Finite 𝒟C-groups

Abstract

Let G be a group and [Formula: see text]. G is said to be a [Formula: see text]-group if [Formula: see text] is a chain under set inclusion. In this paper, we prove that a finite [Formula: see text]-group is a semidirect product of a Sylow p-subgroup and an abelian [Formula: see text]-subgroup. For the case of G being a finite p-group, we obtain an optimal upper bound of [Formula: see text] for a [Formula: see text] p-group G. We also prove that a [Formula: see text] p-group is metabelian when [Formula: see text] and provide an example showing that a non-abelian [Formula: see text] p-group is not necessarily metabelian when [Formula: see text]. In particular, [Formula: see text] 2-groups are characterized.