# Finite Groups with a Trivial Chermak-Delgado Subgroup

## TL;DR

This paper studies finite groups with trivial Chermak-Delgado subgroups, characterizing their lattice structures, decompositions, and introducing CD-minimal groups, with examples illustrating diverse lattice configurations.

## Contribution

It introduces the concept of CD-minimal groups, analyzes their lattice properties, and provides a classification framework along with explicit examples.

## Key findings

- Groups with trivial Chermak-Delgado subgroups can decompose based on their lattice structure.
- The paper establishes properties of Chermak-Delgado lattices in CD-minimal groups.
- Constructs twelve examples of CD-minimal groups with varied lattice structures.

## 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 Chermak-Delgado subgroup of $G$. This paper concerns groups with a trivial Chermak-Delgado subgroup. We prove that if the Chermak-Delgado lattice of such a group is lattice isomorphic to a Cartesian product of lattices, then the group splits as a direct product, with the Chermak-Delgado lattice of each direct factor being lattice isomorphic to one of the lattices in the Cartesian product. We establish many properties of such groups and properties of subgroups in the Chermak-Delgado lattice. We define a CD-minimal group to be an indecomposable group with a trivial Chermak-Delgado subgroup. We establish lattice theoretic properties of Chermak-Delgado lattices of CD-minimal groups. We prove an extension theorem for CD-minimal groups, and use the theorem to produce twelve examples of CD-minimal groups, each having different CD lattices. Curiously, quasi-antichain $p$-group lattices play a major role in the author's constructions.