Subgroup perfect codes of $ S_n $ in Cayley graphs

Abstract

A perfect code in a graph $\Gamma = (V, E)$ 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.