Subgroup Perfect Codes of $\mathcal{A}_t$-Groups and Their Applications

TL;DR

This paper classifies subgroup perfect codes in certain finite groups called _t-groups for t=0,1, and characterizes these codes in groups with abelian Sylow 2-subgroups, advancing understanding of perfect codes in algebraic structures.

Contribution

It provides a complete classification of subgroup perfect codes in _t-groups for t=0,1, and characterizes codes in groups with abelian Sylow 2-subgroups, extending previous work.

Findings

Complete classification of subgroup perfect codes in _t-groups for t=0,1

Characterization of subgroup perfect codes in groups with abelian Sylow 2-subgroups

Reduction of subgroup perfect code study to p-groups, especially 2-groups

Abstract

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 subgroup $H$ of a group $G$ is called a subgroup perfect code of $G$ if $H$ is a perfect code in some Cayley graph of $G$. Recently, Zhang reveals that the study of subgroup perfect codes of finite groups naturally reduces to the case of $p$-groups, especially $2$-groups. Based on the combined works of Berkovich, Janko and Zhang, every $p$-group is an $\mathcal{A}_t$-group. In this work, we establish a complete classification of subgroup perfect codes of $\mathcal{A}_t$-groups for $t \in\{0, 1\}$. Moreover, subgroup perfect codes of finite groups with abelian Sylow $2$-subgroups are also characterized.