Perfect codes in Cayley graphs of abelian groups

A perfect code in a graph $Γ= (V, E)$ is a subset C of V such that no two vertices in C are adjacent and every vertex in V C is adjacent to exactly one vertex in C. A total perfect code in $Γ$ is a subset C of V such that every vertex of $Γ$ 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.