A subset $C$ of the vertex set of a graph $\Gamma$ is called a perfect code in $\Gamma$ if every vertex of $\Gamma$ is at distance no more than $1$ to exactly one vertex of $C$. A subset $C$ of a group $G$ is called a perfect code of $G$ if $C$ is a perfect code in some Cayley graph of $G$. In this paper we give sufficient and necessary conditions for a subgroup $H$ of a finite group $G$ to be a perfect code of $G$. Based on this, we determine the finite groups that have no nontrivial subgroup as a perfect code, which answers a question by Ma, Walls, Wang and Zhou.

中文翻译：

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Cayley图中子群完美码的表征

如果 $\Gamma$ 的每个顶点到 $C$ 的一个顶点的距离不超过 $1$，则图 $\Gamma$ 的顶点集的子集 $C$ 被称为 $\Gamma$ 中的完美代码. 如果 $C$ 是 $G$ 的某个 Cayley 图中的完美代码，则组 $G$ 的子集 $C$ 被称为 $G$ 的完美代码。在本文中，我们给出了有限群$G$ 的一个子群$H$ 是$G$ 的完美码的充分必要条件。在此基础上，我们确定没有非平凡子群的有限群为完美码，回答了 Ma、Walls、Wang 和 Zhou 的问题。