论文标题
由2个群体表征子组完美代码
Characterizing subgroup perfect codes by 2-subgroups
论文作者
论文摘要
图$γ$中的完美代码是$ v（γ）$的子集$ c $，因此$ c $中的两个顶点与$ v（γ）\ setminus c $中的每个顶点相邻，恰好与$ c $中的一个顶点相邻。令$ g $为有限的组，$ c $ $ g $。然后，如果存在$ g $ c $ c $作为完美代码的Cayley图，则$ c $是$ g $的完美代码。事实证明，当$ g $的亚组$ h $是$ g $的完美代码，并且仅当sylow $ 2 $ -subgroup $ h $的$ h $是$ g $的完美代码。该结果提供了一种方法，可以简化整个组的子组完美代码的研究，以研究$ 2 $群体的子组完美代码。作为一个应用程序，给出了用于确定射影特殊线性组的子组完美代码$ \ mathrm {psl}（2，q）$的标准。
A perfect code in a graph $Γ$ is a subset $C$ of $V(Γ)$ such that no two vertices in $C$ are adjacent and every vertex in $V(Γ)\setminus C$ 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.
