🤖 AI Summary
This study investigates the characterization of subgroups of the symmetric group $S_n$ that serve as perfect codes in Cayley graphs, with a focus on cyclic 2-subgroups. By integrating techniques from group theory, graph theory, and combinatorial coding theory, the work systematically explores the relationship between the algebraic structure of subgroups and their coding properties in Cayley graphs. It provides the first complete classification of cyclic 2-subgroups of $S_n$ that can be realized as perfect codes, covering both abelian and non-abelian cases, and extends these results to broader classes of subgroups. The findings yield precise structural characterizations supported by numerous explicit examples, thereby advancing the interplay between algebraic graph theory and coding theory.
📝 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.