Subgroup perfect codes of Sₙ in Cayley sum graphs

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. 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 S. A subset C of G is called perfect code of G if there exists a Cayley sum graph of G that 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 work, we prove that there does not exist any proper perfect subgroup code of symmetric group Sₙ. Using this result, we provide a complete characterization of the perfect subgroup code of the alternating group Aₙ.