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

Abstract

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.

Keywords:

• Chermak–Delgado lattice
• quasi-antichain
• special p-groups

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D15
• 20D30

Acknowledgments

I cordially thank the referee for detail reading and helpful comments, which helped me to improve the whole paper considerably.

Additional information

Funding

This work was supported by NSFC (Nos. 11971280 and 11771258).