Chermak–Delgado Lattice Extension Theorems

Abstract

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 maximal Chermak–Delgado measure; this set is a moduar sublattice within the subgroup lattice of G. In this paper we provide an example of a p-group P, for any prime p, where 𝒞𝒟(P) is lattice isomorphic to 2 copies of ℳ₂ (a quasiantichain of width 2) that are adjoined maximum-to-minimum. We introduce terminology to describe this structure, called a 2-string of 2-diamonds, and we also give two constructions for generalizing the example. The first generalization results in a p-group with Chermak–Delgado lattice that, for any positive integers n and l, is a 2l-string of n-dimensional cubes adjoined maximum-to-minimum and the second generalization gives a construction for a p-group with Chermak–Delgado lattice that is a 2l-string of ℳ _p+1 (quasiantichains, each of width p + 1) adjoined maximum-to-minimum.

Key Words:

• Lattices of subgroups
• p-Groups

2010 Mathematics Subject Classification:

• 20D15
• 20D30
• 20E15

ACKNOWLEDGMENTS

The authors would like to thank Ben Brewster of Binghamton University for posing the original challenge to find a group P with 𝒞𝒟(P) a 2-string of 2-diamonds. The second and fourth authors would like to thank Qinhai Zhang and Shanxi Normal University for the gracious invitation and support during their visit.

Notes

Communicated by P. Tiep.