Central Products and the Chermak-Delgado Lattice

TL;DR

This paper investigates the structure of Chermak-Delgado lattices in finite groups, especially how they behave under central products, providing new insights into their composition and properties.

Contribution

It proves that the Chermak-Delgado lattice of a central product contains the product of the lattices of its factors and explores their height relationships.

Findings

Chermak-Delgado lattice of a central product contains the product of the factors' lattices

Provides bounds on element heights in the Chermak-Delgado lattice

Uses central product structure to analyze lattice properties

Abstract

The Chermak-Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups. We prove that the Chermak-Delgado lattice of a central product contains the product of the Chermak-Delgado lattices of the relevant central factors. Furthermore, we obtain information about heights of elements in the Chermak-Delgado lattice relative to their heights in the Chermak-Delgado lattices of central factors. We also explore how the central product can be used as a tool in investigating Chermak-Delgado lattices.