Groups whose Chermak–Delgado lattice is a subgroup lattice of an elementary abelian p-group

The Chermak–Delgado lattice of a finite group G is a self-dual sublattice of the subgroup lattice of G. In this paper, we focus on finite groups whose Chermak–Delgado lattice is a subgroup lattice of an elementary abelian p-group. We prove that such groups are nilpotent of class 2. We also prove that, for any elementary abelian p-group E, there exists a finite group G such that the Chermak–Delgado lattice of G is a subgroup lattice of E.