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arXiv · 2507.15206

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

Huye chen·Binbin Li·Jingjian Li·Hao Yu
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Abstract

A subset $C$ of the vertex set of a graph $Γ$ is called a perfect code in $Γ$ if every vertex of $Γ$ 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.

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Huye chen, Binbin Li, Jingjian Li, Hao Yu. 2025-07-21. Subgroup Perfect Codes of $\mathcal{A}_t$-Groups and Their Applications. https://arxiv.org/abs/2507.15206

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