A perfect code in a graph $Γ= (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a subgroup perfect code of $G$ if there exists a Cayley graph of $G$ which admits $H$ as a perfect code. In this work, we present a classification of cyclic 2-subgroup perfect codes in $ S_n$. We analyze these subgroup codes, detailing their structure and properties. We extend our discussion to various classes of subgroup codes in the symmetric group $ S_n $, encompassing both commutative and non-commutative cases. We provide numerous examples to illustrate and support our findings.

翻译：在图$Γ= (V, E)$中，完美码是顶点集$V$的一个子集$C$，满足$C$中任意两个顶点不相邻，且$V \setminus C$中的每个顶点都恰好与$C$中的一个顶点相邻。若存在群$G$的某个Cayley图以子群$H$作为完美码，则称$H$为$G$的子群完美码。本工作给出了$S_n$中循环2-子群完美码的完整分类。我们系统分析了这些子群码的结构与性质，并将讨论扩展到对称群$S_n$中各类子群码，涵盖交换与非交换情形。文中提供了大量示例以阐释并佐证所得结论。