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

A subset $C$ of the vertex set of a graph $\mathrm{\Gamma }$ is called a perfect code in $\mathrm{\Gamma }$ if every vertex of $\mathrm{\Gamma }$ 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.