Which maximal subgroups are perfect codes?

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 subgroup H of a group G is called a subgroup perfect code of G if it is a perfect code in some Cayley graph of G. In this paper, we undertake a systematic study of which maximal subgroups of a group can be perfect codes. Our approach highlights a characterization of subgroup perfect codes in terms of their ``local'' complements.