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| Paper IPM / M / 17923 |
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| Abstract: | |
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Let $\Gamma = ( V(\Gamma) , E(\Gamma) )$ be a graph. A subset $C$ of $V(\Gamma) $ is called a perfect code of $\Gamma$, when $C$ is an independent set and
every vertex of $V(\Gamma) \setminus C$
is adjacent to exactly one vertex in $C $.
Let $\Gamma=\Cay (G,S)$ be a Cayley graph of a finite group $G$. A subset $C$ of $G$ is called a perfect code of $G$, when there exists a Cayley graph $\Gamma$ of $G$ such that $C$ is a perfect code of $\Gamma$.
Recently, groups
whose set of all subgroup perfect codes forms a chain are classified. Also, groups with no proper non-trivial subgroup perfect code are characterized. In this paper,
we generalize it and classify groups whose set of all non-perfect code subgroups forms a chain.
It is proved that if $G$ has a normal Sylow 2-subgroup, then a subgroup $H$ of $G$ is a perfect code of $G$ if and only if its Sylow 2-subgroup is a perfect code of $G$.
In the rest of this paper, we show that the same result holds for a 2-nilpotent group $G$, i.e., a subgroup $H$ of a 2-nilpotent group $G$ is a perfect code of $G$ if and only if a Sylow 2-subgroup of $H$ is a perfect code of $G$.
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