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 \(E\).