Characterizing subgroup perfect codes by 2-subgroups

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A perfect code in a graph $$\Gamma $$ is a subset C of $$V(\Gamma )$$ such that no two vertices in C are adjacent and every vertex in $$V(\Gamma ){\setminus } C$$ is adjacent to exactly one vertex in C. Let G be a finite group and C a subset of G. Then C is said to be a perfect code of G if there exists a Cayley graph of G admiting C as a perfect code. It is proved that a subgroup H of G is a perfect code of G if and only if a Sylow 2-subgroup of H is a perfect code of G. This result provides a way to simplify the study of subgroup perfect codes of general groups to the study of subgroup perfect codes of 2-groups. As an application, a criterion for determining subgroup perfect codes of projective special linear groups $$\textrm{PSL}(2,q)$$ is given.