Perfect codes in Cayley graphs of abelian groups

A perfect code in a graph $\mathrm{\Gamma }=(V,E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V\setminus C$ is adjacent to exactly one vertex in $C$. A total perfect code in $\mathrm{\Gamma }$ is a subset $C$ of $V$ such that every vertex of $\mathrm{\Gamma }$ is adjacent to exactly one vertex in $C$. In this paper we prove several results on perfect codes and total perfect codes in Cayley graphs of finite abelian groups.