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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.15206 |
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| _version_ | 1866918099044794368 |
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| author | chen, Huye Li, Binbin Li, Jingjian Yu, Hao |
| author_facet | chen, Huye Li, Binbin Li, Jingjian Yu, Hao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15206 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Subgroup Perfect Codes of $\mathcal{A}_t$-Groups and Their Applications chen, Huye Li, Binbin Li, Jingjian Yu, Hao Combinatorics 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. |
| title | Subgroup Perfect Codes of $\mathcal{A}_t$-Groups and Their Applications |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.15206 |