Characterizing subgroup perfect codes by 2-subgroups

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