Characterizing subgroup perfect codes by 2-subgroups

TL;DR

This paper characterizes subgroup perfect codes in finite groups via Sylow 2-subgroups, simplifying their study and providing criteria for specific groups like PSL(2,q).

Contribution

It establishes that subgroup perfect codes are determined by Sylow 2-subgroups, reducing the problem to the study of 2-groups and applying this to groups like PSL(2,q).

Findings

A subgroup is a perfect code iff its Sylow 2-subgroup is a perfect code.

Simplifies the analysis of subgroup perfect codes to 2-group cases.

Provides a criterion for subgroup perfect codes in PSL(2,q).

Abstract

A perfect code in a graph $\Gamma$ is a subset $C$ of $V(\Gamma)$ such that no two vertices in $C$ are adjacent and every vertex in $V(\Gamma)\setminus C$ is adjacent to exactly one vertex in $C$. Let $G$ be a finite group and $C$ a subset of $G$. Then $C$ is said to be a perfect code of $G$ if there exists a Cayley graph of $G$ admiting $C$ as a perfect code. It is proved that a subgroup $H$ of $G$ is a perfect code of $G$ if and only if a Sylow $2$-subgroup of $H$ is a perfect code of $G$. This result provides a way to simplify the study of subgroup perfect codes of general groups to the study of subgroup perfect codes of $2$-groups. As an application, a criterion for determining subgroup perfect codes of projective special linear groups $\mathrm{PSL}(2,q)$ is given.