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

@inproceedings{An2021GroupsWC, title={Groups whose Chermak-Delgado lattice is a subgroup lattice of an elementary abelian \$p\$-group}, author={Lijian An}, year={2021} }

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 ChermakDelgado 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.

#### References

SHOWING 1-10 OF 18 REFERENCES

Two classes of finite groups whose Chermak-Delgado lattice is a chain of length zero

- Mathematics
- 2018

ABSTRACT It is an open question in the study of Chermak-Delgado lattices precisely which finite groups G have the property that 𝒞𝒟(G) is a chain of length 0. In this note, we determine two classes… Expand

Quasi-antichain Chermak–Delgado lattices of finite groups

- Mathematics
- 2014

The Chermak–Delgado lattice of a finite group is a dual, modular sublattice of the subgroup lattice of the group. This paper considers groups with a quasi-antichain interval in the Chermak–Delgado… Expand

Groups Whose Chermak-Delgado Lattice is a Chain

- Mathematics
- 2013

For a finite group G with subgroup H the Chermak-Delgado measure of H in G refer to the product of the order of H with the order of its centralizer, C_G(H). The set of all subgroups with maximal… Expand

Finite groups with a trivial Chermak–Delgado subgroup

- Mathematics
- 2017

Abstract The Chermak–Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups of G. The least element of the Chermak–Delgado lattice of G is known as the… Expand

Chermak–Delgado simple groups

- Mathematics
- 2015

ABSTRACT This paper provides the first steps in classifying the finite solvable groups having Property A, which is a property involving abelian normal subgroups. We see that this classification is… Expand

A note on the Chermak-Delgado lattice of a finite group

- Mathematics
- 2016

In this note we describe the structure of finite groups G whose Chermak-Delgado lattice is the interval [G/Z(G)] = {H \in L(G) \mid Z(G)\leq H\leq G}.

Chermak–Delgado Lattice Extension Theorems

- Mathematics
- 2013

If G is a finite group with subgroup H, then the Chermak–Delgado measure of H (in G) is defined as |H||C G (H)|. The Chermak–Delgado lattice of G, denoted 𝒞𝒟(G), is the set of all subgroups with… Expand

SOME GROUPS WITH COMPUTABLE CHERMAK–DELGADO LATTICES

- Mathematics
- Bulletin of the Australian Mathematical Society
- 2012

Abstract Let G be a finite group and let H≤G. We refer to |H||CG(H)| as the Chermak–Delgado measure ofH with respect to G. Originally described by Chermak and Delgado, the collection of all subgroups… Expand

The Chermak–Delgado measure in finite p-groups

- Mathematics
- 2018

Abstract Let G be a finite group and let H ≤ G . The Chermak–Delgado measure ( [7] ) of H is defined as the number | H ‖ C G ( H ) | . The subgroups of G having maximum measure, the F 1 ( G )… Expand

Centrally large subgroups of finite p-groups

- Mathematics
- 2006

Let S be a finite p-group. We say that an abelian subgroup A of S is a large abelian subgroup of S if |A|⩾|A∗| for every abelian subgroup A∗ of S. We say that a subgroup Q of S is a centrally large… Expand