Characterization of subgroup perfect codes in Cayley graphs

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 subset $C$ of a group $G$ is called a perfect code of $G$ if $C$ is a perfect code in some Cayley graph of $G$. In this paper we give sufficient and necessary conditions for a subgroup $H$ of a finite group $G$ to be a perfect code of $G$. Based on this, we determine the finite groups that have no nontrivial subgroup as a perfect code, which answers a question by Ma, Walls, Wang and Zhou.