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Paper IPM / M / 17923

School of Mathematics
Title:	The influence of non-perfect code subgroups on the structure of the groups
Author(s):	Zeinab Akhlaghi (Joint with Y. Khaefi and Beh. Khosravi)
Status:	Published
Journal:	J. Algebra Appl.
Vol.:	24
Year:	2024
Pages:	1-8
Supported by:	IPM

Abstract:

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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“School of Mathematics”

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