{𝒫}(G)$有限群的$G$在一个领域$ķ$简单当且仅当$G$是有限多个 2 阶循环群的直接和。最后，我们证明了 Leavitt 路径代数${\mathrm{大号}}_{ķ}({\mathcal{𝒫}}^{*}(G))$是质数环当且仅当$G$要么是循环的$p$-群或广义四元数$2$-团体。