Completely realizable groups

Abstract

Given a construction [Formula: see text] on groups, we say that a group [Formula: see text] is [Formula: see text]-realisable if there is a group [Formula: see text] such that [Formula: see text], and completely[Formula: see text]-realisable if there is a group [Formula: see text] such that [Formula: see text] and every subgroup of [Formula: see text] is isomorphic to [Formula: see text] for some subgroup [Formula: see text] of [Formula: see text] and vice versa. In this paper, we determine completely [Formula: see text]-realisable groups. We also study [Formula: see text]-realisable groups for [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] denote the center, the Fitting subgroup, the Chermak–Delgado subgroup, the derived subgroup and the Frattini subgroup of the group [Formula: see text], respectively.