The influence of non-perfect code subgroups on the structure of groups

Let [Formula: see text] be a graph. A subset [Formula: see text] of [Formula: see text] is called a perfect code of [Formula: see text], when [Formula: see text] is an independent set and every vertex of [Formula: see text] is adjacent to exactly one vertex in [Formula: see text]. Let [Formula: see text] be a Cayley graph of a finite group [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is called a perfect code of [Formula: see text], when there exists a Cayley graph [Formula: see text] of [Formula: see text] such that [Formula: see text] is a perfect code of [Formula: see text]. Recently, groups whose set of all subgroup perfect codes forms a chain are classified. Also, groups with no proper nontrivial 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.