Chermak-Delgado Lattice Extension Theorems

TL;DR

This paper explores the structure of Chermak-Delgado lattices in finite p-groups, providing explicit examples and generalizations of lattice configurations such as 2-strings of diamonds and cubes, revealing complex lattice behaviors.

Contribution

It introduces new lattice structures called 2-strings of 2-diamonds and generalizes these to more complex configurations within Chermak-Delgado lattices of p-groups.

Findings

Constructed p-groups with Chermak-Delgado lattices as 2-strings of 2-diamonds.

Generalized examples to 2l-strings of n-dimensional cubes.

Provided methods to create p-groups with Chermak-Delgado lattices as 2l-strings of quasiantichains.

Abstract

In a finite group G with subgroup H, the Chermak-Delgado measure of H (in G) is defined as the product of the order of H with the order of the centralizer of 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 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 M_4 (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+3) (quasiantichains, each of width p + 1) adjoined maximum-to-minimum.