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Paper IPM / M / 17072  


Abstract:  
Let Ã?ÃÂ´ = Cay(G, S) be a Cayley graph on a ÃÂ¯ÃÂ¬Ã¯Â¿Â½nite group G. A perfect code in Ã?ÃÂ´ is a subset C of G such that every vertex in G \ C is adjacent to exactly one vertex in C and vertices of C are not adjacent to each other. In Zhang and Zhou (Eur J Comb 91:103228, 2021) it is proved that if H is a subgroup of G whose Sylow 2subgroup is a perfect code of G, then H is a perfect code of G.AlsotheyprovedthatifG is a metabelian group and H is a normal subgroup of G, then H is a perfect code of G if and only if a Sylow 2subgroup of H is aperfect code of G. As a generalization, we prove that this result holds for each ÃÂ¯ÃÂ¬Ã¯Â¿Â½nite group G. Also they proved that if G is a nilpotent group and H is a subgroup of G, then H is a perfect code of G if and only if the Sylow 2subgroup of H is a perfect code of G.Wegeneralize this result by proving that the same result holds for every group with a normal Sylow 2subgroup. In the rest of the paper, we classify groups whose set of all subgroup perfect codes forms a chain of subgroups and also we determine groups with exactly two proper nontrivial subgroup perfect codes.
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