Subgroup perfect codes in Cayley sum graphs

Let \(\Gamma \) be a graph with vertex set V. If a subset C of V is independent in \(\Gamma \) and every vertex in \(V\setminus C\) is adjacent to exactly one vertex in C, then C is called a perfect code of \(\Gamma \). Let G be a finite group and let S be a square-free normal subset of G. The Cayley sum graph of G with respect to S is a simple graph with vertex set G and two vertices x and y are adjacent if \(xy\in S\). A subset C of G is called a perfect code of G if there exists a Cayley sum graph of G which admits C as a perfect code. In particular, if a subgroup of G is a perfect code of G, then the subgroup is called a subgroup perfect code of G. In this paper, we give a necessary and sufficient condition for a non-trivial subgroup of an abelian group with non-trivial Sylow 2-subgroup to be a subgroup perfect code of the group. This reduces the problem of determining when a given subgroup of an abelian group is a perfect code to the case of abelian 2-groups. As an application, we classify the abelian groups whose every non-trivial subgroup is a subgroup perfect code. Moreover, we determine all subgroup perfect codes of a cyclic group, a dihedral group and a generalized quaternion group.